Bifurcation Theory and Applications

•^

Bifurcation Theory and Applications

u(x,t, if) of (2.1) converges to v\ in L2(0,1), i.e. lim \\u{-,t,tp)-vx\\L2

t—>oo

=0.

However, when Ao < A the stability will be lost. Then it is nature to study two questions: (1) from what values of A, the steady state v\ will lose its stability, and (2) if Ao is the point from which v\ loses its stability, then what behaviors of the solutions u(x, t) of (2.1) near v\ may occur for t —* oo. The problem (1) is related to the steady state bifurcations introduced in Chapter 1, and both problems (1) and (2) are central topics of dynamic bifurcation. We end this introduction with two remarks. First, let u = v + v\ in (2.1) and (2.2). Then the stability and bifurcation problems for (2.1) and (2.2) near v\ are equivalent respectively to the following problems:

1

v=0 v{x,0) = ip(x)

( 2i3 )

at a; = 0,1, Vz€(0,l),

r g + 2 U A V + u 2 = o v, e( o,D,

{2A)

at a; = 0 , 1 ,

[v = 0

near v = 0, where ip —

^(0,1)0^(0,1),

# 1 / 2 = #1(0,1),

H = L2(0,l),

where H2(0,1) and HQ(0, 1) are the usual Sobolev spaces. It is clear that Hi C Hx/2 C H are dense and compact inclusions. We define the mappings L\ = -A + B\ : Hi —* H and G : Hx —> H by Av = - ^ ,

Bxv = 2vxv,

Gv = v2,

v e Hi.

(2.5)

Introduction to Dynamic Bifurcation

23

The equations (2.3) and (2.4) are then reformulated into the following operator forms respectively -=L,u

+ Gu,

u€Hlt

( 2 6 )

«(0) = V, and (2.7)

L\u + Gu = 0.

We know that A : Hi —» H is a linear homeomorphism, B\ : Hi —> H a linear compact operator, and G : H\ —> i7 a compact operator. Moreover, we see that -BA : ffi/a -^ H a bounded linear operator, G : Hi/2 —> H a bounded continuous operator, G(u) =

o(\\u\\1/2).

We notice that the equation (2.7) can be rewritten as u-A-1Bu-A~1Gu = 0,

u£H,

which has the same form as (1.13). In fact, many differential equations in Physics, Chemistry and Biology can be put into the abstract form given in (2.6) and (2.7). 2.2 2.2.1

Semi-groups of Linear Operators Introduction

As an example, we continue to consider the equation (2.3). By the SturmLiouville theorem, for each |A| < A, there exist a sequence of eigenvalues {Afc} with Afc —> +oo as k —* oo and eigenvectors {cf>k} C Hi such that

{

d24> — k +2vxk = -\k<j>k,

4>k(0) = 0*(1) = 0, and {4>k} C H constitutes an orthogonal basis of H.

{2g)

24

Bifurcation Theory and Applications

Let u = Y^kLi xk4>k- The equation (2.3) can be written as / I T = -**** + **(«).

*= i,2,..,

(29)

l*fc(0) = Vfc, where V* = fo i> • fadx, Gk(u) = / ^ u2kdx. We know that the solution of (2.9) can be expressed by Jo which implies that the solution of (2.3) is given by oo

u{x,t) = Y,e~Xkt^k+

-t oo e Xk{t T)G u

Y, ~

•/o

fc=l

~

^ )^dT.

(2.10)

fc=l

We find from (2.10) a family of linear bounded operators T(t) : H —» H defined by CXI

OO

A

T(tty = 5>- *Vfe&,

V t > 0 , V = X)V-fc^*.

(2-11)

fc=i fe=i

which enjoy the following semi-group properties: ' T(0) = id : £T -^ F , <

T(t + s)=T(t)T(s),

(212)

With the definition (2.11), the solution of (2.3) is written as u = T(t)ip + f T(t- T)G(u)dT.

(2.13)

Jo More generally, if for a linear operator L : Hi —> H there exists a one parameter family of linear bounded operators T(t) : H —> H satisfying (2.12), then the solution of the nonlinear equation (d^=Lu

+

G(u),

( 2 i 4 )

I «(0) = V, can be expressed by (2.13). The family of linear bounded operators T(t) is called a semi-group of linear operators.

25

Introduction to Dynamic Bifurcation

2.2.2

Strongly continuous

semi-groups

Definition 2.1 Let X be a Banach space, T(t) : X -> X (0 < t < oo) a one parameter family of bounded linear operators. T(t) is called a strongly continuous semi-group of linear operators, if (1) T(0) = id : X -> X the identity, (2) T{t + s) = T(t)T(s) for every *, s > 0, and (3) lim(_o T(*)a: = a; for each x eX. Let Xi be a Banach space, X\ C X a dense inclusion, and L : Xi —> X a bounded linear operator. We say that L : Xi —> X generates a strongly continuous semi-group T(t) : X —» X, if Lx = ]imT{t)x~X t—o i

= j-T(t)* , dt t=o

VxeXj.

In this case, the operator L : Xi —> X is also called a generator of T(i). The following is the Hille-Yosida theorem. Theorem 2.1 A linear bounded operator L : Xi —+ X generates a strongly continuous semi-group T(t), t > 0 if and only if (1) L is a closed operator, i.e. if xn —» £0, Lxn —> 2/0 wi X , £/ten x n —> zo in X\ and LXQ = 3/0 • ^ T/iere are Ao > 0 and C > 1 suc/i t/iat (Ao, +00) C p(L), and ||(Aid - L ) - " | | < C(A - Ao)-",

V A > Ao, n = 1,2, • • • ,

where p(L) is the resolvent set of L defined by p(L) = {A € C I (Aid - L)-1 : X -» X bounded) .

(2.15)

Remark 2.1 If L : Xi —> X is a linear completely continuous field, then L satisfies Condition (1) in Theorem 2.1. Condition (2) implies that there is a real number Ao > 0, such that for all eigenvalues A € C of L, ReX < Ao. A strongly continuous semi-group T(t) : X —» X generated by L : Xi —» X (Xi C X densely) has the following basic properties. Theorem 2.2 Let T(t) : X —» X (t > 0) be a strongly continuous semigroup generated by L : Xi —> X . Then

26

Bifurcation Theory and Applications

(1) For any UQ £ X\, the following initial value problem ( du

[ u(0) = u0, has a unique solution u(t) = T(t)u0 € Cl([Q, oo), X) n C°([0, oo), Xx). (2) If the equation (2.13) has a solution u(t) £ X\, then the solution u(t) satisfies (2.14). (3) For any xeX, f*T(s)xds € Xu and T(s)xds) = T(t)x - x.

L([

(4) For xeXi,_ T(t)x 6 Xi and T{t)x - T(s)x = f T(r)LxdT = f LT{r)xdT. Js Js 2.2.3

Sectorial operators and analytic semi-groups

The strongly continuous semi-group T(t) : X —> X has a limitation that in general T{t)x $. Xi for t > 0 and x S X. However, for the semigroup of linear operators given by (2.11) the situation is different. Let Xx = H2{0,1) n ^ ( 0 , 1 ) , and X = L2(0,1). Then, t -» T(t)x is analytic for 0 < t and x = 23^=1 xk4>k £ X, and oo

T(t)x

= Y^e~Xktxk
<=Xi,

Vt > 0, x e X.

(2.16)

fc=i

Moreover, for t h e operator L = A — B : X i —> X defined b y (2.5), where L = —C is a generator of t h e semi-group T(t) in (2.16), we can define fractional power operators Ca with domains Xa = D(Ca) (a G R ) as follows oo oo 1

{

fc=l r oo

< H^ii _

fc=l

\^\l»xl Lfc=l

oo

cax = YlKxkk £X, V i e i o , fc=i

J

-I i/2

(2.17)

Introduction to Dynamic Bifurcation

27

provided that the eigenvalues of (2.8) satisfy 0 < A^, for any k = 1,2, It is clear that Xa C Xp (a > (3) is a dense and compact inclusion, the norms || • || and || • ||o are equivalent to those of H2(0,1) and L2(0,1) respectively. Furthermore, we have T(t):X^Xa

Vt>0, aeR1,

T(t)Cax = CaT(t)x

Vi£l o ,

nfi _ r/3 _ pot _ rjx+P

£a

Another important property is \\CaT(t)\\
Vt>0,

which follows from the following computation: ||£ a T(*)||= sup \\£aT(t)x\\ lkllo=i

= sup

r oo

-11/2

Tx^e-^'xll

ll*llo=l U = 1 < max (A£e<

(A(

J Al

=- )*) • e" Al *

Cat-ae-Xlt.

The generalization of this case leads to the concepts of analytic semigroups and sectorial operators. Definition 2.2 A strongly continuous semi-group T{t) : X —> X is called an analytic semi-group, if t —» T(t)a; is analytic for 0 < t < oo and x £ X. An operator L : Xi —> • X is called a sectorial operator if L generates an analytic semi-group. Theorem 2.3 only if

A linear bounded operator L : X\ —• X is sectorial if and

(1) L is a closed operator. (2) There exist 0 < 6 < TT/2, a real number Ao, and a constant C > 1, such that SAo,0 = { A e C

I | a r 5 ( A - A o ) | < | + ^}c/9(L),

and IKAid-L)-1^—^—

|A — A o |

VA G SXote, A ^ Ao,

28

Bifurcation Theory and Applications

where p(L) is the resolvent set of L defined by (2.15). 2.2.4

Powers of linear operators

In this subsection we define fractional powers of sectorial operators. Let L : X\ —» X be a sectorial operator which generates an analytic semi-group {e*L}t>o, with the sector S\Oig C p{L) for some real number Ao < 0. For the operator L = — L, and a > 0, we define 1 P°°

£~a = -f- / ta~le-tcdt, r(a) Jo where T(a) = /0°°

(2.18)

f^e^dt.

Theorem 2.4 C~a (a > 0) is a linear bounded operator on X, which is one to one and satisfies C~a . C-P = C-(a+P),

Va,/3>0.

Definition 2.3 With C as above, we define Ca = inverse of C~a (a > 0), £° = id : X -f X, Xa = D{Ca) = R(£-a). It is easy to verify that if L = —C = —A + B : X\ —> X as defined by (2.5), then Ca and Xa defined by Definition 2.3 are coincident with (2.17). In fact, by (2.16), for x = Y%Li xk4>k tc

tL

OO

e~ x = e x = ^ e-Xktxkcj>k. fc=i By (2.18),

OO

fc=i

29

Introduction to Dynamic Bifurcation

Hence, we deduce that oo

fc=l

{

oo

oo

1

fc=i

fc=i

J

Theorem 2.5 .Le£ L : X\ ^> X be a sectorial operator, which generates an analytic semi-group T(t). If the sector S\oj C p(L) for some Ao < 0, then for the operators Ca (C = —L) we have (1) T{t) : X -> Xa for each t > 0 and a € E 1 . (2) T{t)Cax = CaT{t)x V i e X a . (3) For each t > 0, the operator CaT(t) is bounded on X, and \\CaT{t)\\ < Carae-St, (4) LetO
for some 6>0.

and x e Xa. Then ||r(f)x-x|| < Mata\\Cax\\.

Theorem 2.6 Let L : X\ —> X be a sectorial operator with S\Ote C p(L) for some Ao < 0, and if B is a linear operator such that BC~a : X —> X is bounded for some 0 < a < 1, then —L + B is also a sectorial operator. Remark 2.2 As X\ c X is a compact inclusion, Xa C Xp is compact for a > (3. In this case, Theorem 2.6 implies that if L : X\ —> X is a sectorial operator, B : Xa —> X is linear bounded for some 0 < a < 1, then L + 5 is a sectorial operator. 2.3

Dissipative Dynamical Systems

In this section, we introduce briefly dynamical systems associated with partial or ordinary differential equations, which can be considered as an abstract operator equation on Banach spaces. More precisely, let X\ and X be two Banach spaces, and X\ C X be a dense inclusion. Consider an abstract operator equation

( «(0) = uo,

(2.19)

30

Bifurcation Theory and Applications

where F : X\ —> X is a continuous mapping. Let {S(t)}t>o be an operator semi-group generated by (2.19), which enjoys the following properties S(t) : X —> X a continuous (nonlinear) operator for each t > 0, 5(0) = id (identity on X), S(t + s) = S(t)-S(s),

t,s>0.

Then the solution of (2.19) can be expressed as u(t,uo)

Vi>0.

= S(t)uo,

Definition 2.4 A set S C X is called an invariant set of (2.19) if S(t)E = E (V t > 0). An invariant set E C X of (2.19) is called an attractor if E is compact, and there exists a neighborhood U C X of E, such that for any •uo £ U we have dist(5(t)uo, E) —> 0 in X-norm as t -> oo.

(2.20)

The largest open set U satisfying (2.20) is called the basin of attraction of S, or we say that S attracts U. Especially, if the basin of attraction of E is X, the E is called a global attractor of (2.19). For a set S C X, we define the w-limit set of E by

u,(S) = f| (JS(t)£, s>0

t>s

where the closure is taken in X. Likewise, when it exists, the a-limit set of S C X is defined by

a(S) = f| \JS(-t)Z. s>0

t>s

Lemma 2.1 Suppose that for some subset E C X, E ^ 0, and for some t0 > 0, the set Ut>to5(t)S is relatively compact in X. Then the w-limit set w(E) is nonempty, compact, and invariant. If the sets S(—t)T, (t > 0) are nonempty and for some to > 0, Ut>t0S(—£)E is relatively compact, then the a-limit set a(E) is nonempty, compact, and invariant. Definition 2.5 Let E C X be a subset and U an open set containing E. We say that E is absorbing in U if the orbit of any bounded set of U enters into E after a certain time (which may depend on the set). Namely, for any

Introduction to Dynamic Bifurcation

31

bounded set B C U, there exists a to = to(B) such that for all t > to(B), S(t)B C E. Lemma 2.2 Suppose that the operators S(t) are uniformly compact for t large, i.e. for any bounded set B there exists to such that Dt>toS(t)B is relatively compact in X. We also assume that there exists an open set U and a bounded subset BofU such that B is absorbing in U. Then the wlimit set of B, S = w{B) is an attractor which attracts the bounded sets of U, and it is the maximum attractor in U. Furthermore, if U is connected, then E is connected too. In the following, we consider the stability of the attractors. Let Sv(t) be a family of perturbation of the semi-group S(t), which depend on a parameter r), 0 < 77 < rjo. Let Xn C X (0 < 77 < rjo) be a family of closed subspaces with I)

Xrj is dense in X, and

Srj(t) : Xn —> Xn a continuous operator semi-group. Assume that the semi-group {<S'(£)}t>o has an attractor E which attracts a neighborhood U c X of S. We also assume that for each bounded interval /C(0,oo), sup sup dist(Sr^(i)u0, S(t)u0) —> 0 as r) —> 0, uoet/nx, t€i where the distance of two sets A and B C X is denned by dist(A, B) = sup inf ||x - y\\Xx€AyeB

(2.21)

Theorem 2.7 Under the above assumptions, if for each rj > 0, 5^(t) has an attractor T,v which attracts UnXv, where U is a neighborhood o/E,,nS independent off], then T,n converges to E in the distance (2.21): dist(S,,,I!)—»0

asrj^O.

32

Bifurcation Theory and Applications

2.4

Center Manifold Theorems

2.4.1

Center and stable manifolds in R"

We consider the system of ordinary differential equations

{

dx — =Ax + G1(x,y,X),

(2.22)

f

-ft=By + G2{x,y,\),

where x e Rm, y e Rn~m (0 < m < n), A and B are the m x m and (n — m)x (n — m) matrices respectively, Gi(x,y, A) (i = 1, 2) are continuous on A, and Cr (r > 1) on (x, y) € Rm x M"- m , and Gi(ar,y, A) =0(1*1,15,1)

(i = 1,2),

V A e R1.

(2.23)

The following is the center manifold theorem. Theorem 2.8 Suppose that all eigenvalues of A have nonnegative (resp. non-positive) real parts, and all eigenvalues of B have negative (resp. positive) real parts. Then for the system (2.22) with the condition (2.23), there exists a Cr function, called the center manifold function, h(-, A) : D —> Rn-m,

D C Rm a neighborhood ofx = 0,

such that h(x, A) is continuous on A, and (1) h{Q1\)=Q,h'x{Q,\)=O; (2) the set Mx = {(x,y) \xeDcRm,

y = h(x,X)},

called the center manifold, is a local invariant manifold of (2.22); (3) if M\ is positively invariant (resp. negatively invariant), namely z(t,tp) € M\ (resp. z(—t,ip) G MA) V t > 0, then M\ is an attracting set of (2.22) (resp. a repelling set), i.e. there is a neighborhood U c M " of M\, as ip € U we have lim dist(z(t, ip), M\) = 0 t—>oo

(resp. lim dist(z(-t,i/>), M\) = 0), t—>oo

where z(t, ip) = (x(t, ip),y(t, ip)) is the solution of (2.22) with the initial value z(O,ip) = ip.

Introduction to Dynamic Bifurcation

33

Property (1) means that the center manifold M\ C W1 is tangent to the eigenspace M.m of A at z = (x, y) = 0. Although, as we know, the local center manifold M\ may not be unique, the following theorem makes it applicable. Theorem 2.9 There is a neighborhood C / c R n of z = 0 such that every invariant set of (2.22) in U belongs to the intersection of all local center manifolds in U. In the following, we introduce the stable manifold theorem. Theorem 2.10 Let all eigenvalues of A have positive real parts, and all eigenvalues of B have negative real parts. Then there exist two unique manifolds Mu and Ms, called the unstable manifold and stable manifold of (2.22) at z = 0, which are characterized by

Mu = lz e Rn I lim S(-t)z = o\ , Ms = lz G W1 I lim S(t)z = OJ , I

t—>oo

J

where S(t) is the semi-group generated by (2.22). Moreover, Mu and Ms are tangent to the eigenspaces of A and B respectively at z = 0: TZ=OMU = R m ,

TZ=OMS = K"- m .

Therefore, the stable manifold Ms and the unstable manifold Mu are transversal at z = 0. Consider the following general system ' dx — < ^=By

=Ax-\-G1{x,y,z,\), + G2(x,y,z,X),

dz — =Cz +

(2.24)

G3(x,y,z,\),

where x G E m , y £ Rk, z € R n , Gi(x,y,z,\) = o(\x\, \y\, \z\) (t = 1,2,3). We have the following invariant manifold theorem. Theorem 2.11 Let the eigenvalues of A have positive real parts, the eigenvalues of B have negative real parts, and the eigenvalues of C have zero real parts. Then, the system (2.24) has three locally invariant manifolds Mu, Ms and Mc, which are tangent to the eigenspaces of A, B, C

34

Bifurcation Theory and Applications

respectively:

T'

A/Tu

u>=O-»w

lUm

T1

— IK ,

H/f3

TD>^

= K ,

1W=QM

T1

A/fC

TO7"1

=K

J.w=oM

,

where Mu is the unstable manifold, Ms is the stable manifold, Mc is the center manifold. Moreover, if Gi (i = 1,2,3) are Cr (r > 1) then Mu, M", Mc are Cr. 2.4.2

Center manifolds for infinite dimensional

systems

Now, we introduce the infinite dimensional version of center manifolds theorems. Let H and Hi be two Hilbert spaces, and Hi C H be a dense inclusion embedding. We consider the nonlinear evolution equation given by tiG^AeR1,

f -^ = Lxu + G(u,A),

1 «(0) = u0,

(2.25)

where L\ : Hi —> H are parameterized linear completely continuous fields depending continuously on A £ R1, which are defined by < A : Hi —> H \ B\ : Hi —> H

linear homeomorphism, parameterized linear compact operators.

(2.26)

We first consider the case in which L\ : H\ —> H is a sectorial operator. Then, one can define fractional power operators L" for a £ l ' with domain Ha = D{L$). Furthermore, we assume that the nonlinear terms G(-,A) : He —> H for some 0 < 8 < 1 are a family of parameterized Cr (r > 1) bounded operators continuously depending on the parameter A g l 1 , such that G{u, A) = o(Htf,),

(0 < 6 < 1), V A G R1.

(2.27)

We assume that the spaces Hi and H can be decomposed into •Hi=E$®E$, tiixnEi < oo, < H = E$@E2, E2 = closure of E% in H,

(2.28)

Introduction to Dynamic Bifurcation

35

for A near Ao, where E^, E2 are the invariant subspaces of L\, i.e. L\ can be decomposed into L\ = C\ ® C2 such that for any A near Ao,

( L2 = L\\E\ . t,2 —> b2 ,

(2.29)

where the eigenvalues of C2 possess negative real parts, and the eigenvalues of L\ possess nonnegative real parts at A = Ao. Thus, for A near Ao, equation (2.25) can be written as

{%=C*1x + Gl(x,y,\),
(2.30)

where u = x + y £ Hi, x £ E$, y £ E%, G»(x,j/,A) = P{G(u, A), and Pi : H —» Ef are canonical projections. We have the following center manifold theorem. Theorem 2.12 Assume (2.26)—(2.28). Then there exists a neighborhood of Ao given by |A — Ao| < 5 for some S > 0, a neighborhood O\ C E$ of x = 0, and a C1 function h(-,X) : O\ —> E^iO) depending continuously on X, where E^O) is the completion of E2 in the Hg-norm, with 0 < 6 < 1 as in (2.27), such that (1) ft(0,A) = 0, ^(0,A) = 0, (2) the set Mx = {(x,y) £H\x£Ox,y

= h(x,X) G E%(9)} ,

(2.31)

called the center manifolds, are locally invariant for (2.25), i.e. for each u0 £ Mx, ux(t,u0)£Mx,

V0
for some i(wo) > 0, where u\(t,uo) is the solution of (2.25); (3) if (x\(t),y\(t)) is a solution of (2.30), then there is a f3\ > 0 and k\ > 0 with kx depending on (x\(0),yx(0)) such that

WyxW-hixxW^WnKkxe-P**.

36

Bifurcation Theory and Applications

If we only consider the existence of the local center manifold, then the conditions in (2.29) can be modified in the following fashion. Let the operator L\ = Lx ® £-2 > a n d £2 be decomposed into / fX

rX

ffi

rX

*-2 — *"21 ^ ^22» £/ 2 = &21 © - ^ 2 2 '

. ^

= ^0^2,

(2.32)

dim .E^i = d i m ^ < 00, , ^2i : ^2i ~* ^2i

are

invariant (i = 1,2),

such that at A = Ao [ eigenvalues of C\ : E^ —> E± have zero real parts, < eigenvalues of £31 : -^21 ~* ^21 have positive real parts, and ( eigenvalues of ££2 : -^22 ~* ^22

nave

(2.33)

negative real parts.

Theorem 2.13 Assume (2.26)—(2.28), (2.32) and (2.33). conclusions (1) and (2) in Theorem 2.12 hold true.

Then the

We now consider the case where L\ : Hi —> H generates a strongly continuous semi-group of linear operators. Assume that G(-,X) : H —> H are Cr (r > 1) bounded continuously depending on A s R 1 , and G(u, A) = o(||u|| JJ),

VAGK1.

(2.34)

Also, we assume that for any uo S Hi, the equation (2.25) has a unique solution u{t) £ Hi for all t>0. Theorem 2.14 Assume that L\ : Hi —» H generates a strongly continuous semi-group of linear operators, and the conditions (2.26), (2.28), (2.29) and (2.34) hold true. If the strongly continuous semi-group S\(t) generated by £2 satisfies

II5A(t)II < Kxe-a^,

(2.35)

for some constants K\ >l,a\>0, then there exists a C1 center manifold function h{-, A) : B\ —> E2 such that the assertions (1)—(3) in Theorem 2.12 hold, in which the center manifold M\ is replaced by

MA = {(x,y) £H\x£BxcEi,y

= h(x,y) G E$ C H] .

37

Introduction to Dynamic Bifurcation

Remark 2.3 Theorem 2.14 can be proved by using the same method as in [Henry, 1981], which is useful in the dynamic bifurcation of nonlinear wave equations. 2.4.3

Construction of center manifolds

In later discussions, it is necessary to show how the center manifold is constructed. In this subsection we give a sketch of the proof for Theorem 2.12. Let pE:Ef^> [0,1] be a C°° cut-off function defined by H V ;

f 1

if ||x|| < e,

\ 0

if ||x|| > 2s,

for some e > 0. We denote CQ>l{E$,E${6)) = {h : E$ -> E%{0) \ h{0) = 0, h is Lipschitz} . We need to find a function h € C°'x(^E^, E^B)) satisfying h(-)= [

J — oo

e-c^Pe(x(T,-))G2(x(T,-),h(x(r,.)))dT

(2.36)

where x(t,xo) is a solution of the ordinary differential equation

{

f/r

-=C>x

+ Mx)Gl{xMx),X),

(237)

x(0) = xo,

Then, it is easy to see that the function y(t,h(x0)) = h(x(t,x0)) = / J—oo

= /

J—oo

= f

J — oo

e-c*Tpe-G2(x(T,x(t,xo)),h)dT e-c*Tpe-G2(x(t

+ T,x0),h(x(t + T,x0)))dT

ec^t-^pE(x(T,xo))G2(x(T,xo),h(x(T,xo)))dT

satisfies the equation f ^ = £$V + \ 1,(0) = h(x0).

Pe(x(t,xo))G2(x(t,xo),y),

38

Bifurcation Theory and Applications

Thus, (x(t,xo),h(x(t,xo))) is a local solution of (2.30), and the manifold given by (2.31) is locally invariant for (2.25). Hence, the existence of the center manifold of (2.25) is referred to as the existence of the fixed point of the mapping

F : C°>l{El,E${6)) —> C 0 ' 1 ^ , ££(
F(h)=f

(2.38)

J — oo

By the assertion (3) in Theorem 2.5 for a sectorial operator, one can prove that under the condition (2.27) the mapping F denned by (2.38) is a contraction operator for £ > 0 small. For the details of the proof, see [Temam, 1997; Henry, 1981], When the nonlinear terms G(-, A) in (2.25) satisfy (2.34) and the linear part satisfies the condition (2.35), we can derive Theorem 2.14 in the same fashion.

2.5

Hopf Bifurcation

We consider the ordinary differential equations as follows dr

(2.39)

— =Ax + G(x,X), where i g l " (n > 2), A £ R1, A a parametrized n x n matrix

(

an(A) ••• a i n (A)\ ;

;

.

ani(A) ••• ann(X)J

and G : R" x R1 -> R" a Cr (r > 1) mapping with G(x, A) = o(|:r|),

uniformly for A in any bounded set of R1.

(2.40)

Let all eigenvalues (counting the multiplicity) of A be given by /?i(A),

Introduction to Dynamic Bifurcation

39

f32(X),- • • , /? n (A). Assume t h a t ' ft (A) = 02 (A) = a(A) + ip(A)

near A = A o ,

«(Ao) = 0, (2-41)

< da(X0) A»(Ao) ^ 0, Re/?j(Ao)^0

Vj>3.

(2.42)

Theorem 2.15 (Hopf Bifurcation). Assume (240)—(242). Then, equation (2.39) bifurcates from (re, A) = (0, Ao) a periodic orbit F(A), i.e. there exists a sequence of periodic orbits T\n of (2.39) such that

{

lim Xn = A o ,

lim d(Txn,0)=

n—>oo

lim max

\x\=0.

(2.43)

n—*oo x£Fxn

To see this theorem, with a coordinate transformation, we decompose the equation (2.39) into

{

- ^ =a(\)xi

£

+ p(\)x2 + G1(x1,x2,y, X),

—1 = -p(X)xi + a(X)x2 + G2(xux2,y, X), at ^=By + G3(xl,x2,y,X),

(2-44) (2.45)

where B is an (n — 2) x (n — 2) matrix, which has the eigenvalues /? 3 (A),.-.,/3 n (A). By the center manifold theorem (Theorem 2.11), for (2.44) and (2.45) there is a Cr center manifold function h\ : U\ —> R

,

where £l\ C K2 is a neighborhood of (x\, x2) = 0, such that y(t) = hx{x1{t),x2(t)) is a solution of (2.45), where {x\(t),x2(t)) is a solution of (2.44). Thus, the bifurcation of periodic orbits of (2.39) is reduced to the existence of

40

Bifurcation Theory and Applications

periodic solutions of the following equations

{

—!- = a(X)x! + p(X)x2 + Gi(xi,x2, hx(xi,x2), A), dx2

(2'46)

— = -p(X)x1 +a(X)x2 +G2(xi,x2,h\(x!,x2),X). Prom (2.46), the Hopf bifurcation theorem can be easily derived from the Poincare-Bendixon theorem. 2.6

Notes

2.2 We refer the readers to [Pazy, 1983; Henry, 1981] for sectorial operators, and semi-groups of linear operators. 2.3 For more detailed discussions on infinite dynamical systems theory, the readers are referred to [Temam, 1997; Vishik, 1992]. 2.4 The proofs of Theorems 2.8-2.11 can be found in [Chow and Hale, 1982; Hirsch et al., 1977; Iooss and Joseph, 1980]. The infinite dimensional center manifold theorems, Theorems 2.12 and 2.13, can be found in, among others, [Henry, 1981; Temam, 1997]. 2.5 The Hopf bifurcation theorem is introduced in many basic textbooks on ordinary differential equations and dynamical systems; see, among others, [Chow and Hale, 1982; Guckenheimer and Holmes, 1983]. Poincare-Bendixon theorem used in the sketch of the proof of the Hopf bifurcation theorem can be found in these references as well.

Chapter 3

Reduction Procedures and Stability

In this chapter, we focus on the following three topics (1) spectral theorems, (2) reduction procedures, and (3) asymptotic stability. These are crucial components of the bifurcation theory developed in this book.

3.1

3.1.1

Spectrum Theory of Linear Completely Continuous Fields Eigenvalues of linear completely continuous fields

Let H and Hi be two Hilbert spaces, and Hi C H be a dense and compact inclusion. A linear operator L : Hi —> H is called a completely continuous field if L = -A + B : Hi -> H, A : Hi —> H a linear homeomorphism, B : Hi —> H a linear compact operator. A number A = a + i/3 G C is called an eigenvalue of a linear operator L : Hi —> H if there exist x, y G Hi with x ^ 0 such that Lz = \z,

(z = x + iy), 41

(3.1)

42

Bifurcation Theory and Applications

and the space

Ex= U {x,yeH! | (L - A)"z = 0, z = x + iy} is called the eigenspace of L at A, and x, y £ E\ are called eigenvectors1 of L. Alternatively, (3.1) can be equivalently expressed as Lx = ax — j3y, Ly = /3x + ay. Definition 3.1 A linear mapping L* : Hf —» H is called the conjugate operator of L : Hi —> H, if (Lx,y)H

= {x,L*y)H,

V x,j/ G H±.

Here iJj c i? is a dense inclusion. A linear operator L : Hi —» H is called symmetric if L — L*. We remark here that in this definition, we first treat L as an closed and densely defined unbounded operator from H to itself with domain Hi. Then L* : H —> H make sense, and has domain H$ C H. In all applications presented in this book, however, we have Hi = Hf. Definition 3.2 We say that a linear operator L : Hi —> H has a complete eigenvalue sequence {Afe} C C if each eigenspace Ek at \k is finite dimensional, and all eigenvectors of L constitute a basis of H. We know that a linear completely continuous field L = —A + B : Hi —> H is a Predholm operator with index zero, and the spectrum of L consists of eigenvalues. Furthermore, the following properties are classical; see [Kato, 1995]. Theorem 3.1 Then

Let L : Hi —» H be a linear completely continuous field.

(1) the spectrum of L consists of eigenvalues; (2) each eigenvalue A € C is isolated, and the corresponding eigenspace E\ is finite dimensional; (3) L and its conjugate operator L* have the same eigenvalue, and the corresponding eigenspaces have the same dimension dimI?A — dim.E£; and throughout this book, eigenvectors are always referred to (generalized) eigenvectors.

Reduction Procedures and Stability

43

(4) if L is symmetric, then the eigenvalues of L are real. The following theorem is on the complete spectrum of symmetric operators. Although this theorem is essentially known, we shall give a proof because this is in a slightly different form from the classical one. We remark here that the proof presented here connects the eigenvalue problem of L with that of L~x o / : H\ —> H\ between the same space H\. With this in our disposal, Theorem 3.1 is obvious. Theorem 3.2 Let L : Hi —» H be a linear completely continuous field, and L be a sectorial operator. If L is symmetric, then L has a complete eigenvalue sequence {Afc} C R1, and all eigenvectors {
(3.2)

Thus, we have (L~1oIx,y)i = (L(L-1oIx),Ly)H = {x,Ly)n (by the symmetry of L)

= {Lx, y)x 1

= {Lx,L(L- oI)y)H = {x,(L-1oI)y)i. We assume without loss of generality that all eigenvalues of L are positive. Hence, the operator L"1 o I : Hi —» Hi is symmetric and compact with respect to the inner product (3.2). For any x £ Hi with x ^ 0, {L~1olx,x)i

= (x,Lx)H = (l}l*x,LV*x)H >0.

Therefore, L~xol: Hi —» Hi has a complete eigenvalue sequence {pk} c R1

44

Bifurcation Theory and Applications

w i t h pk —> 0 a s k —• oo: pk
(3.3)

and the eigenvectors {fk} form an orthogonal basis of Hi under the inner • product (3.2). Hence the theorem follows from (3.3). Remark 3.1 It is still an unknown question if the complete spectrum theorem holds true for general symmetric linear completely continuous fields L : H1 -> H. 3.1.2

Spectral theorems

We begin with finite dimensional linear operators. Let M be an n x n matrix, and M* its conjugate matrix. Let /?j (j = 1,... ,n) be all eigenvalues of M (counting multiplicities). Vectors £j £ R n (1 < j ' < n) are called eigenvectors of M if there exist 1 < kj < rrij with m,j being the multiplicity of /3j, such that {M-Pj)k^j=Q,

(3.4)

when 13j are real numbers, and f ( M - ^ ) f c i ( ^ + ^ + i ) = 0, \(M-&+1)fc'&-»6+i)=0,

(3.5)

when /3j = /3 J + 1 are complex numbers, with /3J+i being the complex conjugate of 13j. Let £,• = (£ji, • • • ,^jn)* be eigenvectors of M such that (3.6)

MP = PJ, where J is the Jordan form of M, and / & i 6 i - " £m \ P=

^

^

-

^.

(3.7)

V^ln ^2n • • • £,nn/ Then it is easy to see that Mt(P-iy

= (p-i)tj*,

(3.8)

45

Reduction Procedures and Stability

where J* is the transpose of J, which is also the Jordan form of conjugate matrix (transpose) M* = M*. Hence if we set ( P - 1 ) * - (6, •••,1™), then £i, • • • ,£ n a re eigenvectors of M*,

P-1

=

;

W

,

(3.9)

and

/-^p-n'jfc-.e.)-

&1&2

- * " . (-0,

We note that equations (3.6) and (3.8) are equivalent to the classical Fredholm alternative theorem. Therefore, from (3.10) we get a theorem as follows, which is considered as a unified version of the Fredholm alternative theorem and the Jordan theorem. Theorem 3.3 Let f3j•, (j = 1, • • • ,n) be the eigenvalues of annxn matrix M. Then we can take the eigenvectors {£j £ Rn \ 1 < j < n} of M and eigenvectors {£,• € Rn | 1 < j < n} of M* such that

&,l) = sij. The Fredholm-Jordan theorem can be generalized to general linear completely continuous fields. For this purpose, consider a linear completely continuous field L — —A-\- B : H\ —• H. As in the proof of Theorem 3.2, we assume that L has a compact inverse given by

L-^Ii-A

+ B)-1 :H->H,

where I : Hi —* H is the inclusion mapping. Therefore, the eigenvalue problem of L can be equivalently written in the following form (L'1 - Pk)m^k = 0 ,

(3k = X^\

for some m > 1.

(3.11)

46

Bifurcation Theory and Applications

Let H C be the complexified space of H given by akuk | uk € H, ak € C I

H = H®C=<^2

I

fc

J

with inner product ( u .u)# = ((wi.vi)j/ + (u2,U2>i/) + i((u2,^i)H- (UI,V2)H)

for u = ui + iu 2 , w = vi + iv2 € H C. We are now in position to state the spectral theorem for general linear completely continuous fields. Theorem 3.4 Let L = — A + B : Hi —» H be a linear completely continuous field. Then the following assertions hold true. (1) If {Afc | k > 1} C C are eigenvalues of L, then we can take the eigenvectors {tpk} C Hi of L and eigenvectors {ip*k} C H\ of the conjugate operator L* such that (
(3.12)

(2) H can be decomposed into the following direct sum

(

H = E1®E2, Ei = the closure of span {ipk | k > 1} in H, E2 = {veH\(v,
(3.13)

Vfc>l};

(3) Ei and E2 are invariant spaces of L~l, and lim \\L-nv\\^n

=0

\/veE2;

n-*oo

(4) Let Pi,---f3k in the order

£ C be eigenvalues of L~x (counting the multiplicity) \Pi\ > | / 3 2 | > ••• > | / H V»i, • • • , 4>Z G H ® C be the

corresponding eigenvectors of (L~i)* = L*" 1 , and let E^ = span {V-I,---,^}, (E*k= 0 , (3.14) Pk+i= ||u||=l,uJ_EJ

then there is an eigenvalue (3k+i & C of L~l with |/3jt+i| = Pfc+i, ow^ l/3fc+i| < |)8fc|.

Reduction Procedures and Stability

47

Remark 3.2 By (3.12) and (3.13), for any u G # , we have the generalized Fourier expansion u = ^2 uWk + v,

v£E2,

uk = (u,
(3.15)

k

In particular, if the operator L = —A + B : H\ —> H has a complete eigenvalue sequence {Afe} c C with eigenvectors {tpk} C H\, then we have the following complete Fourier expansion oo U = ^UkVk, k=l

Uk = {u,
Remark 3.3 Let X be a Banach space, X* its dual space, and B : X - » X a compact linear operator with its conjugate operator B* : X* —» X*. Then the above spectral theorem holds true as well for the eigenvalue problem Bx = \x. Proof of Theorem 3.4- We proceed in several steps as follows. STEP 1. It is known that the space H can be decomposed into direct sums of invariant spaces of L~x as follows

'H=

E1®E2,

H = E{®E* , 1 22 Ei = the closure of span {ip^ | k > 1} in H,

(3.16)

E± = the closure of span {ip*k \ k > 1} in H. By the spectral radius theorem, we have f lim ||L- n H|^ / n = 0 ,. I n"°° [ lim^ ||L*- n u||^ /n = 0

Vv£E2, Vv G E\.

(3.17)

Assertion (3) follows. STEP 2. Let {£i, • • • ,£OT} c Hx C be the eigenvectors of L corre-

sponding to an eigenvalue A £ C, and {T?I,• • • ,rfM} be the eigenvectors of L* corresponding to an eigenvalue p 6 C. Set & = &i +1&2,

Vj = Vji + iVj2> 1 < ^ < "i, 1 < j < M.

We shall prove that if A ^ p, (Zik,rijr)H

= 0,

V f c , r = l , 2 , \<£<m,

1< j < M,

(3.18)

48

Bifurcation Theory and Applications

and (v,
(v*,l,veE2,v*

eE%

(3.19) (3.20)

(tek,Z*r)H = 5ej-6kr.

For simplicity, we consider only the case where m = M = 2, and the geometric multiplicities of A and p are one; while the general case can be proved in the same fashion. Then we have L-^i = A-^i, 1

L- ^

1

= p- ^,

L~% = A - ^ 2 + £ 6 , 1

1

L*- rll=p- r,t+1r&.

(3.21)

(3.22)

for some e,y G R 1 . When A ^ p , it follows from (3.21) and (3.22) t h a t (£i, »72)# = (^11 + ^12 > V21 + iV22)H

= \(L-1t1,r,Z)A = *{Zi,L

V2/H 1

=

*P~ (ZI,V2)H

= 0, *P~1(ZI,V2)H

(ZUV*2)H =

= 0, which imply that (£ifc,*72r)ff = 0 ,

k,r = 1,2.

Similarly, we can induce from (3.21) and (3.22) that <&,»?;>£ =°>

&,V])H

= 0,

1 < t, J < 2.

Thus, we have proved (3.18), and (3.19) can be obtained in the same fashion from (3.11) and (3.17). STEP 3. When restricted to the generalized eigenspace with a particular eigenvalue, the problem is then reduced to Theorem 3.3. Hence Assertion (1) holds true. 4. P R O O F OF ASSERTION (4). It is clear that the numbers defined by (3.14) satisfy 0 < p < HL"1!!. By the spectral radius theorem, if STEP

49

Reduction Procedures and Stability

p > 0 then L~l must have at least an eigenvalue /3 G C. Let fij e C (j > 1) be the eigenvalues of L"1 in the order IAI > Iftl > • • • > l&l > |&+i| > • • • ,

(3-23)

and {^j} C H C be the eigenvectors of L'1, then by Assertion (1)—(3), for any u G H ® C with u J_ I?£ (A; > 0), u can be expressed as

u= Y^, UJ^J + u> j>fc+i

Urn \\L-nv\\1/n

n—>-oo

= 0.

For simplicity, we assume that the eigenvalues (3j are simple, and other cases can be proved in the same fashion. Then we have

\{L-nu,u)6\= / ^

^UjiPj + L~nv, Y,

\j>k+l

=

i>k+l

Yl aj^ + (L-nv,u)H ,

j>k+\

Vrfi+A I

H

(3.24)

where i>k+l

Then by (3.23) and (3.24) we obtain that \/3k+i\ — pk+iThe proof of the theorem is complete.

•

Remark 3.4 By the proof of Theorem 3.4 we see that if ipk is an eigenvector of L corresponding to /3jt, and tpj is an eigenvector of L* corresponding to f3j with /3fc 7^ /3j, then (£} of L and 2 The general case can be considered with an index permutation, and the same results hold true.

50

Bifurcation Theory and Applications

L* respectively corresponding to the same eigenvalue /?, such that

Lip2 = Py>2 +£
(3.25)

_L*
(326)

<™»*{*l *! = * 3.1.3

Asymptotic

'

properties of eigenvalues

We consider a class of linear completely continuous fields L = — A + B : Hi —> H. We assume that A is symmetric, and there exists a real eigenvalue sequence {pk} C R1 and an eigenvector sequence {tfk} C Hi of A:

{

A
Pkfk,

0 < pi < pi < • • • ,

(3.27)

lim pk = oo,

k—^oo

such that {pk} is an orthogonal basis of H. It is easy to see that A is a sectorial operator, and we can define the fractional powers as follows oo

oo

Aax = J2 PZxk
Vx

fc=l

Let

i

(3.28)

k=l

oo

X&H

= Yl xWk e H.

X

oo

"^

= Y,xwk,Y.pk*xl<°°\' fc=l

fe=l

J

(3-29)

51

Reduction Procedures and Stability

endowed with the inner product oo

=J2plaxkyk.

(x,y)Ha

fc=i

We also assume that B : H-y —> H is bounded for some 0 < 7 < 1.

(3.30)

By Theorem 2.6, the operator L = -A + B satisfying (3.27) and (3.30) is sectorial. The eigenvalues of L have the following asymptotic property. Theorem 3.5 Let L = -A + B :HX->H satisfy (3.27) and (3.30). If L has an infinite number of eigenvalues Xk — o-k + ipk £ C, then ak —> —oo when k —> oo, and for the number 7 < 1 as in (3.30) we have lim r ^ L Proof. satisfy

=

o,

V 7 < 5 < 1.

(3.31)

Let Zfc = Zfc + iyk be the eigenvector corresponding to Xk, and

{

- Axk + Bxk = akxk + pkyk, - Ayk + Byk = -pkxk + akyk.

yo.oZj

We infer from (3.32) that ak

(Axk,A2exk) + (Ayk,A2eyk) (Bxk,A2exk) + (Byk,A2eyk) (xk,A™xk) + (yk, A»yk) (xk, A26xk) + (yk, A2°yk) _ INIg + i - (Bxk,A™xk) - (Byk,A™yk)

=

~ Pk

_

~

'

iwii

{Bxk,A29yk)-{Byk,A™xk)

'

Iklli

where 0 < 9 < 1/2 is a constant to be specified later. By (3.30) we get |(B^,A 2 V>l
52

Bifurcation Theory and Applications

Hence W <MJi*»,

,3.33,

\\zk\\o

K l a K | a

II**K + , - «B|I IWWWI» '

h B

IWI! + , + l|B|IIMI-rll»lla. • IMS '

(3 34)

'

<3 35)

'

Since L is a sectorial operator, if the number of eigenvalues A^ of L is infinite, then limfc_oo a^ = limfc_oo ReA^ = —oo. We derive from (3.35) that -+e

\\zk\\e

,oo

(fc->oo).

(3.36)

We take 6 > 0 as follows

\e + \ = i

2 < [0 = 7

if7>J,

1 ifO<7<-.

(3-37)

Since 0 < 7 < 1, we obtain from (3.37) that 0 < 6 < \, and f i + 0>max{ 7 ,20}, < ?

[-+0>min{7,20}.

(3-38)

We take ||z fc || e+ i = 1, V k G N. Then by (3.38) we have 1 > |N|7,

1 > INfcllM-

It follows from (3.36) that \\zk\\g —* 0 (fc —» 00), which implies that Zk —> 0 in Hi+0. Because if a «—> iJ^ is compact for all /? < a, by (3.38) we infer that 2 I l * f c | | 7 - I l * * l l 2 » — > 0 , fc - > o o .

(3.39)

Based on (3.33), (3.34) and (3.39), there is a K > 0 large enough such that

53

Reduction Procedures and Stability

for all k>K, 2\\B\\ | N | 7 | N | 2 e

\Pk\ =

,

'

'

,

(by (3.37)) 2\\B\\-\\zk\\2e 25 1 20 z IIZL.II ll z *:||i 9llllzt-ll *:|l e " '

+

By t h e interpolatio n of space s (see [Temam, 1997]),

H2e =

[Hi+e,He]e,

where e is determined by (1 - e){\ + 9) + eO = 29, i.e. £ = 1 - 20. We have the interpolatio n inequality • I _ it ^ / i I I , ||20 || _ | | l - 2 0 ||2fc||20 S ^0||2fc||i + 0||Zfc||0 i

where Ce > 0 is a constant. Thus, we obtain from (3.40) tha t

J^L<2||fl||.Cfl,

Vk>K,

which implies (3.31). The proof is complete. 3.1.4

D

Generic properties

We present in this subsection a density theorem on operator s with complete spectrum, and a genericity theorem for simple eigenvalues. The spectra l completeness is a very interesting and difficult problem. By Theorem 3.2 we know tha t if L : Hi —» H is a symmetric sectoria l operator , then L has a complete eigenvalue sequence. But, for genera l sectoria l operators , we don't know if the spectra l completeness theorem is still true. However, we have the following density theorem on operator s with complete spectrum. Let L = -A + B : Hi -> H. We assume tha t A : Hi —» H is a sectoria l operator , A has a complete eigenvalue sequence {A^} C C, ReAfc > 0 V k > 1, B:Hi-*H

satisfies (3.30).

' '

'

54

Bifurcation Theory and Applications

Let S(HUH) = {L:Hi^H SA(HlfH)

= {Le

satisfies (3.41) and (3.30)} ,

S(HUH)

\A:Hi-+H\a

fixed}

.

Then we have the following density theorem for completion spectrum. Theorem 3.6 There exists o dense set D C S{H\,H) (resp. D C SA{HI,H)) such that each operator L £ D has a complete eigenvalue sequence. Remark 3.5 In fact, it is reasonable to conjecture that each L € S(Hi,H) satisfying (3.41) and (3.30) has a complete eigenvalue sequence. Proof of Theorem 3.6. Let L = -A + B £ S(Hi,H), and { H be the canonical projection defined by n PnU =

^UHPi. *=1

As PnB : H\ —> H is finite rank, the operator Ln = —A + PnB has a complete eigenvalue sequence for any n € N. In addition, since B : Hi —» H is compact, we have Yanio\\B-PnB\\s{Hun)=Q. Thus we obtain

J m ^ ||L - Ln\\S(HltH) = JLjm, IIB - PnB\\s{HuH)

= 0.

Therefore, the proof of the theorem is complete.

•

Now we consider the genericity of simple eigenvalues. An eigenvalue A € C of L : Hi —> H is called simple if the eigenspace E\ has dimension (l if A = real, dim^A = < [ 2 if A = complex. Let the eigenvalues {Afc} of a linear operator L G S(Hi,H) be given in the following order |Ai| < |A2| < • • • <

|Afc| < | A f c + i | < • • •

.

The following is the genericity theorem for simple eigenvalues.

Reduction Procedures and Stability

55

Theorem 3.7 For any integer m > 1, there is an open and dense set O c S(Hi,H) (resp. O C SA(Hi,H)) such that each L G O has at least m eigenvalues, and the first m eigenvalues {\j \ j = 1, • • • , m} are simple. Proof. By the classical theory of linear operators, each isolated eigenvalue depends continuously on the operators [Kato, 1995]. Therefore, the set O of all linear operators having at least m eigenvalues is open and dense in S(HUH) (resp. in SA(HUH)). Let O C O be the set of all linear operators whose first m eigenvalues are simple. Obviously, O is an open set. We shall prove that for any L = —A + B G O, there is a sequence Ln = —A + Bn G O, which converges to Lin S(Hi, H). Let {fk} be the eigenvectors of L. Without loss of generality, we assume that the first eigenvalues of L have multiplicity two, i.e. Ai = A2, and L
Lif2 = A1V2 + ipi.

(3.42)

By (3.15), for any u £ H, u = J2k ukfk + v, v G Ei- We define a linear operator Tn : Hi —> H by T nu = - u i v i . n Then we infer from (3.42) that the first two eigenvalues Ai and Ai + ^ of the operator Ln = L + Tn are simple, i.e. Ln(p2 - nipi) = \i( L in S(Hi,H) asn->oo. We can prove in the same fashion that for each n G N there exists a Bn : H\ —* H satisfying (3.30) such that Bn -> B in S{H\,H) as n -> 00, and the first • m eigenvalues of L n = - A + Bn are simple. This proof is complete. Remark 3.6 Theorem 3.7 shows that the bifurcation from a simple eigenvalue is generic. However, the study of bifurcation from eigenvalues with high multiplicities is still very important. For instance, there are many problems in science and engineering with bifurcations occurring at eigenvalues with even multiplicities.

56

3.2

Bifurcation Theory and Applications

Reduction Methods

We consider the following nonlinear evolution equation

| g = L>« + G (.,A), I »(0) - «„,

(343)

where L\ : H\ —> H is a parameterized linear bounded operator, G(-,\) : ffi -> ff a C (r > 1) mapping depending continuously on A 6 I 1 , and Hi c H is a dense and compact inclusion. 3.2.1

Reduction procedures

Reduction to center manifolds To investigate the stability and bifurcation of (3.43), it is crucial to reduce it to the center manifold. We start with a canonical reduction of (3.43). Let L\ = — A + B\ : Hi —» H be a linear completely continuous field, and G(-, A) : Hi —> H be a C°° operator, which can be expressed as oo

(3.44)

G(u,\) = YJGn(u,X), n=k

for some k > 2, where Gn : Hi x • • • x Hi —» if is an n-multiple linear m a p p i n g , a n d Gn(u,X)

= Gn(u, • • •

,u,X).

Let /3i(X) G C be the eigenvalues (counting the multiplicity) of L\, and ej(A) and e*(A) be the eigenvectors of L\ and L*x, respectively, corresponding to A (A). Let {hi} be a canonical orthogonal basis of H, and let oo

oo

3= 1

3= 1

Then {ej} and {e*} generate two infinite matrices (in 6 i - - A ^ = U12 62 • • • I

f =

(in Hi • • • \ ^1*2 & • • • .

Let L and £* be two infinite matrices induced by L\ and L*x respectively, £ = (ay), C* = (aijY,

a,ij -

(L\hj,hi)H.

57

Reduction Procedures and Stability

Definition 3.3 The eigenvectors {e^} and {e*} of L\ and L*x are called canonical if under an orthogonal basis they satisfy

d = ^J, £T = r A

(3-45)

where J = J\ is the Jordan matrix

h .

J = Jx=

,

V°

(3.46)

/

and Jjt are the Jordan blocks defined by //? £k Jk=

"'

0 \ "" '••

if

£k

/?jfc+i = • • • = /3jfc+mfc = /?, and by

/ a -p p a

ek 0 0 £fc

Jfe=

0 \

•••

e* 0

a

\ 0

P

o ' £k

—p

aj

if /3jfc+i = • • • = f3jk+mk =a + ip. It is easy to see that the Jordan matrix J is independent of the choice of the orthogonal basis {hi} C H. By Theorem 3.4 and Remark 3.4, if {e,} and {e*} are the canonical eigenvectors of L\ and L\, then ( =0

^ f 0

iti^j,

if i = j .

(3.47)

58

Bifurcation Theory and Applications

Let {ei} and {e*} be canonical. By (3.15), for any u e H,

u = 'Y2xkek+v,

v £ E2 .

k

Then, by (3.45)—(3.47), the equation (3.43) can be expressed as

/zA

/x{\ 1 * 2

^

= JA

*2

/gi(u,X)\ +


= Lxv + PG(u, A),

veE2,

(3.48)

(3.49)

where L\ = L\\E2 '• -^2 —* -E2, P '• H —> ^2 the projection, and flfc(tt,A)=,

\N

(ek,e*k)H

(G(u,X),el)H.

Now, we return to consider the reduction of (3.43) to the center manifold. Assume that the eigenvalues ft (A) satisfy

{

<0

if A < Ao,

(3.50) = 0

ifA = A0,

>0

ifA>A 0 ,

(Re^{\0)>0 [ReP^Xo) < 0

l
(3.50)

m<j<m + n, m + n<j.

Let there be a center manifold of (3.43) near A = Ao with the center manifold function given by S(.,A):fi—>££, (3.52) where fl C Em is a neighborhood of u = 0, and Em = ij^Xiei(X)

(x1,---,xm)€Rm\,

(u,e*(\))H = 0, 1 < * < m l .

Ei = \ueH

Thus, under the conditions (3.44) and (3.50)—(3.52), the equations (3.48) and (3.49) are reduced to the center manifold in the following form dx = JmXX + g(x,X), at

1r

(3.53)

59

Reduction Procedures and Stability

where Jm\ is the Jordan matrix corresponding to the first m eigenvalues /3 1 (A),-.-,/3 m (A)ofL A ,

,gm(um + $(um,A),A))*,

g(x,X) = (gi(um + $(um,\),\),---

gi(u, A) are as in (3.48), and um = J2f=i xier In some applications, the stability and bifurcation of (3.43) at (a;, A) = (0, Ao) is determined by the first approximation of (3.53), given by dx — = JmXX + Fk(x,X),

(3.54)

where k > 2 is given in (3.44), and Fk is a ^-multilinear vector field given by

(Fk(x,X) = (fki(x,X),---Jkm(x,X)y,

e

<355>

(«^<^HJ>4 'i- ' Second approximation

We see that the first approximation (3.54) of (3.53) is independent of the center manifold function (3.52). There are many cases where we need to consider the second approximation of (3.53). Assume that the linear operator L\ : Hi —> H has a complete eigenvalue sequence {/3j(A)} C C In this case, the equation (3.43) is equivalent to the system of (3.48). First, we compute the strongly continuous semi-group T(t) generated by L\. Let u = Y^=\ xk(t)ek be a solution of the linear equation AW

I dt

I «(0) = V,

'

(3.56)

where tp = Yl'kLi fk^k- Then, the equation (3.56) is equivalent to ( dx_ \ dt ~ for x = (xi(t),X2(t), itly as follows.

JxX

'

(3.57)

•••)*. The solution of (3.57) can be calculated explic-

60

Bifurcation Theory and Applications

When /^(A) = • • • = /3 J+m (A) = /3(A) = real, and L\ej+k

= (3j(\)ei+k

£j = 0, ej+k ^ 0, k > 0,

+ ej+kej+k-i,

we have k

xi+k(t) = ip^e?* + J2m £i+*tefj+e-iept.

(3.58)

l=i '

When j3j+k = Pj+k+1

= a + ip, k = 0, • • • , 2m, m > 0, and = Pjej+k +

L\ej+k L\ej+k+i

£j+kej+k-2,

= Pjej+k+i

+£j+kej+k-i,

we have

f

I xj+k

fc

1

+ tar,-+fc+1 = ^+ke^^

+ ^ - £ , + , t'^ + / _ i e (-+*/')*,

<

f=1

«•

(cs.oy)

[ V"j+fc = Vj+fc + ifj+k+iThe semi-group T(i) : H —» i7 is given by oo

r(i)^ = ^ ^ ( f ) e / t , fc=i

(3.60)

with xfe(t) given by (3.58) and (3.59). It follows from (3.58)—(3.60) that if Re/3j < 0 (j = 1,2, • • •), then there exist M > 1, 5 > 0 such that \\T(t)\\<Me-St,

Vi>0.

(3.61)

We are now in position to describe the second approximation of (3.53). For simplicity, we only consider the case where f3j are real with L\ej — /3,e., (j = 1,2, •••), and /?i = • • • = (3m- Then, the second approximation of (3.53) is given by /x[\

//3(A)

{':} ={

0 \

/Xl\

••

i \+F2(x,\) + F3(x,\)+F(x,\), (3.62)

where F2 and F3 are as in (3.55), F(x, A) = {fx{x, A), • • • , fm(x, A))*, and MXA)=

U^ \ek,ek)H

£ ^*^^.je=l

(3-63)

Reduction Procedures and Stability

61

Here OO

a

ije=

j

Z)

Tm

ft

\le

e*\

(G2{ej,ee,X),e*n)H

x (G 2 (e*,e n , A) + G 2 (e n ,e*, A),e*k)n, and G2(u, w) is the 2-multiple linear operator as in (3.44). We prove (3.62) and (3.63) in the following. By (3.53) we have

^

=/3i(A)*fc + 5fc ( f > j e j + (s,A),Aj

(3.64)

where x = YliiLi x%£i- The center manifold function OO

*(z,A)= Y,

*n(a:,A)c

n=m+l

satisfies (2.36), i.e. *n(i,A)=/

e-^p £ (G(z(r,x)+$,A), e ;)^r,

(3.65)

where z(i, x) = X)£li zs(*i 2;)e2 satisfies

( z4(0) = a:*. Hence Zi(t, x) = Xie0lt + o(\x\).

(3.66)

Inserting (3.66) into (3.65), by (3.44) we get m

x

*n(rr, A) = Y, ^ i,e=i

\

J

0

-°°

e^-^Tdr{G2(euee,

A),e*n)H +o(\x\2). (3.67)

62

Bifurcation Theory and Applications

Thus, from (3.64) and (3.67) it follows that

-I

+ Yl a'ljiXiXjxA + o(|z|3), i,j,e=i

J

which yields the second approximation (3.62) and (3.63). Prom the above discussions, it is easy to obtain the following theorem, providing an approximation of the center manifold function. This approximation will be used in applications in later chapters of the book. Theorem 3.8 Under the conditions (3.50) and (3.51), if L\ is a sectorial operator and G satisfies (3.44), then the center manifold function $(x, A) can be expressed as * ( * , A) = (-£$)-1P2Gk(x,

A) + O(\Re(3(\)\ • \\x\\k) + o(\\x\\k),

(3.68)

where £% «s given by (2.29), P2 : H —> E2 the canonical projection, x e E$, and f3(X) = (/3i(A), • • • ,/3m(A)) the eigenvalues of C\.

3.2.2

Morse index of nondegenerate singular points

We say that v G Hi is an equilibrium point, or a steady state of (3.43), if v satisfies Lxv + G{v,X)^0.

(3.69)

In order to investigate the stability and dynamic bifurcation of (3.43), it is necessary to consider the geometric properties of singularities of (3.69). To this end, we shall use the Lyapunov-Schmidt method to reduce the equation (3.69) to a finite dimensional algebraic equation, then consider the singularities of the reduced equation. Let the operator G(-, A) : Hi —> H be compact and satisfy (3.44). Then, for any uo £ Hi, the derivative operator DUG(UQ, A) : Hi —> H is a linear compact operator.

Reduction Procedures and Stability

63

Definition 3.4 A singular point UQ e H\ of (3.69) is nondegenerate, or is regular, if the derivative operator Lx + DuG(u0, X):H1—^H

(3.70)

of L\ + G at UQ is a linear homeomorphism. Definition 3.5 Let UQ £ H\ be a nondegenerate singular point of (3.69), and for all eigenvalues of the operator defined by (3.70), Re0i ^ 0. We define the Morse index of uo by k = number of eigenvalues of (3.70) having positive real part. By Theorems 3.11 and 3.12, if the Morse index k = 0 of a nondegenerate singular point UQ of (3.69), then uo is a locally asymptotic stable equilibrium point of (3.43). If the Morse index k > 1, then the singular point u0 is called a saddle point of (3.43). It is known that a saddle point u0 of (3.43) with Morse index k hasfc-dimensionalunstable manifold and a stable manifold with co-dimension k in H. Let (3.50) and (3.51) hold, and { e j and {e*} are canonical eigenvectors of L\ and L^. Near A = Ao, the equation (3.69) has the decomposition as follows

/*i\ (gi{x + $,\)\ Jmx : + : =0,

\W

(3.71)

\gm(x + $,\)J

£x$ + PG(x + $,\) = 0,

(3.72)

where Jm\ is the Jordan matrix corresponding to the first m eigenvale E ues /?!,••• ,/3m of Lx, gk{u,X) as in (3.48), x = YX=ixiei m = span{ei,--- , e m } , $ G E^ = {u e Hx | (u,e*)H = 0, 1 < i < m}, and Cx = LX\E± : Ei —> Ei, the closure of E^ in H, P : H —> E^ the projection. By (3.51), the linear operator Cx is invertible. By the implicit function theorem, near (x,\) = (0, Ao) there is a solution of (3.72) $ = $(z,A),

x€Em-

(3.73)

64

Bifurcation Theory and Applications

Putting (3.73) in (3.71), we get the following reduced equation of (3.69) /*i\ Jmx

:

/ s i ( x + *(x,A),A)\ +

\xj

:

= 0.

(3.74)

\gm(x + $(x,\),\)J

Under the conditions (3.50) and (3.51), the equation (3.69) may cause bifurcation from (0, Ao). By Theorem 1.9, if x(X) is a bifurcation solution of (3.74), then w(A) = x(X) + $(z(A), A) is a bifurcation solution of (3.69). The following theorem is useful to verify the regularity of a bifurcated solution of (3.69). Theorem 3.9 Let u(X) = x(X) + $(x, A) be a bifurcation solution of (3.69) from (0,Ao). Then u(X) is a nondegenerate singular point of (3.69) if and only if x(A) is a nondegenerate singular point of (3.74). Proof.

The derivative operator of (3.74) at x\ = x(X) is given by Jmx + Dxg + Dsg o Dx$\x=Xx

: Em —> Em.

(3.75)

On the other hand, the derivative operator (3.70) of (3.69) at UQ = x\ + $(XA,A) is invertible if and only if the following equations have no nonzero solution u = (y,w); (JmX + Dxg) • y + D$g -w = 0,

(3.76)

(Cx + D^PG) • w + DXPG -y = Q,

(3.77)

where y G Em, w 6 E^, and the derivative is taken at tto = x\ + $(x\,X). Since ||uo|| is small near Ao, by (3.44), ||£)$PG|| is also small. Hence the operator

B = Cx + D*PG{u0, X) :H!—>H is invertible. Thus it follows from (3.77) that w = -B~1oDxPG-y.

(3.78)

Putting (3.78) in (3.76) we obtain (Jmx + Dxg - D*g o B'1 o DXPG) • y = 0. We deduce from (3.72) that £>**•!/= ( - 5 - 1 o DXPG) • y.

(3.79)

Reduction Procedures and Stability

65

Therefore the operator (3.76) is invertible if and only if (3.79) has no nonzero solution in Em. The proof is complete. • In general both the vector fields in (3.53) and (3.74) are not the same because the center manifold function is different from the implicit function. Since the Morse index of a singular point is also characterized by its stable and unstable manifolds, if the Morse index of a singular point XQ of (3.53) is k for A near Ao, then under the conditions (3.50) and (3.51), the Morse index of singular point UQ = XQ + $(xo, A) of (3.69) is k + n, n as in (3.51). But, this assertion is not generally true for the equations (3.69) and (3.74). However, for the first-order and second-order approximations of (3.74), we have the following result. The equation (3.74) can be rewritten as oo

JmXX + Y,Fp(X>X)=0> p=k

( 3 ' 8 °)

where k > 2 is given as in (3.44), x G R m , and Fp is a p-multiple linear function of x. The following equation

Jmxx+

k+N-l

J2 Fp(x,X)=0

(3.81)

P=k

is called an iV'-order approximation of (3.74). Theorem 3.10 Each nondegenerate singular point of (3.81) uniquely corresponds to a nondegenerate singular point of (3.69) near (0, Ao). If XQ is a nondegenerate singular point of (3.81) with N = 1, and XQ has Morse index k (k > 0), then the corresponding singular point u0 of (3.69) has Morse index k + n, where n as in (3.51). The proof of Theorem 3.10 is trivial. For N = 2, if the first m eigenvalues of L\ are real at Ao, then Theorem 3.10 holds true also, because the vector field in (3.81) differs with that of (3.53) in a sufficiently small quantity near (0, Ao). We remark here that the reason why we are concerned with the topological structure of singular points of (3.81) rather than that of (3.53) is that the computation of the implicit functions is relatively easier than that of the center manifold functions.

66

Bifurcation Theory and Applications

3.3

Asymptotic Stability at Critical States

In the attractor bifurcation theory to be addressed in the later chapters, we shall see that the asymptotic stability of equilibrium points at critical states, i.e. at A = Ao in (3.50) and (3.51) with n = 0, is very important. This section is devoted to the problem. 3.3.1

Introduction to the Lyapunov stability

Definition 3.6

Let v e Hi be an equilibrium point of (3.43).

(1) We say that v is stable in H, if there is a neighborhood U C H of v, such that for any wo £ U the solution u(t, uo) of (3.43), if it exists, satisfies \\u(t, UQ) — V\\H —> 0 as uo —> v, V t > 0. (2) v is called locally asymptotically stable in H, if lim \\u(t,u0) — V\\H = 0,

V w0 € U.

t—>oo

lfU = H, then v is called globally asymptotically stable. (3) v is called exponentially stable if there are constants a,fi > Q such that \\u{t,uo)-v\\H
V t > 0.

The following theorem is a classical stability result; see [Henry, 1981]. T h e o r e m 3.11 Let L : Hi —> H be a sectorial operator, G : Ha —> H be Cr (r > 1) for some a < 1, and v G Hi be an equilibrium point of (3.43). If the spectrum a of L + DG(v) satisfies that Rea < -e, for some s > 0, where DG(v) is the derivative operator of G at v, then v is locally asymptotically stable with exponential decay. The following stability theorem is essentially known, which is useful on the stability of nonlinear wave equations. T h e o r e m 3.12 Let L : Hi —> H be a linear bounded operator, which generates a strongly continuous semigroup T(t), and G : H —> H be continuous satisfying \\G(u)\\H < C\\u\\kH,

for some k> 1.

(3.82)

Reduction Procedures and Stability

67

If there are constants M > 1, 5 > 0 such that \\T(t)\\<Me-St,

V*>0

(3.83)

then u = 0 is an equilibrium point of (3.43), which is locally asymptotically stable with exponentially decay. Proof.

The solution of (3.43) reads u(t) = T(t)u0 + f T(tJo

T)G(u)dr.

It follows from (3.83) that IM|jj<Me-"||«o||ff + Af / Jo

e-s^\\G{u)\\HdT.

Let ||u(t)||i/ = e~stz{t). Then we infer from (3.82) that z(t) < M\\uo\\H + MC I Jo

e-S{-k-^rzk{T)dT,

which implies that Z ( * ) < M | | « O | | H + M I - sup

zk(r).

0
Thus, we infer that z(t) is bounded for z(0) = u 0 in a sufficiently small neighborhood of u = 0. Therefore, u — 0 is locally asymptotically stable with exponentially decay. The proof is complete. • Remark 3.7 by

Theorem 3.12 is also valid if the condition (3.82) is replaced G{u) = O(||U||JJ),

3.3.2

Finite dimensional

u G H.

(3.84)

cases

We first consider the 2-D system given by

f^=G1(x1>x2),

{ f

(3.85)

68

Bifurcation Theory and Applications

where G = {GUG2) and G(0,0) = 0. Assume that G - f + g, and div/=|A OXi

+

| ^

= 0

OX2

.

(3.86)

Let /•X2

V(xi,x2)=

/-Xl

fi(0,x2)dx2-

f2(xi,0)dx1. Jo Jo We have the following theorem, which is useful in the stability, of hyperbolic equations at critical states. Theorem 3.13 Assume the condition (3.86). If there exists an open set Q, C M.2, with 0 € f2, such that x = 0 is a unique singular point of G in £1, and (1) V{x) > 0, V x E il, x ^ 0, (2) fi92 - /2S1 < 0 , V i e f i , and (3) div g < 0, V x € Q,, x ^ 0, then x = 0 is asymptotically stable in 0 for (3.85). Proof.

It follows from (3.86) that 8V

dV •5— = J l i

ox2 Hence, by condition (2) we find

7j— = —J2-

ax\

dV(x(t)) _dV dV + ~^^-dx~1Gl dx~22 = -hgi + /1S2 < 0, V x(t) e ft, x(t) a solution of (3.85), which implies, by condition (1), that V(x) is a Lyapunov function. Hence, x = 0 is stable. On the other hand, by (3.86) and condition (3), we know that (3.85) has no periodic orbits in O. Then the theorem follows from the Poincare-Bendixson theorem. • This theorem can be generalized to the higher dimensional systems as follows. Hr •— = G{x), x G R 2n , n > 1, (3.87) OX

w h e r e x = ( z i , - - - ,xn,yx,---

,yn), a n d G ( 0 ) = 0 . L e t

G{x) = JVH(x)+g{x),

69

Reduction Procedures and Stability

where H is a Hamiltonian function with H(0) — 0, and

'-0DIn the same fashion as used in Theorem 3.13, we can obtain Theorem 3.14

If there is an open set O C R2n, with 0 6 0 such that

(1) x = 0 is a unique zero •point of G in O, (2) H{x) > 0 V xeO andx^O, and

(3) E?=i [Wl'9i + I f 5n+i] < 0, V x G fi, x # 0, then H(x) is a Lyapunov function of (3.87), and x = 0 is asymptotically stable in O. Now, we consider the system as follows

{

dxi —— = an^i + ai 2 x 2 + Gi(x), dx

^ 3 ' 88 ^

—1 = a2ixi + a22X2 + G2(x), where Gi(x) = o(|x|), 2 = 1,2, and

j

a i l + a 2 2 = 0

'

[ CtuCt22 - " 1 2 0 2 1 > 0.

(3.89)

The eigenvalues of matrix (ay) are given by /3± = ±Waua22

- ai2«2i •

It is known that the equilibrium point x = 0 of (3.88) must be one of the three cases: a center, a stable focus, an unstable focus. The following theorem can be used in the Hopf bifurcation, which is a corollary of the Poincare-Bendixon theorem. Theorem 3.15 Let O c R2 be a neighborhood of x = 0. Under the condition (3.89), the following assertions hold. (1) If div G = 0 in O, then x = 0 is a center; (2) If div G < 0 (^ 0) in O, then x = 0 is a stable focus. (3) If div G > 0 (^ 0) in O, then x = 0 is an unstable focus.

70

Bifurcation Theory and Applications

3.3.3

An alternative principle for stability

We consider the equation given by diL

(3.90)

— =Lu + G{u)

where L : H\ —> H is symmetric, therefore all eigenvalues of L are real. Let the eigenvalues {fa} of L satisfy

{ Set

H fa

=0

Pj < 0

Eo = {ueH1\Lu

if 1 -< i -< m,.

(391)

if m + 1 < j < oo. = 0},

Ejt = {u e Hi I (u,v)H

=0,VveE0}.

The following theorem is crucial in the stability and bifurcation of the Rayleigh-Benard convection and the Taylor problem which will be discussed in Chapter 10. Theorem 3.16 Let L : H\ —> H be symmetric with spectrum given by (3.91), and G : H\ —> H satisfy the following orthogonal condition (G(u),u)H

= 0,

Vueft.

(3.92)

Then one and only one of the following two assertions holds true: (1) There exists a sequence of invariant sets {Tn} C EQ of (3.90) such that 0^rn,

lim dist(rn,0) = 0.

n—>oo

(2) The trivial equilibrium point u = 0 of (3.90) is locally asymptotically stable under the H-norm. Furthermore, if (3.90) has no invariant sets in EQ except the trivial one {0}, then u = 0 is globally asymptotically stable. Proof.

We proceed in the following four steps.

1. It is easy to see that Assertions (1) and (2) in Theorem 3.16 cannot be true at the same time. STEP

71

Reduction Procedures and Stability

By (3.91) and (3.92), direct energy estimates imply that the solutions u of (3.90) satisfy that °° J2 A N 2 < 0 ,

• jt\\u\\2H = 2=

IMIIf
(3-93) (3.94)

where u = w + v&H = E0®

EQ,

oo

i=m+l m

w = y^Uj € Ei = EQ. It is easy to see that for any

V t! < t2 and
(3.95)

Hence limt_Kx> ||u(t, v)|| exists. STEP 2. For any tp G Hi, we have lira | K * , v ) | | = lim t—>oo

\\v(t,
t—>oo

Then the cj-limit set, which is an invariant set, satisfies that w(((p) is an invariant set, for an ip G w() we have « ( « » Cw(y)) C 5,5

Vi>0.

Hence if ip = v + w e EQ- ® Eo with v ^ 0, then by (3.93), for any t > 0,

IKt,^)||<W = <J, a contradiction. Namely, for any

(3.96)

72

Bifurcation Theory and Applications

STEP 3. If Assertion (2) is false, then there exists u n e H\ with u n - » 0 as n —> oo such tha t 0 $ w(wn) C £0, and

lim dist(w(u n ),0) = 0 .

n—>oo

Namely, Assertion (1) holds true. STEP 4. If Assertion (1) is not true, there exist a neighborhood U C H of 0 such tha t for any <j) £ U, lim ||u(t l ¥ >)||=0 .

t—»oo

Namely, Assertion (2) holds true. The rest par t of the proof is trivial, and the proof is complete. • Remark 3.8 If all eigenvalues j3j < 0 (1 < j < 00) of the linear symmetric operato r L : Hi —* H, then under the condition (3.92), u = 0 must be a globally asymptotically stable equilibrium point of (3.90). In addition, under the conditions of Theorem 3.16, (3.96) is true. Likewise, we have Theorem 3.17 Under the condition (3.91), if G : H\ -+ H satisfies that (GU,U)H < 0 for all u ^ 0 sufficiently small, then u = 0 is a locally asymptotically stable equilibrium point of (3.90). 3.3.4

Dimension

reduction

We consider the equation (3.90) where the linear operato r L : Hi —» H is not necessar y to be symmetric. Assume tha t the conditions (3.44) hold true, and the eigenvalues {(3k} of L satisfy tha t f i *A = O

ifl
[ Rej3j < 0

if m + 1 < j < 00.

(3.98)

As the reduction of (3.43) to (3.53), we can reduce the equation (3.90) to an m-dimensional ordinar y differential equation ^=JmX

+ g(x),

x€Rm,

(3.98)

73

Reduction Procedures and Stability

where Jm is t h e J o r d a n m a t r i x corresponding t o / ? i , - - - ,/3m, a n d g = (fll,-" i£m),

sw

G

* =

Wi+

e

ra( (fr *(fH)' *)H'

Here $ is the center manifold function. By the center manifold theorem (Theorem 2.12) we know that if x = 0 is a locally asymptotically stable equilibrium point of (3.98), then u = 0 is locally asymptotically stable for (3.90). Equation (3.98) can be expressed as .

oo

-£ = Jmx + Y, FP(X)> fc > 2 as in (3.44).

(3.99)

p—k

We take the iVt/l-order approximation of (3.53) as follows H^T1 dx — = Jmx + £ Fp(x),

N>1.

p=k

(3.100)

Theorem 3.18 Assume that for some N > 1 there are constants C, r > 0 and integer M < k + N such that \

Ik+N-l

/ ^ Fp{x) + Jmx,x) <-C\x\M, \ p=k I

V |x| < r.

(3.101)

Then, x = 0 is a locally asymptotically stable equilibrium point of (3.99). Proof.

Since Zplk+N /

(

F

P(X) = o(|af ^ - ^ we have

oo

V

\p=k+N

\

r

F (x) x ) < -

\x\k+N+1

I

for all \x\ < r\, where r\ > 0 is sufficiently small. It follows from (3.101) that for any |x| < r\,

I

\

C

M ^ •Tp(X), F (T\ T — pl-rl (I JTmxT _i_ -\- Y 2_^ x \) < ^ —— | , A \ I P=k

which implies that the theorem holds true.

•

74

3.4

Bifurcation Theory and Applications

Notes

3.1 The spectral theorem, Theorem 3.4, is very useful in the bifurcation theory developed in this book. It is used, together with the construction of center manifold functions and the attractor bifurcation theory, to solve various problems in Physics, Mechanics, Chemistry, Biology, and Engineering. Theorem 3.4 was first introduced in [Ma and Wang, 2004a; Ma and Wang, 2004b]. The asymptotic properties of eigenvalues stated in Theorem 3.5 is given here for the first time. 3.2 Reduction to center manifolds for infinite dimensional systems is a basic tool for studying dynamic bifurcations. The main new ingredient presented here is the first and second order approximations of the center manifold functions, and its applications to various bifurcation and stability problems including applications to superconductivity, to the Rayleigh-Benard convection, and to the Taylor problem; see [Ma and Wang, 2005a; Ma and Wang, 2005e; Ma and Wang, 2005b]. Note that Theorem 3.8 was introduced by the authors in [Ma and Wang, 2004b]. 3.3 This section is based on the authors' recent work. In particular, the global stability theorem, Theorem 3.15, is crucial for studying the stability and bifurcation of the Benard convection and the Taylor problem [Ma and Wang, 2003a; Ma and Wang, 2004d; Ma and Wang, 2005d].

Chapter 4

Steady State Bifurcations

4.1

4.1.1

Bifurcations from Higher-Order Nondegenerate Singularities Even-order

nondegenerate

singularities

Consider a parameter family of nonlinear operator equations (4.1)

Lxu + G(u,X) = 0,

where L\ : Hi —> H is a completely continuous field, and G(-, A) : Hi —> i7 is a C°° operator having the Taylor expansion as (3.44). Let the eigenvalues (counting multiplicity) of L\ be given by {/?i(A), /?2(A), • • • } with A(A) £ K1 (1 < i < m) such that

{

<0

if A < Ao,

(4.2) = 0

if A = Ao,

> 0

if A > Ao,

V 1 < i < m,

(4.2)

(3j(Xo)^0 Vm + l < j . (4.3) Let {ei, • • • , e r } and {ej, • • • , e*} C Hi be the eigenvectors of L\ and L*x at A = Ao respectively: LXoej = 0, L£oe* = 0 , 1 < j < r. (4.4) Here r < m is the geometric multiplicity of the eigenvalue /3i(A0). We set a

h-jk

=

(Gk(eji,---

where k > 2 is given by (3.44). 75

,ejk,X0),e*)H,

76

Bifurcation Theory and Applications

Definition 4.1 Assume (4.2) and (4.3). The steady state solution u = 0 of (4.1) is called k-th order nondegenerate at A = Ao, if x = (xi, • • • , xr) = 0 is an isolated singular point of the following system of r-dimensional algebraic equations r

E

4-^i---^=°>

l<*
(4.5)

The following two theorems establish bifurcations from even-order nondegenerate singular points. Theorem 4.1 Let the operator L\ + G : Hi —> H satisfy (4.2) and (4-3). If there is an even number k > 2 such that u = 0 is a k-th order nondegenerate singular point of (4-1) at X = Xo, then (u, A) = (0, Ao) must be a bifurcation point of (4-1), and there is at least one bifurcated branch on each side of X = Xo. This theorem establishes the existence of a trans-critical bifurcation, and the following is its global version, in the same spirit of the Rabinowitz global bifurcation theorem. Let E+ = {{u, X) G Hi x R 1 | L\u + G(u, A) = 0, u ^ 0, A > Ao} , I T = {{u, A) G # i x R1 | Lxu + G(u, X) = 0, u ^ 0, A < Ao} . Theorem 4.2 Let Lx : Hi -> H satisfy (3.41), and G(-, A) : Hi -» H be compact. Let S c S (resp. E c S ) be the connected component of S + (resp. S ~ ) containing (0, Ao). Under the conditions of Theorem 4-1, one of the following assertions holds true. (1) E is unbounded, (2) £ contains points (0,Ai) with Ai > Ao (resp. Xi < XQ) such that there are some eigenvalues f3j(X) of L\ at X = Ai with f3j(Xi) = 0, or (3) There exists a point (VQ, /J.) € Hi x R 1 with fi > Ao (resp. /i < Xo) such that

{

0

ifX>fi

(resp. A < n),

(vo,fi) i/A = /i, Ti(A) + T2(A) if Ao < A < \i (resp. p < A < A o ;, tufeere T ^ A Q ) = (0,A 0 ), T2(A0) ^ (0,A 0 ), Ti(X) ^ %, andY2{X) + 0. Remark 4.1 Assertion (3) in Theorem 4.2 implies that there is a bifurcation of (4.1) from (vo,p), and there are at least two bifurcated branches

77

Steady State Bifurcations

Fi(A) and ^(A) on this side of A < /x or \x < A, as shown in Figure 4.1. Also H

v0

/

!

^\

x

Fig. 4.1 Bifurcation from (vo, p) •

we mention that this bifurcation as shown in Figure 4.1 occurs frequently in hydrodynamic equations, which will be investigated in the forthcoming book on the hydrodynamic stability and bifurcation by the authors.The type of bifurcation shown in Assertion (3) is called saddle-node bifurcation. Proof of Theorem ^.1. We proceed with the Lyapunov-Schmidt method, together with the Brouwer degree theory. 1. Consider (4.1). Let {ttfi(A), • • • ,wm(X)} c Hi and {wl(X), • • • , w^,(A)} C Hi be the eigenvectors of L\ and L\ respectively, corresponding to eigenvalues given by (4.2), i.e. STEP

(L A -A(A)r^(A)=0, (LX-ft(A)r'<(A) = 0, for some rij > 1, m, > 1 (1 < i < m), such that (wi,w*)H = 5ij. By the normalization of the Lyapunov-Schmidt reduction given by

78

Bifurcation Theory and Applications

(3.74), we reduce the equation (4.1) to the following equation xeRm.

(4.6)

,gkm(x,\)Y,

(4.7)

Jmxx + g(x,X)=0, Here g(x, A) = Fk(x, A) + f(x, A) is given by Fk{x,X)

= (gkl(x,X),--fc

/ ( i , A) = o(|a:| ).

(4.8)

for some k > 2, where m

9ki{x, A) =

5Z

x

i i ' ' ' x ^ (Gfc(wji. • • • . wjk.

X),w*)H.

Since the Jordan matrix J m A has /?i(A),--- ,/3m(X) as its eigenvalues which satisfy (4.2), we have the index formula f(-l)m ifA Ao. To complete the proof of the theorem, it suffices to show that ind( J mAo + 50, Ao), 0) = even.

(4.9)

We proceed in three steps to prove (4.9). STEP 2. We first consider the case where Jm\0 = 0, i.e. r = m, the geometric multiplicity equals the algebraic multiplicity at A0By assumption, Fk : R m —> R m is a nondegenerate fc-th order multilinear mapping. There exists a number (3 > 0 such that

\Fk(ax,Xo)\>P\a\k,

V x G R m , \x\ = 1.

(4.10)

It follows from (4.8) and (4.10) that for any r > 0 sufficiently small, Fk(x, Ao) + tf{x, Ao) ^ 0,

Vx

GM

m

, |x| = r, 0 < t < 1.

By the homotopy invariance property of the Brouwer degree, we have ind(J mAo +sO,A 0 ),0) =deg(J m A o +gXo,BR,0)

(4.11)

= deg(F fc (.,A 0 ),B R ,0), where BR = {x G R m | |a;| < R}. By Theorem 1.5, we have deg(Ffc(-,Xo),BR,0) = even,

if k = even.

(4.12)

79

Steady State Bifurcations

Hence (4.9) follows from (4.11) and (4.12). S T E P 3 . C A S E WHERE m = 3, r = 2 AND k = 2. To present the

main idea of the proof, we proceed with the special case where the algebraic multiplicity m = 3, the geometric multiplicity r = 2 at A = Ao, and Fk(x, A) is a second order multi-linear mapping. Let L\Bwi = 0, L\ow2 = 0, L\ow3 — w2. By (3.25) and (3.26), we see that the dual eigenvectors of {wi,W2,wz} satisfying {W^W^H = hj are given by

LIM = 0, L*Xowl =wl, L*xM = 0. Therefore, we have ei =w1, e2 =w2,

el = i y * , e*2 =w%,

(4.13)

and

=

Jm\ox

/ 0 0 0 \ (Xl\ 001 u

\0 0 0/ \xj

=

/ o \ i3 ,

\o)

m = 3.

(4.14)

By k = 2, /

-FM^Ao) =

3

3

b x x

3

hxx

J2 ij * i> J2 l i i, \*J = 1

i,j=l

>

Yl % i i,j=l

where &fj =

\ '

h XiX

(G2{wi,wj),w})H.

)

( 4 - 15 )

80

Bifurcation Theory and Applications

Let T : K 3 —> R 3 be a linear homeomorphism defined by

/I 0 0\ fXl\ Tx=\00l\

,

\x2

VOIO/ VW

/°\

TJ3AOZ = 1 0

I.

Then we derive from (4.14), (4.15) and (4.7) that T(J3\0 + 9\0)x = T(J3Xo + F2)x + Tf(x, Ao), Eh^XiXj x3 + E

3

J=1

\,

(4.16) (4.17)

bfjXiXjJ

2

(4.18)

Tf(x,\0)=o(\x\ ).

By assumption, (x\, X2) = 0 is an isolated singular point of (4.5) with r = 2 and k = 2, which, by (4.13), reads 2

J2aeijxixj=0, a

t=l,2,

ij = {G2{ei,ej),el)H

= b}j,

a*j = {G2(ei,ej),eZ))H

= b3ij.

Thus, we infer from (4.17) and (4.18) that for any 0 < t < 1, x = (xi,X2,xs) = 0 is an isolated zero point of the following equation / E L = 1 ahX*Xi + tX* E?=l(fci3 + Hi)Xi + *&33^\ £ j j = i a?.x i x i + te3 Ef=i(&« + hli)*i + ^33^3 + *Tf(x, Ao) = 0. V

x

3 + tElj=lbijxixj

)

(4.19) Based on the homotopy invariance of Brouwer degree, it follows from (4.16)-(4.19) that deg(T( J3Xo +gXo), BR, 0) = deg(tf, BR, 0),

(4.20)

81

Steady State Bifurcations

where R > 0 is sufficiently small, and /EL=i

a

ijxixi\

*(*) = E U i 4 ^ ' • According to Theorems 1.2 and 1.3, it follows from (4.16)-(4.19) that deg(T(J3A0 + 9xo),BR, 0) = ind(T, 0) • deg(J3Ao + g, BR, 0) = (-l)deg(J3Ao +9,BR,0),

(4.21)

and deg(K, BR, 0) - deg(^, S f i n l 2 , 0 ) ,

(4.22)

where

/V2

A^!,^;- I

2

a1 a;-a; A 2

I •

By assumption, {x\,x
(4.23)

Finally, the index formula (4.9) follows from (4.20)-(4.23). STEP 4. GENERAL CASE. Let the Jordan matrix Jm\0 have the following form

/° \o

Jl

°\

/01 0

00 1

'••

JJ \0

m

The space R can be decomposed into

JmXoV = 0,

Vj/6lr.

0\

1

0/

82

Bifurcation Theory and Applications

Let P : R m -> R r be the canonical projection, T : Rm -* R m given by //

0\

/0 ..- 0 1\ 1 ••• 0 0 %= . . . . .

7\ .

T= \0

,

\ 0 ••• 1 0 /

TJ

It is clear that T is a linear homeomorphism, and TJm\ohr = °,

TJ OT A 0 |K—- = *m-r the identity,

:

where ^ 1 ; . . .

J t

(: -

aj-i-j^Ji '"" ^

- r)

{

,

a r e

^

i n

(4.24)

(4.25)

(4-5)-

As in the proof of (4.20)-(4.23), we can deduce from (4.24) and (4.25) that deg(JmAo +gXo,BR,0) = (-l)m~r deg(T(JmXa + gxo),BR,0) = (-l)m~r deg(TJmAo + TFk, BR, 0) =

(-l)m-rdeg(PTFk\Rr,BRnRr,0).

Since PTFk\w : W —* M.r is an even mapping, and by assumption x = (xi, • • • ,xr) = 0 is an isolated singular point of PTFk\Rr, we have ind(P2Tfc|R,-,0) = deg(PTFk\Rr,BR

nRr,0)

= even. Thus (4.9) follows. The proof of this theorem is complete.

•

Proof of Theorem J^.2. It is known that the Leray-Schauder degree theory is valid for the completely continuous fields L\ + G : H\ —> H, provided that L\ satisfies (3.41). The eigenvalues /3j(A) of L\ with Re/3,- > 0 are finite, and if L\ : H\ —> H is invertible, then we have ind(-(L A + G), 0) = deg(-(L A + G), BR, 0) = (-1)",

(4.26)

where n is the sum of all algebraic multiplicities of the real eigenvalues /3 i (A)>0ofL A .

83

Steady State Bifurcations

By similar properties for the Leray-Schauder degree theory as those for the Brouwer degree used in the proof of Theorem 3.13, we can show that under the conditions of Theorem 4.2, the topological degree of L\ + G at A = Ao is an even number, i.e. deg(-(L Ao +G),BR,0)

= (-l)n+m-r = even.

deg(PTFk\Ur,BRnW,O) (4.27)

where W, TPFk are the same as in (4.25). Then the rest of the proof of this theorem is routine using (4.26) and (4.27). The proof is complete. • Remark 4.2 In fact,in the proof of Theorems 4.1 and 4.2, we have shown the following result for the Leray-Schauder degree. Let L\ + G : H\ —> H be a completely continuous field, and G(u, A) = w °(ll l|ffi)- If there are {wi,-•• ,wm} C H\ such that L^Wi = 0,

1 < i < m, Hi > 1,

and u = 0 is afc-thorder nondegenerate singular point of L\o + G(-, Ao) for some k > 2, then deg(-(L Ao + G(-, Ao)), BR, 0) = (-1)" deg( JXom +g,BRn

E, 0)

where n is the sum of all algebraic multiplicities of the real eigenvalues /?j(A0) > 0 of L\o, E = span {ei, • • • , e m } , JXom the Jordan form of L\o restricted on E, and g = — PG\E : E -* E, and P : H —> E the canonical projection. 4.1.2

Bifurcation at geometric simple eigenvalues: r = 1

In this subsection, we address the case where the geometric multiplicity of the eigenvalue is one, i.e. r = 1 in (4.4). Definition 4.2 Let T(X) C Hi be a branch bifurcated from (uo,Ao) of (4.1). F(A) is called regular if for any |A — Ao| > 0 sufficiently small each singular point v\ € F(A) of (4.1) is nondegenerate. Theorem 4.3 the number

Assume the conditions (4-2), (4.3) and r = 1 in (4.4). If a = (Gk(e1,\0),e*1)H^0,

k>2,

(4.28)

where Gk is given by (3.44), then the following assertions hold true.

84

Bifurcation Theory and Applications

(1) For the case where k is even in (4-28), there exists a unique bifurcated branch of (4-1) on each side of Xo. (2) For the case where k is odd and m is even, if a < 0, then (4-1) has no bifurcation from (0,Ao), and if a > 0 then there are exactly two branches bifurcated from (0, Ao) on each side of XQ. (3) For the case where k is odd and m is odd, (4-1) has no bifurcated branch on X < Ao (resp. on X > Xo) and has exactly two branches on X > Ao (resp. on X < Xo) if a < 0 (resp. a > 0). (4) Each branch F(A) bifurcated from (0, Ao) for (4-1) is regular, and the singular points u\ G T(A) can be expressed as ux = ± la" 1 /?!(A) • ••(3m(X)\1/(k~l) 1

+ 0 (|a- /?i(A)---/3 r o (A)|

Proof.

Let {wi,•••,

I LXow1

w m}

ej

(4.29)

1/(fc 1)

- ).

and {w*, • • • , w*^} C H\ satisfy

= 0, LXow2

= wi,---

,LXowm

=

\ Llow*m = 0, L ^ i i C . ! = < , - • - , L*Xowl

wm-i, =w*2.

(4.30)

From (3.25) and (3.26), we see that f =0

iii^j,

<""'•"'> { ^ 0 B . - i

(4 3I)

'

In this case, the bifurcation equations (4.6) can be written in the following form //MA) 1 0 . . 0 /32(A) 1 ••• ' . ^0

0

0 0 .

0 ••• Pm(X)J

where g(x, A) is the same as in (4.6).

\ /

M M +9(x,X)=0, ^

m

'

(4.32)

85

Steady State Bifurcations

We see from (4.30) that ei = wi and e\ = w^. By (4.7), (4.8) and (4.31), we infer from (4.32) that •x2 =

-f31(\)x1+O(\x\k),

' x m = -/3 m _ 1 (A)rr m _ 1 +0(|x| / £ ),

a(X)xk1 = -f3m(X)xm +

fe-l

(4.33)

m

O(^2'£\x1\i\xj\k-i),

i=0 j=2

where a(Ao) = a. It follows from (4.33) that near A = Ao, a(X)xk1-1 = (-l) m /?i(A) • • • 0 m (A) + od^(A) • • • /3m(A)|).

(4.34)

Thus, Assertions (1)—(3) and (4.29) follow from (4.2), (4.28) and (4.34). Assertion (4) can be derived from the following determinant of the first order approximation of the Jacobian of (4.32) and Theorem 3.9: / ft (A) det

0

1 /?2 (A)

0 ••• 1 •••

0 \ 0

W " 1 0 0 ••• /3m(\)J = 0i-"0ro-(-irafcc*-1 = (1 - fc)0i • • • 0m + o(|0i • • • (3m\) ^0

(by (4.34))

for A ^ Ao (by (4.2)).

The proof of this theorem is complete. 4.1.3

Bifurcation with r = k = 2.

Let r = 2 in (4.4), Gfc be bilinear (i.e. k = 2), and a n = (G 2 (ei,ei,A),ei)tf, a22 = (G 2 (e 2 ,e2,A),ei)H, 012 = (G 2 (ei, e 2 , A), e*)^ + (G 2 (e 2 , ei, A), ej)#, 6ii = (G 2 (ei,ei,A),e;)K, {G2(e2,e2,\),e^)H, b22 = bi2 = (G 2 (ei, e 2 , A), e ^ ^ + (G 2 (e 2 , eu A), e|>.

•

86

Bifurcation Theory and Applications

We have the following bifurcation theorem. Theorem 4.4 Let the conditions (4-2), (4-3) and r = 2 in (4-4) hold. If u = 0 is a second-order nondegenerate singular point of L\ + G at X = \Q, and the two vectors (an,ai2, a i3) and (611,612,^13) are linearly independent, then we have the following assertions. (1) There are at most three branches of (4-1) bifurcated from (0,Ao) on each side of X = Ao. (2) If the bifurcated branches on A > Ao (resp. on X < Xo) are regular, then the number of branches on this side is either 1 or 3. (3) On any given side, the number of branches is 3, then these 3 branches must be regular. (4) If the number of branches on one side is 2, then one of them is regular. Proof.

We proceed in several steps as follows.

S T E P 1. We only have to prove the case where m = r = 2. For the case where m > r = 2, as in the proof of Theorem 4.3, the m-dimensional bifurcation equations (4.6) can be treated as if they are two-dimensional equations, and the proof given below for the case where m = r = 2 can then be applied as well. As m = r = 2 in (4.2) and (4.4), the bifurcation equations (4.6) can be rewritten as follows

f /3i(A)xi + aux\ + ai2xix2 + a22x\ + o{\x\2) = 0, \ /32(A)x2 + bxlx\ + b12x1x2 + b22x\ + o{\x\2) = 0,

(4.35)

where x = (xi,x2). 2. We now show that there is an £ > 0 such that for any C°° function (fi(x), f2(x)) with |/i(x)| +1/2(2;)! small near x = 0, the following equations STEP

anxl + a12iia;2 + ^ z ! + fi(x) + o(|x|2) = 0, 2

buxl + b12Xlx2 + b22x\ + f2{x) + o(\x\ ) = 0,

(4.36) (4.37)

have at most four solutions in \x\ < e. Since the second order terms in (4.36) and (4.37) are second-order nondegenerate, at least one of the coefficients a n , 022, ^n and i>22 is not zero. Without loss of generality, we assume that a22 ^ 0. Then we obtain from

87

Steady State Bifurcations

(4.36) that X

*

=

~~2a~ Xl

±

2^~ V ^

2

~ 4 a i i a 2 2 ) x i ~ 4a 22/i( a: ) + °(N 2 ))

which, by the implicit function theorem, implies that

^-s^ss*

(4 38)

-

where A = y ( a i 2 -4ana 2 2)a: i + g(xi) + o(\xx\2) ff(xi)-»0

if

/i(s)->0 .

Inserting (4.38) into (4.37), we have ax\ + b22g(xi) + Aal2f2(x) + o(\x\2) = ^jfttiA,

(4.39)

where a = 4a 22 (a 2 2&n - 011622) + 2ai2(ai 2 6 2 2 - 022612), f3 = 2(022612 - 012622)We infer then from (4.38) an d (4.39) tha t px\ + /iiOn)z 2 + h2{Xl) + o(\Xl\4) = 0,

(4.40)

where p = a2 - f32(a2n - 4 a n a 2 2 ) , h1(x1),h2(x1)^0

if

/ i ( x ) , / 2 ( x ) - » 0.

It is easy to verify that the condition p =f= 0 is equivalent to the following conditions

6 22 4+^ ± +W0,

z± = Z ^ i i ^ S Z ^^ _ ^O 2 2

( (4.40)

By the assumption of this theorem, conditions in (4.41) ar e valid. Therefore, p ^ 0, which implies that (4.40) has at most four rea l solutions satisfying that xi -> 0 as fx(x), f2(x) -> 0. On the other hand, each solution of (4.40) correspond s to one of the signs ± in (4.39) and (4.38). Hence equations (4.36) and (4.37) have at most four rea l solutions near x = 0. Namely, Assertion (1) is proved.

88

Bifurcation Theory and Applications STEP

3.

P R O O F OF ASSERTION

(2). Let FA : M2 -» R2 be defined by

p ix\ _ fPiWxi + a i i x i + a-\ix\xi + 022^2 + O (M 2 A M ' \/32(X)x2 + bux\ + b12x1x2 + b22x\ + o(\x\2)) ' By Theorem 3.9, the singular points z(\) of (4.35) near x = 0 is regular providing the bifurcated branches of (4.1) from (0,Ao) is regular. Hence, we have ind(Fx,z(\)) = ±l

(4.42)

for | A — Ao| 7^ 0 sufficiently small. In addition, we know that if A ^ Ao, ind(FA,O) = sign/?i(A)ft(A) = 1,

(4.43)

and, by (4.9), we have Y^ ind(F A , ^(A)) = even,

(4.44)

\zt\<e

for any e > 0 small. Hence Assertion (2) follows from (4.42)—(4.44) and Assertion (1). 4. P R O O F OF ASSERTIONS (3) AND (4). We know from Step 2 that the solutions of (4.36) and (4.37) are one to one corresponding to those of (4.40). Let x0 6 i 1 be a solution of (4.40) near x = 0. Then (4.40) can be expressed as STEP

(x - xo)kg(x)

= 0,

k>\.

(4.45)

We shall prove that a solution z0 £ R2 of (4.36) and (4.37) corresponding to XQ is nondegenerate if and only if k = 1 in (4.45). Obviously, if k > 2 in (4.45), under a perturbation there are more solutions of (4.36) and (4.37) near ZQ. Suppose zo is degenerate, i.e. detDF(zo) = 0, where DF(zo) stands for the Jacobian matrix at F, and f o u i j + ai2xix2 + a22x\ + fi(x) + o(|a;|2), \ bnxl + 612X1X2 + 622X2 + / 2 (x) + o(|x| 2 ).

(4.46)

89

Steady State Bifurcations

Under a suitable coordinate system, DF(z0) must be one of the following three forms

(4.47)

DF{zo)=(^ DF(zo)=(°0

°Qy o),

(4.48) (4.49)

for some a ^ 0. Since / i , f2 are C°° sufficiently small function, and the quadratic form in F are second-order nondegenerate, for both the cases of (4.47) and (4.48) it is known that with a linear perturbation the equation (4.36) and (4.37) will yield one more solution near z0, which implies k > 2 in (4.45). We proceed for th case of (4.49). Near z0, F can be expressed as

p(y) = {ay2 + ; u y J + c ; * y m + c ; i + ° ^

duVx + di2yiy2 + d22yi + o{\y\i),

[

(4.50)

where y = x — ZQ. The zero point y = 0 of (4.50) stands for the zero point ZQ of F. Now we consider the perturbation equation <*yi = -(cnyf £J/i = ~(dnyl

+ c12y!y2 + c22yl) + o(|y| 2 ), 2

+ d12ylV2 + d22y\) + o(\y\ ),

(4.51) (4.52)

for e small. By the implicit function theorem, we obtain from (4.51) a solution y2 = /(yi). Then (4.52) implies that eyi = -(dnyf + d12yif(yi)

+ d22f2(yi)) + o(\yi\2).

(4.53)

B y t h e s e c o n d - o r d e r n o n d e g e n e r a c y of F(y), t h e r e exist a ^ O a n d m > 2 such t h a t (4.53) b e c o m e s eyi=ay^

+ o(\yi\m).

(4.54)

We infer from (4.54) that equations (4.51) and(4.52) has two solutions (0,0) and (yi,f(yi)), where

90

Bifurcation Theory and Applications

and £ is taken such that e • a > 0. Thus, we prove that if z0 is a degenerate zero point of (4.35) and (4.37), then the exponential k > 2 of the corresponding solution XQ in (4.45). We are now in a position to prove Assertions (3)—(4). If (4.35) has three bifurcated solutions (resp. two bifurcated solutions) from (0, Ao), then the corresponding solutions Xi ^ 0, i = 1,2,3, (resp. Xj ^ 0, j = 1,2) of (4.40) can be expressed as (a; - xi)(x - x2){x - x3)xgi(x)

= 0,

2

(resp. (x — xi) (x — X2)xg\{x) = 0, which implies that Assertion (3) (resp. Assertion (4)) holds true. The proof of this theorem is complete. 4.1.4

•

Reduction to potential operators

We consider the case where the equation (4.1) is reduced to an equation with the first fc-multiple linear operator being potential. We assume that the eigenvalues of L\ in (4.2) satisfy /?i(A) = - - - = / M A ) = / ? ( A )

nearA = A0,

(4.55)

and m = r the geometric multiplicity. Let the reduced equations (3.74) be expressed as P(\)xi+Fi1(x)+o(\x\ki)=0,

(l
(4.56)

where fci > k, k > 2 is the same as in (3.44) of Chapter 3, and 771

X

Fid ( ) =

Yl

a

h--imxh

•' • xi^ •

We also assume that the operator F^ = (F^ ,•••, FJ^) is a potential operator, i.e. there is a function V(x) such that

(4.57) By the Krasnoselskii theorem, (x,X) = (0,Ao) must be a bifurcation point of the following equations, which are the n-th order approximation of (4.56), n = ki-k + l: P(\)Xi

+ Flkl (x) = 0 ,

(1 < t < m).

(4.58)

91

Steady State Bifurcations

We have the following theorem. Theorem 4.5 Let the conditions (4.2), (4.3) and (4.55)—(4-56) hold. If (4-58) has r bifurcated branches Fj(A) (1 < i < r) on a given side of A = Ao, which satisfy that for zx 6 Fj(A), ind(/?(A)id + F f c l I ZA)^O

(4.59)

then the equation (4-1) has at least r bifurcated branches on the side. Proof. Since the vector field Fkl is fci-multiple linear, the solutions zx = (xi(A), • • • ,x m (A)) G I\,(A) have the form

Xi{\)

1

= c^CA)! ^-!),

m

Yla^ °' t=l

( 46 °)

where on are constants independent of A. Since zx S Tj(X) is an isolated zero point of (4.58), by (4.60) there exist constants 5 > 0 and a > 0, such that z\ is a unique singular point of (4.58)

in Bs(z\), and \p(\)x

+ Fkl (a:)| > a I / J M I ^ * 1 - 1 ' ,

V i e dBs(zx),

(4.61)

for | A — Ao | ^ 0 small, where

Bs(zx) = [x G R m | \x - z A | < 5|/3(A)|1/}. It follows from (4.60) and (4.61) that for the equation (4.56) we have

P{\)x + Fkl (x) + tf{x) ^ 0, where f(x) = o(\x\kl). obtain that

V 0 < t < 1, a; G dB5(zx),

By the homotopy invariance of Brouwer degree we

deg(/3(A)id + Fkl + /,B s (z\),0) = ind(/3 id + Fkl,zx).

(4.62)

We infer from (4.59) and (4.62) that the equation (4.56) has at least a • solution near zx for all |A — Ao| ^ 0 small. The theorem is proved. Corollary 4.1 Assume (4.2), (4.3) and (4.55)—(4.57). If x = 0 is an isolated zero point of Fkl with hi = odd, and the bifurcated branches of (4-58) are finite, then (4-1) has at least 2m bifurcated branches on both sides of A = AQ .

92

Bifurcation Theory and Applications

Proof. Because there are at least m pairs of bifurcated singular points, which are the minimax points of the functional | | x | 2 + V(x), the minimax points must have a nonzero index. The proof is complete. •

4.2 4.2.1

Alternative Method Introduction

Let X be a Banach space, I : I - » I a linear compact operator, and h and G : X —> X be continuous mappings. We consider the bifurcation of the following types of equations x - \{L + h)x + G{x) = 0,

(4.63)

x - Xh(x) -Lx + G{x) = 0,

(4.64)

x - XLx - h(x) + G(x) = 0,

(4.65)

x - \h{x) + G(x) = 0.

(4.66)

For this case, the classical bifurcation theory is not accessible. In this section, we present a method, called the alternative method, to deal with the bifurcation of the equations (4.63)—(4.66). To show how the alternative method works, we discuss the bifurcation in the following assumptions L,h : X —> X

linear compact operators,

G : X —» X

compact,

G(x)=o(\\x\\). Consider the following parameterized linear operators id-\iL-\2h:XxIxI—>X,

7 = [0,l].

Assume that Ai = 1 is not an eigenvalue of L, and Ai = p-1 is a unique eigenvalue of L in [l,oo) with odd algebraic multiplicity. In the parameter space (Ai, A2) € I x I, the index of id — AiL — A2/1 at x = 0 is shown as in Figure 4.2. By the homotopy invariance of the Leray-Schauder degree, there exists a curve 70 C / x I, which divides the square I x I into two domains U\ and

93

Steady State Bifurcations

1

_,

J.

ui

X

X /°

/ y * o '^o*

y'

•

P

*i

^ ^

(i. *")

*i

D

Pig. 4.2

17%, such that

• ^

x T

xhn\

J

1

ind(id — X\L — A2fi, 0) = <

\ -1

if(Ai,A 2 )€C/ 1 ,

if(Ai,A 2 )Gl7 2 ,

which implies that • ACA ^ T M.j.nn\ I1 md(id — \\L — A2/i + G, 0) = < ^ \ -1

if(Ai,A 2 )et/i, if(A 1 ) A 2 )et/ 2 .

Thus, one can see that if the curve 70 is the case as shown in Figure 4.2(a), then the number Ao in the intersection of 70 with the diagonal of / x / is a bifurcation point of (4.63). If 70 is the case as shown in Figure 4.2(b), then the number A* in (1,A*) G 70 is a bifurcation point of (4.64). Obviously, the curve 70 must pass through either the diagonals {(A, A) I 0 < A < 1} or the line segment {1} x [0,1]. Hence we can claim that (i) there must be one of the two equations (4.63) and (4.64) having a bifurcation point Ao £ [0,1]; (ii) at least one of the three equations (4.64)— (4.66) has a bifurcation point in [0,1]. In the following subsections, we shall discuss the alternative method for the case where h : X —> X may be a nonlinear operator.

94

Bifurcation Theory and Applications

4.2.2

Alternative bifurcation theorems

We consider the following equations x + L(x, A) + h(x, A) + G(x) = 0,

(4.67)

x + L(x, 1) + h(x, A) + G(x) = 0,

(4.68)

x + L(x, A) + h(x, 1) + G(x) = 0,

(4.69)

x + h(x, A) + G{x) = 0.

(4.70)

Suppose that £(•, A) : X —* X is a linear compact operator with L(x, 0) = 0, h : X x M.1 —> X a compact operator with /i(:r, 0) = /i(0, A) = 0, G : X —> X a compact operator with G{x) = o(||a;||). We say that Ao £ R 1 is an eigenvalue of L : X x R 1 —> X, if there exists x € X with x 7^ 0 such that a; + L(a;,Ao) = 0. Theorem 4.6 If\ = lis not an eigenvalue of L, and the sum of algebraic multiplicities of eigenvalues of L in (0,1) is an odd number, then we have the following assertions: (1) At least one of the two equations (4-67) and (4-68) has a bifurcation point in [0,1]. (2) At least one of the three equations (4-68)—(4-70) has a bifurcation point in [0,1]. (3) If \ = 0 is a bifurcation point as above, then there is a bifurcated branch on the side of A > 0. The following theorem is a global version. Let S = {(x, X)eXx(0,oo)\x

+ L{x, 1) + h(x, A) + G{x) = 0, x ^ 0} .

Theorem 4.7 Let the conditions of Theorem 4-6 hold true. Assume that the following equation has no bifurcation point in [0,1] for any R>1, x + L{x,X) + h(x,XR)+G(x)=0

(4.71)

and the set of bifurcation points of (4-68) is bounded in [0, oo); then there exists a bifurcation point Ao > 0 of (4-68) such that one of the following two assertions holds true.

95

Steady State Bifurcations

(1) The connected component Eo of £ containing (0, Ao) is unbounded in X x (0, oo), or (2) So contains a nonzero solution of the equation x + L(x, 1) + G{x) = 0.

(4.72)

Proof of Theorem 4-6. By assumption, there is a constant R > 0 small such that for 0 < r < R we have deg(id + Lx + G, £ r , 0) = deg(id + Lu Br, 0) -

(-if.

where L\ =£,(-, 1), and /? the sum of algebraic multiplicity of all eigenvalues of L\ in (0,1), which is an odd number. Hence, deg(id + Li + G,jBr,0) = - l ,

V 0 < r < R.

(4.73)

On the other hand, we know that deg(id + G,5 P ,0) = l.

(4.74)

Suppose that 0 ^ (id + L\ + h\ + G)(dBr) for r > 0 small, otherwise X = 1 is a bifurcation point of (4.67). We also assume that deg(id + I a + / i i + G , j B r , 0 ) = l,

V 0 < r < S,

(4.75)

for some 5 > 0. Let Ht = id + i i + h{-, t) + G. Then Ho = id + Lx + G, Hi=id + Lx + hl + G. By the homotopy invariance, it follows from (4.73) and (4.75) that for any 0 < r < 5, there are xr e dBr and 0 < Ar < 1 such that xr + L{xr, 1) + h(xr, Xr) + G(xr) = 0. Obviously, the numbers Ao = lim Ar,

Ai = lim Ar,

T—>0

r-+0

are the bifurcation points of (4.68) in [0,1], and if Ao = 0, the bifurcated branch is on the side of Ao < A. If (4.75) is false, then we have deg(id + L i + / i i + G , 5 r , 0 ) ^ l ,

V 0 < r < 5.

(4.76)

96

Bifurcation Theory and Applications

Let ( Ht = id+ L(;t) + h(;t)+G, I Ho = id + G, [ Hi = id + Li + /ii + G.

(4.77)

Thus, in the same fashion as above, it follows from (4.74) and (4.76) that (4.67) has a bifurcation point Ao in [0,1], and if Ao = 0 the bifurcated branch is on Ao < A. The assertions (1) and (3) are proved. We know that if (4.68) has no bifurcation point in [0,1], then (4.76) holds. By (4.77), if (4.70) has no bifurcation point in [0,1], it follows from (4.74) that deg(id + hi + G, Br, 0) = 1,

(4.78)

for all r > 0 small. Thus, one can derive from (4.76) and (4.78) that (4.69) must have a bifurcation point Ao in [0,1], and if Ao = 0 the bifurcated branch is on Ao < A. The proof is complete. • Proof of Theorem 4..7. By assumption, from Theorem 4.6 we see that the equation (4.68) has a bifurcation point in [0,1]. Let {Afc} C [0, 00) be the set of bifurcation points of (4.68). Based on the conditions, there is a number 0 < A, such that {A^} C [0, A). Denote by Ck the connected component of £ containing (x, A) = (0, Afc), and

C0 = {JCk, k

C = C 0 U({0}x[0, J R]), QR = BRX [0,R], which are as shown in Figure 4.3. Assume that Co is bounded in X x [0,00). Then, there is a R > 1 such that Co e QR, and Co n 8QR = (f> provided Assertion (2) of this theorem is false. Since (4.71) has no bifurcation point in [0,1], we take e > 0 sufficiently small such that id + Li + G and id + L\ + h(-, R) + G has no nonzero points in Be. Let D = dQR\{B£ x {0} UB £ x {R}).

97

Steady State Bifurcations X \ B

R

QR

^

^

^

^

y

_^

C

^

/

/ /

/ c kk

Fig. 4.3

If Assertion (2) is false, C and DUj (7 = S\C 0 ) are two disjoint closed sets. Therefore, there exists a relative open set V C X x [0,R] such that

CcV, F n 7 = 0, VnD = 0, which implies that (see Figure 4.3), VO,VRCBE,

dVn(Zu({O}x(O,R)))=0,

(4.79)

where V\ is the section of V at A £ [0, R]. By the homotopy invariance and the excision property, from (4.73) and (4.79) it follows that

det(id + Ll + h(-,R) + G,VR,0) = deg(id + Z 1 +G,V o ,0) = -1.

(4.80)

98

Bifurcation Theory and Applications

Because VR C Be and id + L\ + h{-, R) + G has no nonzero point in BE, we have

deg(id + L1 + h(;R) + G,VR,O)=deg(id + L1 + h(.,R)+G,B£,O).

(4.81)

On the other hand, by assumption, we have deg(id + L(-, A) + h(-, XR) + G, B£, 0) = const.,

V 0 < A < 1.

Hence deg(id + Li + h(-, R) + G, Be, 0) = deg(id + G, Be, 0) = 1,

(4.82)

a contradiction to (4.80) and (4.81). The proof is complete.

4.2.3

General

•

principle

Let X, Y be two Banach spaces. We consider the equation given by (4.83)

Ax + h{x,X)=0. Suppose that

h : X x R 1 —> Y is a compact operator with h(0, A) = h(x, 0) = 0, A = L + G : X —>7a completely continuous field, i.e. L : X - » 7 a linear homeomorphism, G : X —> Y a compact operator with G(x) = o(||a;||). T h e o r e m 4.8 Under the above assumptions, at least one of the following assertions holds: (1) The equation (4-83) has a bifurcation point in [0,1], and if Ao = 0 is a bifurcation point, then there is a bifurcated branch on the side of X > 0. (2) For any compact operator f : X —> V with /(0) ^ 0, there is a constant S > 0 such that for all \e\ < S the equation A(x) + h(x, 1) = ef{x) has a solution x{e) £ X with x(e) —> 0, as £ —> 0. Proof.

We first assume t h a t deg(>l + / i ( - , l ) , f l r , 0 ) = 0 ,

V r > 0 small.

(4.84)

99

Steady State Bifurcations

By assumption, we have deg(A, Br, 0) = deg(L, Br, 0) ± 0.

(4.85)

Assertion (1) follows from (4.84) and (4.85). If (4.84) is false, then

deg(A + h(-,l),Br,0)^0,

Vr > 0 small,

which implies the assertion (2). The proof is complete.

D

The above theorem is a general principle of the alternative method. The following theorem gives a global version. Assume that (Ai) There exists a real number Ai and a nondecreasing function r\ > 0 of A > Ai such that for any s > 0 there is / G X, \\f\\ = e, and A{x) + h(x, A) = / ,

A > Ai

has no solution in Brx. (A2) There is a real number Ao < A and e > 0 such that for any 0 < t < 1 the equation

Ax + th(x,X0) = 0 has no nonzero solution in Be. Let £ = {(x, A) e X x R | Ax + h(x, A) = 0, x ^ 0} . Theorem 4.9 Under the assumptions of Theorem 4-8 and (A\)—(A^), the equation (4-83) has a bifurcation point in [Ao, Ai]. Moreover, if the set of bifurcation points of (4-83) is bounded in [Ao,oo), then there exists a bifurcated point A* € [Ao,oo) of (4-83) such that the connected component C of S containing (x,X) = (0,A*) satisfies at least one of the following assertions (1) C is unbounded in X x [Ao,oo). (2) C contains a solution (XQ,\Q) of the equation A(x0) + h(x0, Ao) = 0,

x0 ± 0.

The proof of Theorem 4.9 is similar to that of Theorem 4.7.

100

4.3

Bifurcation Theory and Applications

Bifurcation from Homogeneous Terms

In Section 4.2, we consider the bifurcation from a nonlinear term. In this section, we shall continue to investigate the problem. Let X, Y be two Banach spaces. We consider the bifurcation equation given by Ax + XBx + G(x, A) = 0,

x € X,

(4.86)

where A, B, G\ : X —> Y are continuous operators such that I A(Xx) = XkA(x) s [ B(Xx) = XpB(x)

for A > 0 and some k > 0, for A > 0 and some p > 0 with p ^ k,

I G (XM-Plx, X) = o (\X\k^k-pA { ) x / N \G(\1'<*-»x,\-1)=o(\\\k'
(4.87) '

K

iik>p, ( 4 - 88 ) if p>k,

for a fixed x € X with x ^ 0. Let x0 € X with xo ^ 0 be a solution of the equation A(x0) + B{x0) = 0.

(4.89)

It is easy to verify that a;(A) = A 1 /' fc ~ p )x 0 (A > 0) is a solution of the following equation A(x) + XB{x) = 0,

0 < A < oo.

(4.90)

Moreover, as p > k, x(X) = X~1^p~k^xo is a bifurcation solution of (4.90) from (x, A) = (0,+oo); and as k > p, x(X) is a bifurcation solution from (x,X) = (0,0). Prom this simple example, we are motivated to prove the following more general result. Theorem 4.10 Let xo ^ 0 be a solution of (4-89). Under the conditions (4-87) and (4-88), if A, B : X -+ Y is differentiable at x0 e X, and the derivative operator DA(x0) + DB(x0)

:X^Y

is a linear homeomorphism, then the following assertions hold: (1) If p < k, the equation (4-86) has a bifurcated branch (a;(A),A) (A > 0) from (x,X) = (0,0), and the solutions in the branch near X — 0 can be

101

Steady State Bifurcations

expressed as

( x(X) = A-1/(p~fc)x0 + \-1/{p-kh(\)

VA > 0,

1 z(X) is continuous on X, and z(0) = 0. (2) Ifp < k, the equation (4-86) has a bifurcated branch (x(A),A) (A > 0) from (x, A) = (0, +oo), and the solutions in the branch near A = +oo can be expressed as ( x(X) = X-Vtr-Vxo ]

+ X~l^p-kh{X)

VA > 0,

lim z(X) = 0.

Proof. We first proceed with the case of 0 < p < k. Let the solutions of (4.86) near A = 0 be expressed as X(X)

= A ^ - ^ r r o + z(X)).

(4.91)

Putting (4.91) into (4.86), by (4.87) we find A{xo+z) + B{xo-+z) + X-k^k-^G(x1'^~'P\xo+z),Xsj

=0. (4.92)

By the differentiability of A and B at i 0 , near z = 0 the equation (4.92) can be written as (DA(x0) + DB(xo))z + g(z, A) + o (|| z ||) = 0,

(4.93)

where g(z,X) = A-fc/G (X'/^-P^XO

+ z), A) .

Based on (4.88) we find that g(z,0) = 0 for all z £ X, which implies that g : X x R —> Y is differentiable on z at (z, A) = (0,0), and Dxg(0,0) = 0.

(4.94)

By the implicit function theorem, it follows from (4.93) and (4.94) that there exists an unique continuous function z = z(X) with z(0) = 0 satisfying (4.93). Hence (4.91) is a bifurcation solution of (4.86). Now, we consider the case where k < p. Let (3 = X~l. Then the equation (4.86) is equivalent to (3A(x) + B{x) + f3G(x, (3~1) = 0.

(4.95)

102

Bifurcation Theory and Applications

Let the solution of (4.95) near f3 = 0 be of the form xU3)=pVto-Q(xo

+ z(J3)).

(4.96)

Inserting (4.96) in (4.86) we have A(xo + z) + B(xo+z)+p-kttr-VG(j31ttr-V(xo

+ z),f3-^ = 0. (4.97)

In the same fashion as above from (4.97) we can derive Assertion (2). The proof is complete. D Remark 4.3 If the mapping G : X —* Y is independent of A, then the condition (4.88) implies that G(x)=o(\\x\\k).

Remark 4.4 The condition that A > 0 in (4.87) is not essential. Actually, one can also discuss the case where A < 0 in some concrete situations. For the case where k — p in (4.87), the bifurcation of (4.86) is not simpler as k j= p except the linear case. We know that if A + XB is a linear completely continuous field, then the bifurcation point Ao must be an eigenvalue of Ax + XBx = 0. For the nonlinear case, the similar result holds. An operator K : X —> Y is called essential at x = 0 if for any bounded sequence {xn} C X with K(xn) —> 0 (n —» oo), {xn} has a convergent subsequence. Obviously, if K is a completely continuous field, then K is essential at x = 0. Theorem 4.11 Assume the condition (4-87) with p = k. Let the operators A + XB are essential for X € R1, and G(x, A) = o(||x||fe). If Xo £ R1 is a bifurcation point of (4-86) then the equation Ax + XQBX = 0

(4.98)

has a nontrivial solution in X. Proof. Let (x(A0 + t),X0 + t) be a bifurcation solution of (4.86) from (0, AQ). Then we have lim x(A0 +1) = 0.

Steady State Bifurcations

103

Denote by p(t) = \\x(\o+t)\\ z(t)=x{X0+t)/\\x{X0+t)\\. From (4.86) we obtain A(z) + X0B{z) + tB(z) + {3-kG{f3z, A) = 0 which implies HmA(z(t)) + XoB(z(t)) = 0. Since A + XB is essential at x = 0 and \\z{t)\\ = 1, it follows that z(t) —> zo ^ 0 in X, and Zo is a solution of (4.98). The proof is complete. • 4.4

Notes

4.1 This section is based on the authors' recent work [Ma and Wang, 2004a; Ma and Wang, 2004b]. The main theorems in this section are Theorems 4.1 and 4.2, which establish the existence of steady state bifurcation when the parameter crosses an eigenvalue of the linearized problem with even multiplicities. These theorems are proved by calculating the topological degree of the equation, and the bifurcation is obtained when the degree changes as the parameter crosses a critical value. In the classical Rabinowitz and Krasnoselskii theorems, when the eigenvalue of the linear problem has odd multiplicity at the critical parameter, the degree, which are either +1 or —1, changes sign as the parameter crosses the critical value, leading to bifurcation. This is essentially a linear theory as the degree is calculated, using the homotopy property of the degree, by calculating the degree of the linearized operator. When the eigenvalue has even multiplicity, there is no change of the degree for the parameter on the two side of the critical value. The key idea in this article is based on the observation that for a class of nonlinearities, called fc-th (k > 2) order nondegenerate singularities, the degree at the critical parameter is even, creating the discrepancy of the degree, and leading to bifurcation. To carry out this idea, the spectral theorem, Theorem 3.4, plays a crucial role. 4.2 This section is due to [Yu and Ma, 1989; Ma, 1990]. 4.3 Theorems 4.10 and 4.11 are introduced here for the first time.

Chapter 5

Dynamic Bifurcation Theory: Finite Dimensional Case This chapter addresses the dynamic bifurcation theory for finite dimensional systems, developed by the authors based on the notion of bifurcation called attractor bifurcation. Infinite dimensional version of the theory will be presented in the next chapter. The (finite dimensional) theory, together with center manifold reduction procedures, can be applied to both finite dimensional dynamical systems and infinite dimensional dynamical systems; see applications in Chapters 8-10. 5.1

Introduction

A classical example of dynamic bifurcation is the famous Hopf bifurcation. In fact, dynamic bifurcation occurs very often in many scientific and engineering fields. In this section, we give two examples, and briefly describe the basic principle of dynamic bifurcation. 5.1.1

Pendulum in a symmetric magnetic field

We consider a pendulum in a vertical plane forced by a symmetric magnetic field, friction, and gravity; see Figure 5.1. Let the length of this pendulum string be / = 1, and one end of string be tied with a small iron ball with massTO= 1. The small ball moves with friction on a vertical unit circle. On both sides of the small ball there are two symmetrical magnetic plates attracting it, which have the same magnetic magnitude. The distances between the magnetic plates and the downward vertical center of the pendulum are equal, and denoted by r » 1. Intuitively, if the magnitude A of magnetic field on each side plate is smaller than some critical value Ao, i.e. A < Ao, under the action of friction 105

106

Bifurcation Theory and Applications fc-.-«

r

»-

r

i e \'= ]

F

l

^^^-J__-^^^

Pig. 5.1

F2

Pendulum with in a symmetric magnetic field.

and gravity, the pendulum will gradually stop at position 8 = 0, where 9 is the angle between the string of the pendulum and the downward vertical axis. However, as the magnitude A exceeds Ao, two new symmetric equilibrium positions ±6\ ^ 0 (A > Ao) will appear, and the position where the pendulum settles down is determined by its initial state. This is a typical dynamical bifurcation problem. The motion of pendulum satisfies the following initial value problem: d2e -TIT

„

,d9

,

„

= -k-jT - g sin 6 + f cos 6, at at* 0(0) = ai,

e'(o) = a2, where k > 0 is the damping coefficient, g the gravity constant, and / the magnetic force. By the Coulomb law and r » 1,

f = F2-F1 A

A 2

~ ( r - sin 6») 4r A sin 9 =

(r 2 - sin 2 6I)2

~ A sin 6,

(r + sin0) 2

107

Dynamic Bifurcation Theory: Finite Dimensional Case

where A is proportional to the magnitude of magnetic field, and A = 4Ar~3. Thus, the motion equation can be approximatively written as jfl

J2Q

-—r = —k— q sin 9 + A sin 9 cos 6. y dt2 dt Let xi = 9, X2 = 9'. Then, the motion equation is in the form

{

dxi

dx2 —— = — kx2 — g sin xi + A sin xi cos X\, dt with the initial condition xi(0) = au

(5>1)

(5.2)

x2(0) = a2.

By the Taylor expansion Asinxi cosa;! — ^sinxi = (A — g)xi — ~(4A — g)x\ + o(|a;i| 3 ), the system (5.1) near x = 0 is expressed as

(

dx\

(5-3)

f

^ = (A - g)Xl - kx2 - i(4A - g)xl + When A < g, the matrix

o(\Xl\3).

(5.4) has two eigenvalues . A± =

-k ± y/k2 - A(g - A) 2 '

which have negative real parts. In this case the equilibrium point x = 0 of (5.3) is locally asymptotically stable. When A = g, the equations (5.3) are as follows dX! 1

!

±l = -kx2- -gxl

(5 5) +

o(\x1\*).

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Bifurcation Theory and Applications

The eigenvalues of (5.4) are Aj = 0, A2 = — k, i.e. The system (5.5) is at the critical state. We shall apply Theorem 3.13 to show that x = 0 is a locally asymptotically stable equilibrium point of (5.5). Let / = (/i, / 2 ), g = (gi,g2) be given by /i=z2,

h = —^gx\ + o{\x1\3),

9i = 0 ,

g2 = -kx2.

Obviously x = 0 is an isolated singular point of (5.5), and div/ = 0, divg = -k < 0, fi92 ~ /251 = -kxl < 0,

V{x1,x2)=l-xl+l-gx\ + o{\x1\i). It it easy to see that the conditions of Theorem 3.13 are satisfied. Hence x = 0 is locally asymptotically stable for (5.5). When X > g, the system (5.1) bifurcates two new equilibrium points (xf,x%) = (±ex,0) from (0,g), where ex=cos-1{g/\). It is easy to verify that (±#,\,0) are two locally asymptotically stable equilibrium points of (5.1). The above discussion can be summarized as the following theorem. Theorem 5.1 There exists a neighborhood U C R2 of x = 0, such that the following are true. (1) If A < g, x = 0 is locally asymptotically stable for (5.1) in U as shown in Figure 5.2(a), and (2) If A > g, the system (5.1) bifurcates from (0,g) two equilibrium points (±#A,0) which are asymptotically stable in U. Moreover, the open set U is decomposed into two open sets U\ and U%, as shown in Figure 5.2(b),

U^ul + Uf, U{nul = $, OedUlndul with (0A,O) e U\, (—6»A,0) £ Ul, such that

lim (o:i(t,a),a;2(t,a)) = (0A,0) t—•OO

lim (x1(t,a),x2(t,a)) = (-0X,0)

t—*oo

ifa= (QI,Q 2 ) € U\, if a = {a^a?) e Ul,

Dynamic Bifurcation Theory: Finite Dimensional Case

where (xi(t,a),X2(t,a))

109

is the solution of (5.1) and (5.2).

*2

(a)

| *2

U

2

)

(b) Fig. 5.2 (a) If A < p, a; = 0 is an attractor; (b) If A > g, the system bifurcates to two attractors.

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Bifurcation Theory and Applications

5.1.2

Business cycles for Kaldor's model

In this subsection, we consider business cycles in the following Kaldor model:

{

dx —

7

=I(x,y,X)-S(x,y,X),

(5-6)

I =/(*,*, A),

where x is the total social income, y the total social capital, A the industrial technique parameter, I(x, y, A) the investing function, and S(x, y, A) the saving function. For each technique parameter A, the social business has an equilibrium state (xo,yo), xo > 0, y0 > 0, which depends continuously on A. According to economic laws, in a neighborhood of (xo,yo), I and S satisfy that dl dl OS dS 7j- > 0, j j - < 0, 7j- > 0, 7,- > 0. ox ay ox oy By these conditions, in a neighborhood of (xo,yo), I and 5 take the form as follows (I = ax(x-xo)-ax(y-yo)3, (5.7) \S = b\(x - x0) + c\(y - 2/0) + Py{x - x0)3 + jx(y - y0)3, where a\,b\,c\ > 0, depending continuously on A. The parameters here a, /3,7 > 0 are constants, a\ stands for investing intention, and 6A stands for saving intention. Based on (5.6) and (5.7), the Kaldor model is given by

{

- £ = k\(x - x0) - cx(y - 2/0) - Sx(y - y0)3 - Py(x - x0)3,

f

-j- = ax(x - xo) - ax(y - yof, at where k\ = a\ — b\, 5 = a + j . The eigenvalues of the matrix \ax

0J

are given by A± =

k\± y/k\ - 4axc\

•

(5.8)

Dynamic Bifurcation Theory: Finite Dimensional Case

111

Near k\ = a\ — b\ = 0, the eigenvalues of A\ are complex number and Re\± = 7}k\. Hence if k\ < 0, the equilibrium point (xo,2/o) of (5.8) is locally asymptotically stable. If k\ — 0, the eigenvalues A± = ±i\/4a.\c\. We shall prove that (io,2/o) is locally asymptotically stable. To see this, we note that the divergence of the nonlinear terms in (5.8) is non-positive in a neighborhood of (xo,yo). Namely, for any (x — XQ)2 + (y — yo)2 < £ with (x,y) ^ (xo,yo) and with e > 0 sufficiently small, — [-6x(y - y0)3 - /3y(x - x0)3} + Q^[-ctx(y - yo)3} = -[3/3y(x - x0)2 + 3ax{y - y0)2] - S(y - y0)3 <0. Then the local asymptotic stability of (xo,yo) follows from Theorem 3.15. Let Ao € 1 be the critical technique parameter satisfying

{

<0 =0

ifA
>0

if A > Ao.

(5.9)

Under the condition (5.9), by the Hopf bifurcation theorem we know that if A > Ao, the equation (5.8) bifurcates from (xo,yo) to an attractor, which is homeomorphic to a circle S1; see Figure 5.3. From the economics point of view, the S1 attractor bifurcation represents periodic fluctuations of the economic system. When the industrial technique level parameter A < Ao, the investing interest and the social income are lower. Therefore the investment intention is weaker than that of social savings, namely ax < &A- Thus, k\ = a\—b\ < 0, and in this case the social economy fluctuates around the equilibrium point (xo,yo)', as shown in Figure 5.3(a). When the technique level is promoted to exceed the critical value Ao, i.e. A > Ao, the new industrial technology brings the investing interest higher, and in this case the investment intention is stronger than that of social savings. Therefore k\ = a\ — bx > 0; thus the business is transferred from a stable equilibrium state to the form of periodic fluctuation, as shown in Figure 5.3(b), which is that we see today.

112

Bifurcation Theory and Applications

(a)

(b)

Fig. 5.3 (a) When A < Ao, (xo,yo) is stable; (b) when A > Ao, the system bifurcates to a cycle attractor.

5.1.3

Basic principle of attractor bifurcation

The two examples above show that a parameter family of vector fields have an attractor bifurcation if some eigenvalues cross from left to right the imaginary axis leaving the other eigenvalues on the left side, provided that the equilibrium point at the critical parameter is locally asymptotically stable. In later discussions we can see that these are basic conditions for attractor bifurcations. To illustrate the basic principle behind attractor bifurcations, we consider the following equations ^=\X ^

+ Gl(x,y),

= -y + G2(x,y),

(5.10) (5.11)

where x £ R m , y € R"~ m (1 < m < n), and Gi(x,y)

=o(\x\,\y\),

i = 1,2.

It is shown that near A = 0, the flow of (5.10) and (5.11) in a small neighborhood of (x,y) = 0 is squeezed into an m-dimensional surface E\, which is called the center manifold. The m-dimensional surface Y,\ can be expressed by a function y = h(x,X). The flow in E.\ has the same topological structure as the flow in R m of the following equations, which are the projection from Y,\ into R m : (5.12) -£=\x + Gy(x,h(x,\)). at It suffices then to consider the dynamic bifurcation for (5.12). When

Dynamic Bifurcation Theory: Finite Dimensional Case

113

A < 0, near x = 0 the linear term Aa; determines the dynamical behavior of (5.12), and therefore the flow of (5.12) converges to x = 0. When A = 0, assume that x = 0 is locally asymptotically stable for (5.12) at A = 0, i.e. (IT

(5.13)

— =G1(x,h(x,0)). Then when A > 0 is sufficiently small, (5.12) can be rewritten as (IT

— =G 1 (x,/ J (x,0))+A 2 r + X(x,A),

(5.14)

where K(x,X) = Gi(x,h(x,X)) - Gi{x,h{x,Q)) -> 0 as A -> 0. Hence, the flow of (5.14) is a superposition the flow of (5.13) and flow of the following system dx — = Xx + K{x,X)

(A>0).

(5.15)

The flow of (5.15) is outward from x = 0, and this flow of (5.13) is inward to x = 0. Near x = 0, the linear term dictates the flow, and for x away from x — 0, the nonlinear term G\(x, h(x, 0)) dominates the flow structure. Thus the inward flow and outward flow squeeze an attractor near x = 0, as shown in Figure 5.4.

/

Rm

Fig. 5.4 Schematic showing the existence of bifurcation attractor.

114

5.2 5.2.1

Bifurcation Theory and Applications

Attractor Bifurcation Main theorems

We consider the dynamic bifurcation for the finite dimensional systems given by dx — ~ Axx + G(x,X),

AeR, x£Rn

(n>l),

(5.16)

where G : I " x I 1 -> R" is C r (r > 1) on a; 6 Kn, and continuous on AeR 1 , G(x,\)=o(\x\),

VAeR1,

(5.17)

the operator

(

a u (A) ••• ai n (A)\ ani(A) ••• ann(X)J

is an n x n matrix, and ay (A) are continuous functions of A. Let all eigenvalues (counting multiplicities) of A\ be given by

^(A),...,A,(A). We know that the eigenvalues /?i(A) (1 < i < n) are continuous with respect to A. Definition 5.1 (1) We say that the system (5.16) bifurcates from (x,X) = (0,A0) an invariant set Q,\, if there exists a sequence of invariant sets {fi.\n} such that 0 ^ £l\n, and lim An = Ao,

n—*oo

lim d(Q\ ,0) = lim max Ircl = 0.

(2) If the invariant sets Q\ are attractors of (5.16), then the bifurcation is called attractor bifurcation. (3) If the invariant sets Cl\ are attractors of (5.16), which are homotopic equivalent to an m-dimensional sphere, then we say that the system (5.16) has an 5 m -attractor bifurcation at (0, Ao).

Dynamic Bifurcation Theory: Finite Dimensional Case

115

Assume that

{

<0

ifA
= 0

if A = Ao,

>0

ifA>A 0 ,

Re0j(\o) < 0

forl
for m +1 < j < n.

(5.18)

(5.19)

The eigenspace of A\ at Ao is then given by m oo

£0 = U L ) { ^ r I (A, -&(Ao))fc* = °}i=lk=l

It is known that dim EQ = m. The main results in this section are the following 5 m -attractor bifurcation theorems for the finite dimensional system (5.16). Assume that (5.17)-(5.19) hold true, and x = 0 is locally Theorem 5.2 asymptotically stable for (5.16) at X = Ao. Then we have the following assertions. (1) The system (5.16) bifurcates from (0, Ao) to an attractor Q\ for A > Ao and near Ao, with m — 1 < dimf^A < rn, which is connected when m>2. (2) The bifurcated attractor Q,\ is a limit of a sequence of m-dimensional annulus Mk with M^ +1 C Mk, i.e. Cl\ = n ^ M f e . Especially, if Sl\ is a finite simplicial complex, then Cl\ is a deformation retract of mannulus; hence Q,\ has the homotopy type of Sm~1. (3) For any x\ E £l\, x\ can be expressed as x\ =z\+o(\z\\),

Z\GE0;

(4) If the number of singular points of (5.16) in Q.\ is finite, then we have the following index formula:

En x6

A

- J/ / A ^s s f2 tf™= odd, md(-(Ax + G),x) = \ 1 ° if m= even.

Theorem 5.3 Assume that (5.17)-(5.19) hold true and x = 0 is globally asymptotically stable for (5.16) at A = Ao. Then for any bounded open set U C Rn with 0 € U, there is an e > 0 such that if Ao < A < Ao + e, the attractor Q\ of (5.16) bifurcated from (0, Ao) attracts U\T, where F d " is the stable manifold ofx = 0 with dimF = n — m. Especially, if (5.16) has

116

Bifurcation Theory and Applications

global attractors near A = Ao, then e is independent of the bounded open sets U C l " .

Remark 5.1 Theorem 5.2 can be regarded as a united version of both the steady state and dynamic bifurcations. When m = odd, the Sm~lattractor Q.\ must contain singular points of the vector field A\ + G, which is an assertion of the Krasnoselski theorem. When m = 2 and /3i(A0) = /?2(^o) = h this theorem shows the existence of periodic orbits, as in the classical Hopf bifurcation. In addition, one advantage of the theorem here is the stability of the bifurcated periodic solution, which is in general an important, but difficult problem, and which is not available in the classical Hopf bifurcation theorem. Remark 5.2 In general, the attractors QA in Theorem 5.3 are not global. For example, the system

satisfies the conditions of Theorem 5.3 with Ao = 0, and for any A > 0 sufficiently small, Q\ contains two singular points X12 ~ iA 1 / 2 of the function \x — x3(l — e~1^x ). However it is easy to see that for each A > 0 small, Q.\ is certainly not a global attractor since there are two more singular points xifi ~ ±A~1//2, bifurcated from infinity, which are not in Q,\. 5.2.2

Stability of attractors

In order to prove the attractor bifurcation theorems, we first introduce a stability theorem of attractors, which is crucial not only in the proof of Theorems 5.2- 5.3, but also in the proof of perturbation stability theorems for infinite dimensional systems in Ch.6. This theorem is also a supplement of Theorem 2.7 where the existence of attractors for the perturbation dynamical systems is given as an assumption. To proceed, we start with a technical lemma on stability of extended orbits for vector fields. Let v G C r ([/,R") be a vector field where U C Rn is an open set. A curve 7 C U is called an extended orbit of v, if 7 is a union of curves

7= U 7 i 8=1

such that either 7* is an orbit of v, or 7, consists of singular points of v, and if 7, and 7 i + i are orbits of v, then the w-limit set of 7* is the a-limit

Dynamic Bifurcation Theory: Finite Dimensional Case

117

set of 7i+i, w(x) = a(y),

Vz eji,

y€ 7»+i'

Namely, endpoints of 7, are singular points of v, and the starting endpoint of 7J-|_I is the finishing endpoint of 7$; see Figure 5.5.

PI Fig. 5.5 An extended orbit.

Then we have the following stability lemma of extended orbits. The result stated this lemma has been proved in Step 2 of the proof of Lemma 4.5 in [Ma and Wang, 2002]. Here we only state the result as a lemma. Lemma 5.1 Let Vk £ Cr(U, R") be a sequence of vector fields with limfc-KjoVA; = v0 £ Cr(U,M.n). Suppose that jk C U is an extended orbit of Vk with uniformly bounded length, and the starting points p\ of 7^ converge to pi, then the extended orbits 7fc of Vk converge, up to taking a subsequence, to an extended orbit 7 of VQ with starting point p\. Remark 5.3 The stability lemma of extended orbits is useful for orbit analysis of vector fields, which is a basic tool to solve some problems in fluid dynamics, such as the boundary layer separation and the interior separation of fluid flows; see [Ma and Wang, 2004c; Ma and Wang, 2003b; Ma and Wang, 2004f; Ghil et al, 2001; Ghil et al., 2003; Ma and Wang, 2005c]. The following is the stability theorem of attractors. Theorem 5.4 Let vn £ Cr(U, R n ) (r > 1) be a sequence of vector fields such that Yvoin^ooVn = VQ 6 Cr(U, R"). Let £0 C U be an attractor of VQ and D C U be the basin of attraction for So • Then the following assertions hold true. (1) For each n sufficiently large, vn has an attractor S n C D, and lim d(E n ,Eo) = lim sup dist(x, So) = 0 .

118

Bifurcation Theory and Applications

(2) For any bounded open set O C D there is an N sufficiently large such that O is in the basin of attraction of S n for all n > N. Proof. Let Do CC D be a bounded open set with S o C A)- Then the w-limit set for VQ of Do is the attractor So, i.e. So = w(Do).

(5.20)

To complete the proof, by Lemma 2.2 and Theorem 2.7, it suffices to prove that

{

lim d(ujn(Do),Ho) = lim sup dist(x, So) = 0, ^-«-<«»>

w n (A>)= n u t > r S n (0A>, r>0

(521)

~~

where Sn(t) is the operator semigroup generated by vn. By Lemma 2.1, wn(D0) is an invariant set of vn. If (5.21) is false, then at least one of the following two cases must occur: (a) There exist a number 5 > 0 and points pn g uin(Do) such that Vn —* Po £ So (n —» oo) and the extended orbits F n c wn(Do) of un starting at p n with bounded length satisfy that d(r n ,S 0 ) = sup dist(x, So) > 6.

(5.22)

(b) There exist extended orbits Tn C uin{Do) of vn and a number 5 > 0 such that inf inf x - y > 5

Vn 6 N.

(5.23)

For the case (a), by Lemma 5.1, the extended orbits Tn converge to the extended orbit To of vo with the starting point p0 £ So r n —• To

(n -> oo).

(5.24)

It follows from (5.22) and (5.24) that d(T0, So) = sup dist(x, So) > 5 > 0. (5.25) zero On the other hand, So is an invariant set of VQ. Hence To C So, a contradiction with (5.25). If only the case (b) occurs, then by the definition of w-limit sets, we infer from (5.20) and (5.23) that there exist a number $i > 0 and points

Dynamic Bifurcation Theory: Finite Dimensional Case

119

Pn S -Do with pn —» po £ Do (n —> oo) such that the orbits xn (£) of u n with the initial value xn(0) = pn satisfy that dist(in(t),S0)>5il

Vt>0.

(5.26)

By Lemma 5.1, from (5.26) we derive that the (extended) orbit xo(t) of i/o starting at po € Do satisfies dist(zo(*)> So) > Si,

V 0 < « < oo.

Here the inequality holds true for all £ S [0, oo) is due to the fact that for any time T > 0, the arc-lengths of xn(t)

an(T) = f \vn(t)\dt Jo

is uniformly bounded independent of n. It is a contradiction with (5.20). Hence the equality (5.21) holds true. The proof is complete.

5.2.3

•

Proof of Theorems 5.2 and 5.3

It is easy to see that Theorem 5.3 is a direct corollary of Theorems 5.2 and 5.4. Hence we only need to prove Theorem 5.2. We proceed in several steps as follows. S T E P 1. By the canonical reduction procedure, the dynamic bifurcation problem of (5.16) is reduced to the dynamical bifurcation problem of the following equations (5-27)

^=Jxz+g(z,\),

where z e M™, J\ is the Jordan matrix corresponding to the eigenvalues /3i(A), • • • , /3m(A), which satisfy (5.18), and gi(x,y,X)=o(\z\)

VAeM.

For simplicity, we assume that Ao = 0, i.e.

{

< 0

if A < 0,

= 0

if A = 0,

> 0

if A > 0,

1 < i < m.

(5.28)

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Bifurcation Theory and Applications

STEP 2. PROOF OF ASSERTION (1). Consider (5.27), and let v\ = J\ +9(',ty- By assumption, x = 0 is asymptotically stable for (5.16) at Ao = 0. Consequently z = 0 is an attractor of v0. By Theorem 5.4, there exist constants r and Aj > 0 such that for all 0 < A < Ai, the set Br = {z e R m | \z\ < r} is an absorbing set of v\, and the w-limit set

Ax = u\{Br) c B r

(0 < A < Ax)

(5.29)

is an attractor of v\ in some open set D C M.m (Br c D). In addition, by the stable manifold theorem (Theorem 2.10) and (5.28) the unstable manifold W£ of v\ contains an open neighborhood of x = 0 in R m for all A (0 < A < Ai). Prom (5.29) we see that W%cA\CBr,

V0
By the definition of unstable manifolds, we obtain that A\\W%

C D CW1 is an attractor of (5.27) in D\{0},

which implies that Cl\ is an attractor of (5.16),

<

m — 1 < dimf^A < m,

(5-3°)

o ?nx, lim d(fl\,0) = lim sup |a;| = 0, A-.O+ \->o+ x € n A

where £l\ is denned by Qx = {(x,y) eWl\x£Ax\WZ,y

= h(x, A)} ,

and h(x,y) is the center manifold function. Here m — 1 < dimfi,\ follows from Corollary 2 on p.46 in [Hurewicz and Wallman, 1941]. Therefore, Assertion (1) is proved. STEP 3. PROOF OF ASSERTION (2). Let SA = - 4 A \ W " be the attractor

of (5.27) in D\{0}. Obviously, there is an rx > 0 sufficiently small such that the disk Brx C W%, and the annulus Rx = Br\Br;< is absorbing in £>\{0}. Hence, by Lemma 2.2, the attractor SA of (5.27) in £>\{0} is given by

(5.31) T>0 t>T

121

Dynamic Bifurcation Theory: Finite Dimensional Case

Since £ A C R\ is invariant under the action of the semigroup {S\(t)}t>o, for any t0 > 0 £A C Sx(t0)Rx.

(5.32)

It follows from (5.31) and (5.32) that there is a sequence {£„} with tn+i > tn > 0 and tn —> oo (n —> oo) such that ( Sx(tn+1)RX

c

Sx(tn)Rx,

I EA = lim 5A(tn)i2A = n n—*oo

\

n=l

Sx(tn)Rx.

We know that S\(t)R\ is homeomorphic to an m-annulus for any t e i 1 . Hence OA is a limit of a sequence of m-annulus. Let £ A be a finite simplicial complex. We shall prove that EA is a deformation retract of an m-dimensional annulus. Let M C Km be an m-dimensional smooth manifold with boundary. For each point x € dM we define z(x, s) — the point z G M, which lies on the inward normal line starting at x, and the arc length from z to x is s {s > 0). Obviously, z(x, 0) = x. We take a sequence of smooth m-dimensional annulus {Mn} in R\ = Br \ Brx such that

{

SA C M n + 1 c Mn c i?A, Vn > 1, Mn are deformation retracts of R\, and

(5.33)

lim Mn = EA. Moreover, the sequence {Mn} enjoys the following properties n—>oo

i) for any point x € dMn, there exists a number An(a;) > 0 such that for all x,y G 9M n , x ^y, the line segment ^x = {z(x,A) |0 0, z(x, Xn(x)) G dMn+1, if An(a:) = 0( which implies x G 9M n D dMn+1).

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Bifurcation Theory and Applications

These properties can be achieved by letting the smooth manifold Mi = R\ shrink along its inward normal direction, and by taking properly the intercepted manifold Mi (i = 2,3, • • •), inductively. Thus, for any x G dM\ we can define a curve

Lx =

U ^»'

xi=x

n=l

Xn+i = z Xn A x

( ' «( "))-

>

Namely Lx is the union of the line segments tXn, where the endpoint xn+\ of £x is the starting point of £Xn+1Since SA is a finite simplicial complex, the length Lx is finite for any x G dM\\ otherwise the number of simplexes in £A can not be finite. It is easy to see that L x n L y = 0,

Vz ^ y , x,y£

dM1:

and by (5.33) the endpoint qx of Lx = Dn°=1£Xn satisfies lim yn = qx G

SA

(yn £ 4 n ) ,

n—»oo

since Lx has finite length. On the other hand, we see that Ma = £AU( ( J L x ) . \xedMx

)

Then we define a mapping H : Mi x [0,1] —> Mi by H{y,t) = < [p(y,t), y G Lx, where p(y, i) is the point p G Lx such that the arc length along L x from y to p is t x r(y) where r(y) is the length of the arc in Lx from y to qx eT,\, the endpoint of L x . It is clear that H is continuous, and H(-,0) =

id:Mi->Mlt

^(-.IJIMX-^EA,

tfoi = i,i:EA->£A, where i : EA —> Mi is an inclusion mapping. Hence SA is a deformation retract of Mi = -RA- Assertion (2) is proved. STEP 4. PROOF OF ASSERTION (3). By (5.30) we see that the attractor Qx of (5.16) is in the center manifold MA for 0 < A < Ai. Since the

Dynamic Bifurcation Theory: Finite Dimensional Case

123

eigenspace Eo is tangent to M\ (0 < A < Ai) at x = 0, Assertion (3) follows. STEP 5. P R O O F OF ASSERTION (4). By the Brouwer degree theory and the conditions (5.18) and (5.19), the degree of the vector fields in (5.16) satisfies that

VAGC-A^AI),

(5.34)

for some r, Xi > 0. Since £l\ is the maximum attractor of (5.16) in Br\{0}, singular points of A\ + G\ in Br are in Cl\. Then we have

all nonzero

deg(-(Ax + G),Br,0) = l

deg(-(A A + G), Br, 0) = ind(-(Ax

+ Y,

+ G), 0) +

ind(-(Ax + G),Xi).

(5.35)

On the other hand, by (5.18) and (5.19)

{

1

if m = even,

—1 if m = odd, for 0 < A < Ai. Hence Assertion (4) follows from (5.34)-(5.36). The proof of Theorem 5.2 is complete.

5.2.4

Structure of bifurcated

(5-36>

attractors

Pitchfork bifurcation A related interesting question is to address under what conditions the attractors Q\ in Theorem 5.2 are homeomorphic to an (m — l)-dimensional sphere. Here we consider the case where m = 1, which corresponds to the classical pitchfork bifurcation, and is a general version of the example of pendulum in a symmetric magnetic field given by Theorem 5.1. Theorem 5.5 Assume that conditions (5.17)-(5.19) with m = 1 hold true, G(x, A) is analytic at x = 0, and x = 0 is locally asymptotically stable for (5.16) at A = Ao. Then there exists an open set U c K™ with 0 e U, such that if A > Ao the system (5.16) bifurcates from (0,Ao) exactly two equilibrium points x\,X2 € U, and the open set U is decomposed into two open sets U\ and U%, satisfying the following properties

(i)V = u\ + ul,uinui = %,

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Bifurcation Theory and Applications

(2) o e dU\ n dUl, Xi eU{(i = l, 2), (3) for any ye U{ (i = l,2), lim x(t,
t—KX>

where x(t,ip) is the solution of (5.16) with x(O, Ao:

A\x + G(x, A) = 0, x e Rn.

(5.37)

By the Lyapunov-Schmidt reduction, the bifurcation equation of (5.37) near Ao is as follows 0i(\)z + g(z + $(z),\) = O, z e l 1

(5.38)

where g(z, A) = P\G{x, A), Pi : R n —» E\ the cannonical projection, E\ the eigenspace corresponding to f3\, and $(z) is the implicit function defined by (3.73) of Ch.3. Because G is analytic, $(2) and g(z + $(z),\) are analytic at z = 0. Hence there exist a\ ^ 0 and k > 2 such that g(z + $(z), A) has the Taylor expansion as follows g(z +

$(z),\)=axzk+o(\z\k).

By assumption, x = 0 is asymptotically stable for (5.16) when A < Ao. Therefore if A < Ao, the system (5.37) has no nonzero solutions near x = 0, which implies that k is an odd number, and a\ < 0 for A near Ao. By (5.17), the equation (5.38) is given by

0i(\)z + axzk + o(\z\k)=O, which obviously bifurcates from (0, Ao) to exactly two singular points *i.2(A) = ±|/3! (A)/^! 1 /^"!) + 0 (|/3 1 (A)| 1 /(*-D). By Theorem 5.2, the system (5.16) bifurcates from (0, Ao) to an attractor A\ with dim A\ < 1. It is clear that A\ consists of exactly two singular points Xi{\) - (zi(\),$(zi)) £ R" (i = 1,2). By the stable manifold theorem (Theorem 2.10), there is an (n — l)-dimensional stable manifold Ws of (5.16) at x = 0 dividing the open set U into two parts U\ and U% such that Xi G Ulx and X{ attracts U{ (i = 1,2). This proof is complete. •

Dynamic Bifurcation Theory: Finite Dimensional Case

125

Remark 5.4 It is still an open problem whether the analytic condition of vector fields is sufficient for the bifurcated attractor £l\ of (5.16) being homeomorphic to an (m — l)-dimensional sphere when m > 2. Minimal attractors It is easy to see that a subset T\ C Cl\ may be an attractor as well. The minimal attractor T\ contained in £l\ is called the bifurcated minimal attractor of (5.16). In general, the bifurcated attractor Q\ may have no minimal attractors other than itself. We are interested here in the existence problem of minimal attractors. Here we always assume that the conditions of Theorem 5.2 hold. If the bifurcated attractor Q\ of (5.16) is homeomorphic to a sphere Sm, we denote it by £l\ = Sm for simplicity. Theorem 5.6 Under the conditions of Theorem 5.2, we assume that m = 2 and D,\ = S1 contains singular points of (5.16) which are all nondegenerate. Then the following assertions hold true. (1) The number of singular points on£l\ = S1 is 2k for some integer k > 1, and there are exactly k singular points {x^ | 1 < i < k} C £l\ such that each Xi is a bifurcated minimal attractor of (5.16). (2) There is an open set D C R™ which can be decomposed into k open sets Di (1 < i < k) such that fl\ U {0} c D, D = X)*=1 A , A n Dj = 1), with k attractors and k repellors. Let {xi | 1 < i < k} be the attractors, and {ZJ | 1 < i < k} be the repellors. Obviously, {xi \ 1 < i < k} are the minimal attractors in ftA of (5.16), and {z* | 1 < i < k} are the saddle points of (5.16), each of which has an one-dimensional unstable manifold and an (n — 1)dimensional stable manifold. It is easy to see that in the two-dimensional center manifold, there is an open set U which can be decomposed into k open sets Ui (1 < i < k) by the stable manifolds of Zj (see Figure 5.6) such that

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Bifurcation Theory and Applications

Fig. 5.6 The steady states x\ and X2 are minimal attractors, and the circle represents the bifurcated attractor Q\.

0 U = E l i U i , UinUj = (x) is either a singular point or a periodic orbit. If £ contains no periodic orbits, then as a two-dimensional invariant closed set, all points in E are interior points of E. Hence E = S2. By

Dynamic Bifurcation Theory: Finite Dimensional Case

127

assumption we can derive that £l\ = £ = S2; otherwise by the PoincareHopf index theorem, Q\ must contain more than two singular points. Since Q\ = S2 contains only two nondegenerated singular points, by Theorem 5.2, their indices are +1. Therefore, the two singular points are either attractors or repellors in $l\ = S 2 (since Q\ contains no periodic orbits). Due to the Poincare-Bendixon theorem, we can deduce that one of the two singular points is an attractor and another is a repellor, furthermore, the attractor attracts fl\/{xi}, where x\ is the repellor in Cl\. Hence the attractor attracts a neighborhood of x = 0 except the stable manifolds of x\ and x = 0. Thus Assertions (1) and (2) are proved. When the vector field G{x, A) in (5.16) is odd, the two singular points in Cl\ have the same eigenvalues. Hence they have the same local topological structure, which implies, by the above conclusion, that fix must contain a periodic orbit. The proof is complete. • 5.2.5

Generalized Hopf bifurcation

Now, let us consider more general bifurcation. Let the eigenvalues of A\ satisfy

{

< 0 (or > 0)

if A < Ao,

(5.39)

=0 ifA = A0, Vl 0 (or < 0) if A > Ao, Re(3j{\0) ^ 0 V m + 1 < j < n. (5.40) It is known that if m = odd, the system (5.16) must bifurcate from (0, Ao) a singular point. When m = 2, the Hopf bifurcation amounts to saying that if /?i(A) = /32{X) with 7m/3i(A0) ^ 0, then under the conditions (5.39) and (5.40) the system (5.16) bifurcates from (0,Ao) a periodic orbit. Our next question is to see whether (5.16) bifurcates from (0,Ao) an invariant set assuming only (5.39) and (5.40) with m = even are valid, i.e. without the asymptotic stability assumption. In general, as we shall see later, this statement is not true. However, we can still derive a generalized version of the Hopf bifurcation as follows; see also the discussion in Section 6.3. Under the conditions (5.39) and (5.40) we know that there exists an m-dimensional center manifold of (5.16) at A = Ao Mcm = {(x,y) G W1 | x £ Q c M.m, y = h(x,X0)} ,

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Bifurcation Theory and Applications

which is invariant under the flow of (5.16) at A = Ao. We say that the center manifold M™ is stable (resp. is unstable) if the w-limit set (resp. the a-limit set) of M™ is w(Mcm) = 0

(resp. a(M™) = 0).

Theorem 5.8 Let the conditions (5.39) and (5.40) hold. If the center manifold M™ of (5.16) at \ = \0 is stable or unstable, then (5.16) must bifurcate from (0, Ao) an invariant set T\ with A ^ Ao, and T\ has the homotopy type of Sm~1 provided T\ being a finite simplicial complex. Especially, if m = 2 and there are no singular points in F\, then F\ must contain a periodic orbit. The proof of Theorem 5.8 is similar to that of Theorem 5.2; we omit the details. Remark 5.5 If m = 2 in (5.39) and (5.40) and ft (A) = /?2(A) with /mft(A 0 ) 7^ 0, then the center manifold Mc of (5.16) must be one of the three cases: i) stable, ii) unstable, iii) containing infinite periodic orbits which implies the bifurcation of periodic orbits. Hence, the Hopf bifurcation is included in Theorem 5.8. In the following, we give an example which shows that under only conditions (5.39) and (5.40), bifurcations to invariant sets may not occur. We consider the parameterized vector field given by

{

Xxi — x2 — A x2, Xx2 + x\ - 1xxx\ + \2xl.

(5.41)

It is clear that the vector field (5.41) has no bifurcated singular points from (a;, A) = (0,0). Actually, we see from (5.41) that x2vx -

Xlv2

= -{x\ - x\f - X2{x\ + x\) < 0,

for all {x\,X2) ^ 0 and A ^ 0. In addition, by using the polar coordinate system, it is easy to show that (5.41) has no bifurcated periodic orbits from (0,0). The topological structure of vector field (5.41) can be schematically illustrated by Figure 5.7(a)-(c).

Dynamic Bifurcation Theory: Finite Dimensional Case

129

Fig. 5.7

5.3

Invariant Closed Manifolds

Motivated by the 5 m -attractor bifurcation, we study in this section invariant closed manifolds and their stability of a vector field. 5.3.1

Hyperbolic invariant

manifolds

A closed manifold is a compact manifold without boundary. Let v € Cr(D,,Rn) (r > 1) be a vector field and fi C Rn be an open set. Let M C fl be an m-dimensional manifold with 0 < m
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Bifurcation Theory and Applications

Definition 5.2 We say that an invariant manifold M of a vector field v e Cr(Q.,Rn) is /i-stable if there exists a neighborhood O C C r (fi,R") of v such that for any u 6 O, u has a unique invariant manifold M\ which is homeomorphic to M and d(Mi,M) = max dist(a;,M) -> 0 as u —> v.

(5.42)

A point #o € M is called C r (r > 1) if there is a Cr neighborhood O C M of xo • Let M be an m— dimensional invariant manifold of a vector field v € Cr(O,]Rm) and x0 G M is a C 1 point. Then there exists an (n — m)-dimensional normal space NXo of M at xo- We call the vector field u Nx0 °f the projection of u on Nx the normal vector field of v at XQ, and XQ is an origin of NXo. Obviously, a Cr (r > 1) manifold M C 0 is invariant for vector field u 6 Cr(Q.,Wn) if and only if the normal vector field of v at each point xo G M satisfies that vNxo(x0) = 0.

(5.43)

Definition 5.3 Let v £ Cr(fi,]Rn) (r > 1) and M C fi be an invariant manifold of v. We say that M is hyperbolic if (1) M is Cr manifold; (2) for each point xo £ M, the normal vector field VNXO of v is regular at the origin Xo, i.e. the Jacobian matrix DVNXO{XQ) of VJVXO at XQ is nondegenerate; (3) the real parts of all eigenvalues of DVNXO (XO) are nonzero. The concept of hyperbolic invariant manifolds is a natural generalization of hyperbolic singular points and closed orbits. Similar to the stable manifold theorem for singular points, we have the following stable manifold theorem for hyperbolic invariant manifolds. Theorem 5.9 Let M be an m-dimensional hyperbolic invariant manifold of v £ C r (Q,R. n ). Then there exist two unique manifolds Wu and W3 with dim Wu = m + ki, dim Ws = m + k2, fci + k2 = n — m, called the unstable manifold and stable manifold of M, which are characterized by Wu = {z G Rn I lim dist(S(-t)z,M) = 0}, Ws = {zeRn\

t—»oo

lim dist{S{t)z,M) = 0},

t—too

where S(t) is the semigroup generated by the vector feied v. Moreover, Wu and Ws are Cr and transversal on M.

Dynamic Bifurcation Theory: Finite Dimensional Case

131

Proof. Since M is hyperbolic, by Theorem 2.10, for each point xo 6 M the normal vector field VNXQ of v has the unstable manifold W"o and stable manifold W*Q with dim W^o = &i,

dxmW*0=k2,

ki+k2=n-m,

and W£o, W£o are Cr and transversal at xo. It is clear that the sets

wu= U w?0,

w s = U W^

xoeM

xoeM

are the desired manifolds by this theorem. The proof is complete.

•

Prom the stable manifold theorem (Theorem 2.10) we know that for each point XQ £ M, the normal space NXo can be decomposed into a direct sum of two spaces NXo = Ei® E2,

dimEi = k\,

dimE2 — k2,

such that WXo and W£Q are tangent to E\ and E2 at the origin XQ respectively. Moreover, E\ and E2 are the eigenspaces of DVN^^XQ), and the eigenvalues Ai, • • • , X^ on E\ and /?i, • • • ,(3k2 on E2 of DVNXQ (XO) satisfy that ReXi > 0 (1 < i < hi),

Reft < 0 (1 < i < k2).

(5.44)

By Theorem 5.9, we can see that there is a tubular neighborhood O c ! l of M such that f LO(U) n U = Wu n U, a{U) nU = W3nU, and < \ M = LJ(U) na(U) nU,

(5.45) J

where u>(U) and a{U) are the w-limit and a-limit sets generated by v and Wu, W3 are the unstable and stable manifolds of M. In fact, (5.45) can be also achieved from (5.43) and (5.44). Let u 6 Cr(Q,R") with ||u - v\\c- > 0 sufficiently small. Then there exists a sufficiently small neighborhood U of M such that the normal vector field UNX of U on Nx with x £ M has a unique zero point ZQ S U PI A^, and the Jacobian matrix DU^^ of L/jvx at ZQ is nondegenerate: (UN.(ZQ)=0,

{ detUNx (z)

Z0£NX,

X£M,

^0,\fzeNxDU.

(5.46)

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Bifurcation Theory and Applications

Obviously, the below set M = {zxeU\

UNx (zx) = 0, x £ M}

(5.47)

is a manifold homeomorphic to M. Although in general M is not invariant for u, we can expect that there is a unique invariant manifold of u around M which is homeomorphic to M. It is the subject in the next subsection. 5.3.2

S1 attractor bifurcation

In this section, we shall prove that the bifurcated attractor Q,\ of (5.16) from an eigenvalue with multiplicity two is homeomorphic to a circle S1. Let o b e a two-dimensional Cr (r > 1) vector field given by vx(x) = \x-G(x,\), for xeR2.

(5.48)

Here

G(x,X)=Gk(x,X)+o(\x\k), where Gk is afc-multilinearfield,which satisfies CMk+l

< (Gk(x,X),x) < C2\x\k+1,

(5.49)

for some constants Ci > C\ > 0, k = 2m + 1, and m > 1.

Theorem 5.10 Under the condition (5.49), the vector field (5.48) bifurcates from (x, X) = (0,0) on X > 0 to an attractor Cl\, which is homeomorphic to S1. Moreover, one and only one of the following is true. (1) Q\ is a periodic orbit, (2) Q\ consists of only singular points, or (3) Q\ contains at most 2(k + 1) = 4(m + 1) singular points, and has AN + n (N + n > 1) singular points, 2N of which are saddle points, 2N of which are stable node points (possibly degenerate), and n of which have index zero, as shown in Figure 5.8 for N = 1 and n = 2. Proof.

We proceed in the following five steps.

STEP 1. Obviously, (5.49) implies that x = 0 is asymptotically stable for (5.48) at A = 0. Hence, by Theorem 5.2, the vector field v\ bifurcates from (x, X) = (0,0) to an attractor Q\ on A > 0, which has the homology type of a circle S1.

Dynamic Bifurcation Theory: Finite Dimensional Case

"pj I

""

\T

"

133

lp6 "

i

Fig. 5.8 Q\ has AN + n (N = 1 and n = 2 shown here) singular points, where pi, P4 are saddles, P3, P6 are nodes, and p2, ps are singular points with index zero.

2. Let ^A have no singular points. Then, Q.\ must contain at least a periodic orbit. We need to show that Q,\ contains only one periodic orbit. Take the polar coordinate system (xi,x<2) — (r cos6,r sin9). Then the vector field v\ becomes STEP

d r

^rcos^i+sin^2 d6 cos 6v2 — sin 6v\'

(55Q)

We see that cos 6vi = Xr cos2 6 — cos 6gi (r cos 0, r sin 9, A), sin 0^2 = Ar sin 2 6 — sin 0
t = l,2.

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Bifurcation Theory and Applications

By (5.49), (5.50) is rewritten as d r

dO

X

_

r2m(cQS 9

m

+

g i n Qgk2)

+

2m l

r - {sm9gkl-cos9gk2

o ( r 2m)

+ O{r))

'

{(5.52)

' '

Based on (5.49), we have C\ < cos6gki(cos6,sin#, A) + sin#<7fc2(cos#,sin0, A) < C2.

(5.52)

On the other hand, by assumption, Q\ contains a periodic orbit for any A > 0 sufficiently small. Hence 0 < C < sin0gki(cos9,sin9,\)

- cos0gk2(cos6,sin9, A) + O(r),

(5.53)

for any 0 < 9 < 2TT and some constant C > 0. The condition (5.53) amounts to saying tha t the orbits of v\ ar e circula r aroun dx = 0. Let r(9,r 0) be the solution of (5.51) with initial value r(0,r 0 ) = ro . Then we have the following Taylor expansion r2m(9,r0)

= r20m + R(9)-o(\r0\2m),

R(0) = 0.

(5.54)

+ o(r^m),

(5.55)

It follows from (5.51) and (5.54) tha t

= 2n(a\-brlm) where ,2TT

a=

L

h=

L

!

m+o(r)d9> m+o(r)d9>

a{9) = cos9gki + smdgk2, (3{6) = sin6gki - cos9gk2. Pro m (5.55) we see tha t the periodic solutions of v\ near x = 0 corre spond to positive solutions of 2n(aX - br2m) + o{rlm) = 0.

(5.56)

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Dynamic Bifurcation Theory: Finite Dimensional Case

By (5.52) and (5.53), a > 0 and b > 0. Therefore, (5.56) has a unique positive solution near r = 0:

(

> \ i/2m

for any A > 0 sufficiently small. Thus, Q\ has a unique periodic orbit. 3. We claim that if Cl\ contains either finite number of singular points or a circle of singular points, and if it contains finite number of singular points, then there are at most 2(k + 1) of them near x = 0. In fact, if STEP

9i(x,X) = £i g2 (x, A)

12'

then ft A has a cycle of singular points. Otherwise, by (5.49), the number of singular points of v\ is finite. The maximal number of singular points for v\ is determined by the following equation Xx~Gk(x,X) =0.

(5.57)

Since Qk is a fc-multilinear vector field, the singular points of (5.57) must be on the straight lines x-i = zx\, where z satisfies z = gM (*l,S2,A) =

5fci(zi,Z2,A)

^llil^. gki(l,z,X)

(5.58)

The number of solutions of (5.58) is at most k + 1. Since k =odd, the number of solutions of (5.57) is at most 2(fc + 1). STEP

4. Let £l\ contain a cycle S1 of singular points, then we shall see

that nx = S1.

Under the polar coordinate system, we have vr{0, r) = (vx, x) = Xr2 - rk+la{6) + o{rk+1), where a{6) is defined by (5.55). By (5.52),

0 < C\ < a(6) 0 near r = 0. Hence the set n\ = {(0,rx(0)) |

vr(0,rx(0))=O,O<0<2ir}

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Bifurcation Theory and Applications

is homeomorphic to a cycle S1, and all singular points of v near x = 0 are on fl\. It implies that fix C fix- Let

9ki= £

^ijX\4,

(5.59)

1 = 1,2.

i+j=k

By Step 3, we know that X2<7fci = X\gk2- Then we infer from (5.49) that 0 < al0 = a2k_n.

(5.60)

For the singular point (£\, 0) € fix of i>,\, we have div vx(xlt0)

= 2A - fca^xt"1 - a2k_nxk-1

+ o^"1)

= (by aio^i" 1 = A a n d (5-60)) = -(fc-l)A + o(A) <0, for any A > 0 sufficiently small. In the same fashion, for any point x 6 £l\, we take an orthogonal system transformation such that x is on the 5i-axis, then we can prove that div v\{x) < 0, which implies that fl\=flx STEP

\/x G fix,

= S1.

5. fl\ contains finite number of singular points. We show that

flx = S\

By the Brouwer degree theory, it follows from (5.49) that deg(i>A,fi,0) = 1,

|A| > 0

sufficiently small,

in some neighborhood fi C M2 of x = 0. It is known that ind(«A,0) = l,

|A|^0.

Hence we have

(5.61)

J2 md(vx,Zi)=0.

Let z £ fix be a singular point of via. Without loss of generality, we take the orthogonal coordinate system such that z = (xi, 0). Then by (5.49) and (5.59), the Jacobian matrix of vx at z is given by _, , ,

/-(Jfc-l)A + o(A)

Dvx(z)=( V

'

U

'

* \i~

\

, i u ) , k-Ulak0)AJ

2 a

,_... (5-62)

Dynamic Bifurcation Theory: Finite Dimensional Case

137

where a\0 > 0. Obviously, Dv\(z) has an eigenvalue f3 = — (fc—l)A+o(A) 7^ 0. Hence for any singular point z £ £l\ of v\, the index of v\ at z can only be either 1, —1 or 0. It is easy to see that if the index is 1, then z is a stable node point. Let the index of v\ at z be — 1: index(t>A, z) = —1-

(5.63)

When a1_n ^ a\0, z is nondegenerate. Therefore, v\ has a unique unstable manifold at z. When af._n = a\0, divt/A(z) = - ( f c - l ) A + o(A) < 0 .

(5.64)

If the unstable manifold of v\ at z is not unique, then the local structure of v\ at z is topologically equivalent to that as shown in Figure 5.9.

Fig. 5.9

On the other hand, (5.64) means that there is a neighborhood O c R 2 of x = 0, such that div v\(x) < 0,

Vx e O,

which implies that for any open set O C O, \6\ > \6t\,

0
(5.65)

where to > 0 depends on O, Ot = S(t)O, and S(t) is the flow semigroup generated by v\.

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Bifurcation Theory and Applications

However, it is clear that for any open set O C O in domain P as shown in Figure 5.9, the property (5.65) is not true. Therefore, the unstable manifold of v\ at z must be unique We can prove in the same fashion that if the index of v\ at z is 0, then the unstable manifold of v\ at z is also unique. By the Poincare-Bendixson theorem, all unstable manifolds of the singular points of v\ at z with index —1 and 0 are connected to the singular points with index 1 and 0, as shown in Figure 5.8. Thus by the uniqueness of unstable manifolds for each singular point with either index —1 or index 0, the set of all singular points and unstable manifolds is a cycle S1, and Q x = S1. The proof is complete. • The above proof implies also the following theorem. Theorem 5.11 Ifv\ is an m-dimensional C (r > 1) vector field given by (5.48) and satisfies (5.49) for x € K m , then v\ bifurcates from (x, A) = (0,0) to an attractor fi,\. If Q\ contains an (m — 1)-dimensional sphere consisting of singular points ofv\, then Cl\ is homeomorphic to 5 m - 1 .

5.4

Stability of Dynamic Bifurcation

Let Q. C R" be an open set, x = 0 e £1 and A € / = (a, b) C E 1 . We denote the space of parameterized vector fields by C0M(fi x / , R " ) = { t ; : f i x / ^ R r i | «(0, A) = 0, VA € 1} endowed with the norm

He*,. = sup ( J2 \D>\ + \D*°\ + \^Dxv\ ) . (*,A)enxz ^ H = o

)

Obviously, if v £ CQ'1^ X 7,R n ) (k > 1) then v is fc-th differentiate at I E ! 1 and differentiable at A £ / , and v can be expressed as V(;\)=AX + G(;X), where A\ and G{-, A) are the same as in (5.16) and (5.17).

(5.66)

Dynamic Bifurcation Theory: Finite Dimensional Case

139

It is known that simple real and complex eigenvalues of A\ are differentiable on A [Kato, 1995], which can be expanded as (3i{\)=ai

+ ai\ + o(\\\), 1

(l
X

Definition 5.4 Let vu v2 6 C Q ' ^ J,R") and A; £ / (i = 1,2) be a bifurcation point of an invariant set Ti{p) of Vi(x,p). We say that both bifurcation points X\ and A2 have the same structure if v\ and v2 are locally topologically equivalent at Ti(pi) and T2(p2) with p\ — \\ = p2 — X2, i.e. there are neighborhoods Ui C R™ of Ti(pi) and a homeomorphism of (f : U\ —> U2 such that

0 sufficiently small there is a neighborhood O C CQ'1^ X /, R") of v such that for any u £ O, u has a bifurcation point Ai with |Ai — AQ| < s, and Ai and Ao have the same structure. If all bifurcation points of v are stable, we say the v is stable for bifurcation in CQ'X(Q, X /,R n ). Let v e C Q ' 1 ^ x 7,R n ) (k > 1). Ao £ / is called an eigen-parameter with multiplicity m if the eigenvalues /?t(A) of Ax = Dv(0, A) satisfy Re/3i(X0) = 0 , (1 < i < m),

Re/3i(X0) ^ 0, (m + 1 < i < n).

(5.67)

Ao G / is simple if m = 1 for /3i(A0) = 0 and m = 2 for /3i(Ao) = ^(Ao) = i/3 (/? ^ 0). If Ao £ / is simple, and the eigenvalues in (5.67) satisfy ^-Re{3i{X0) ^ 0 , 1 < i < m, aX then Ao is called a regular eigen-parameter.

(5.68)

The following theorem is basic for stable bifurcation. Theorem 5.12 Let v £ Co' (Q. X I,R") and Xo £ I be a simple eigenparameter of v. Then there exists a number b(Xo), called the bifurcation number of v at Ao, which depends continuously on \\v\\c3,i, such that the following assertions hold true. (1) Ao is a stable bifurcation point of v if and only if Ao is regular and the bifurcation number 6(Ao) 7^ 0. (2) If XQ has multiplicity m = 1 and b(Xo) =^= 0, then v(x, A) bifurcates to a unique branch of singular points x(X) (A ^ 0) on each side of X = Ao, which are hyperbolic, and x(X) has Morse index k + 1 for /?i(A) < 0

140

Bifurcation Theory and Applications

and Morse index k for /?i(A) > 0 where k=number of the eigenvalues with Ref3j(\0) > 0. (3) If Ao has multiplicity m = 2, then v(x, A) bifurcates to a unique branch of periodic orbits T\ for Re/3i(X) < 0 when b(Xo) > 0, and for RePx(X) > 0 when b(Xo) < 0, which are hyperbolic. Proof. CASE 1. Ao HAS MULTIPLICITY m = 1. We divide the proof in several steps in the following. STEP 1. The canonical bifurcation equation of the vector field (5.66) is as follows ^ = /31(X)x + G&, $(*, A), A) at By (5.68), Pi(X) satisfies that

{

< 0 (or > 0) =0

if A < Ao, ifA = A0,

>0(or<0)

ifA>A 0 .

(5.69)

By assumption G\ is C 3 at x = 0, we have the Taylor expansion Gi(x, $(i, A), A) = b{X)x2 + c(X)x3 + o(\x\3),

(5.70)

where 92Gi(0,0,A) &5 • 6(A0) is defined as the bifurcation number of v. It is clear that 6(A) continuously depends on A and the norm |M|c2STEP 2. We can see that as b(X0) ^ 0, the equation (5.69) with (5.70) bifurcates on each side of Ao a unique branch of singular points x* (A) which can be expressed as 6(A) =

x*W = -r^hW

+ o(\Pi\).

(5.71)

It is clear that (x*(\),$(x*(\),\)) is a nondegenerate singular point of v at A ^ AoWe shall prove that z(X) = (x*,$(x*,\)) is a hyperbolic singular point of v. v is decomposed into v=(p1(X)x V ~ \Bxy

+ G1{x,y,\), + G2(x,y,X).

Dynamic Bifurcation Theory: Finite Dimensional Case

Then, by (5.70), (5.71) and $(x,\)

141

= o(\x\), we have

Dv(z(X) X) - ^ l ( A ) ° ^1 + / ^ ( ^ ( ^ A ) , ^ _f-f31(X) + o(\f31\) 0{ft) \ -^ O(/3i) Bx + Ofa))'

(5.72)

and signdetDv(z,X) = \ S V \ -signdet5A

W U

(5.73)

if/?i(A)>0.

Since 5 A is hyperbolic, by (5.72) Dv(z,X) is also hyperbolic for any |A — Ao| > 0 small. It follows from (5.73) that = 1

f* + l iWA)<0,

~\fc

if ^(A) > 0.

where ki =number of the eigenvalues of Dv(z, A) with positive real parts. Thus the assertion (2) is proved. STEP 3. SUFFICIENCY IN ASSERTION (1) FOR m = l . Let Ao be regular andfc(Ao)> 0 (for the case of b(Xo) < 0 the proof is the same). Since the eigenvalue /3i(A) is C 1 at Ao, for any e > 0 there exists 5 > 0 such that if «i £ C03a(fl x I,Rn) and |K - «|| c ,.i < S,

(5.74)

the eigenvalues /3i(A) of v and /?i(A) of v\ satisfy

{

A = a(A - Ao) + o(\X - Ao|), a jt 0, ^ = a i ( A - A ! ) + o ( | A - A i | ) , with

(5.75)

|tt! - a | < £ , |A0 — Ai| < e . Hence it follows that there is a neighborhood O C C Q ' 1 ^ X / , W1) oiv such that for any v\ 6 C, t;i has a simple and regular eigen-parameter Ai with |Ai — Ao| < e, and the bifurcation number of v\ at Ai satisfies 6i(Ai) > 0. We say that a linear vector field A is hyperbolic if all eigenvalues of A have nonzero real parts. The number of eigenvalues of A with negative real parts is called index of A. It is known that two linear vector fields A and B are topologically conjugate (or topologically equivalent) if and only if A and B are hyperbolic and have the same index; see [Palis and de Melo, 1982].

142

Bifurcation Theory and Applications

Let zi(A) be the bifurcated branch of v\ from (0, Ai). By Assertion (2), Dv(z,p0) and Dvi(zi,pi) are hyperbolic for p0 ^ Ao and p\ ^\\. On the other hand, by (5.74) and (5.75) we have

(,76)

^^(-^tf\^my

where \\BX-B4K5,

for S > 0 small. Hence it follows from (5.72) and (5.76) that Dv(z,p0) and Dv(zi,pi) with p\ — \\ = po — Ao have the same index. By the HartmanGrobman theorem we derive the sufficient condition in Assertion (1) for m = 1. STEP 4. NECESSITY. When 6(A0) = 0 and c(A0) ^ 0 in (5.70) it is clear that the topological structure of bifurcation of v is different from v\ where the bifurcation number h[\\) ^ 0 with b{\{) —» 0 as vi —> v in CQA(Q, x /,R"). If 6(A0) = c(A0) = 0, then v can be approximated by vx and i>2 in C o ' (O x I, W1), where the bifurcation number 6i(Ai) =fi 0 of v\, and 62(A2) = 0, C2(A2) ^ 0 of U2- Hence Ao is not stable for bifurcation. When Ao is not regular, i.e. /?i(A) = o(|A — Ai|), then v can be approximated by VN in C Q ' 1 ^ X /, R n ), where DVN{0, A) have the eigenvalues near Ao as follows N

^

U

_ j 0, A e [Ao - eN, Ao + £jv], eN > 0, £JV -» 0, AT -> oo, ~ \ ^ 0 , A^[A o -£w,Ao + £jv].

Obviously, the bifurcation structure of VN is different from v at A = AoThus the conclusions for m = 1 are proved. CASE

2. Ao HAS MULTIPLICITY m = 2. We proceed in a few steps as

follows. STEP 1. For simplicity, let v(z, A) can be expressed as (a{\) - 1 0 \ (x{\ v(*,A)= 1 a(A) 0 U a

\ 0

0 Bj

\yj

/Gi(x,j/)\ + G2(x,i/) ,

\G3(x,y)J

where z = (x,y) £Q,x = (xi,x2) e O n R 2 , and y S f i n R " - 2 .

Dynamic Bifurcation Theory: Finite Dimensional Case

143

The reduction equation of v is given by

{

dx\

.

.

(5.77) — =a(A)a:i - x2 + fi(xi,X2), dt

uX2 —r- =Xi +a(X)X2

at

where

{

< 0 (resp. > 0)

if A < Ao,

= 0

ifA = A0,

>0

(resp. < 0)

fi(x)=Gi(x,h(x,\)) and h the center manifold function. By the Taylor expansion we have

(5.77)

+j2{Xl,X2),

ifA>A 0 ,

i = 1,2,

ft(x)= £ a^xlxl + jyL^oVk + o^), J

j=lfe=l

2
yk = hk(x,\)=

x x

(5,78) x 2

Y^ <%s i 2+°(\ \ ), r+s=2

where a

™=

dp+iGAQ 0) to?0x<

f o r 0 < p , g < 3 , i = l,2,

It is clear that the coefficients alpq, fejfe and c£s continuously depend on A and the norm ||u||(73,i. Let xi — rcos6, x^ = rsin#. Then we have dr

dt=cos0^F

dxi

.

+ sme

dx2

-dF'

d9 dx2 . adxx r— ~ cosfl-T- - sm6—r-. dt dt dt

144

Bifurcation Theory and Applications

It follows from (5.77) and (5.78) that dr _ a(X)r2 + r cos 9fi (r cos 9, r sin 9) +r sin 9f2 (r cos 0, r sin 9) d9 r + cos 8f2(r cos 6, r sin 6) — sin Ofi(r cos 8, r sin 8) = [o(A)r + «i(0)r 2 + u2(8)r3 + o(r3)][l + vx{9)r + v2{9)r2 + o(r2)} = a(\)r + (ui + a(X)vi)r2 + (uivi +u2 + av2)r3 + o(r3), (5.79) where ttj, Vi (i — 1,2) are the homogeneous functions with degree i + 2 for cos # and sin #, and continuously depend on the coefficients alpq, bjk and c£s in (5.78). It is easy to check that 2TT

/

(5.80)

[u1(O)+av1(0)]d6 = O.

Let b(X) = [ [«!(0)i;i(0) + «2(fl) + a{\)v2{6)\d9. Jo The number 6(A0) is defined as the bifurcation number of v at Ao. STEP 2. v(z,\) bifurcates a unique branch of periodic orbits F\ for a(X) < 0 as 6(A0) > 0, and for a(X) > 0 as 6(A0) < 0. The above claim is well known. However, for the sake of completion, we here give a proof. Let r(9,X,a) be the solution of (5.77) with initial value r(0,A,a) = a. We know that r(9, A, a) is 3rd differentiable on a > 0. Then r(9, A, a) has the expansion near a = 0 as r(9,X,a)=a + r0{8,X)O(a'2)

(5.81)

Without confusion, we denote r(0, A, a) by r(9,a). On the other hand, by (5.79) we get — — = -a(X) + (MI

+ a«i) + (uiv\ +u2 + av2)r + o(r).

(5.82)

Inserting (5.81) into (5.82), then integrating it, by (5.80) and r(0,a) = a we obtain that

r(27r

;(L"aT'

a)=c(a A ) Q ( A ) + b { x ) a 2 + ( a 2 ) )

-

°

(5 83)

-

Dynamic Bifurcation Theory: Finite Dimensional Case

145

where

{

,2TT

c(a,X)=l

[l +

o(a)r0(9,\)}d6,

C(O,A)=27T.

It is easy to see that each positive solution a > 0 of the equation c(a, X)a(X) + b(X)a2 + o(a2) = 0

(5.84)

corresponds to a periodic orbit of (5.77) passing through (xi,£2) = (o,0). Obviously the positive solution of (5.84) is unique for a(X) < 0, b(X) > 0 and a(X) > 0, b(X) < 0. Step 1 is proved. STEP 3. THE BIFURCATED PERIODIC ORBIT T\ ARE HYPERBOLIC. For

convenience, we first introduce the concepts of the Poincare maps and hyperbolic orbits, which can be found in standard textbooks and monographs on dynamical systems, e.g. see [Palis and de Melo, 1982]. Let 7 be a periodic orbit of a vector field v, x0 € 7 a point. Let E be a section transversal to v through the point XoThe orbit through x$ returns to intersect £ at time T > 0, where T is the period of 7. By the continuity of the flow of v, there exists a neighborhood U C £ of xo, such that for each point x € U the orbit through x returns to intersect S at some time t > 0. Thus we can define a map P : U —> E as that for every point x £ U, P{x) is the first point where the orbit of x returns to intersect S. This map P is called the Poincare map associates the periodic orbit 7; see Figure 5.10. The Poincare map P : U —> P(U) C E is a homeomorphism, and each fixed point of P associates a periodic orbit of v. We say that 7 is a hyperbolic periodic orbit of v, if XQ is a hyperbolic fixed point of the Poincare map P : U —> P(U) C S, i.e. DPXo has no eigenvalues of modulus one. Without loss of generality, we assume that the center manifold M\ of the vector field v(z, A) is the x-plane, i.e. y = h(x, A) = 0 V|A - Ao| > 0 sufficiently small and x e R2. In fact, in the transformation of coordinate system as follows

{

x = x, y = y-h(x,X),

146

Bifurcation Theory and Applications .POO

Fig. 5.10

the vector field v is transformed into the below form v=

(a{\) - 1 0 \ (xA 1 a(X) 0 i2 \ 0 0 BxJ \y)

(Hx{x,y)\ +\H2(x,y)\, \H3(x,y)J

where i7 i (5,y)=G i (£,y + /i(i,A)),

i = 1,2,

tf3(2, y) = BA • ^(5, A) - V/i ~ cti_ (a -IN _ dt ~ \1

a )

X

+ G3(x, y + h),

/FA +

\H2)

^(2,10=0(1x1,151),

'

l
It is clear that both vector fields v and v are topologically equivalent, and the x-plane is a center manifold of v at A. Now we give the Poincare map associated the bifurcated periodic orbits T\ oiv at A. By assumption, the ar-plane (y = 0) is a center manifold of v at A, then T\ C R2 the z-plane. Let Tx be through the point (a(A),0) e R2 with 0 < a(A) < p (p sufficiently small). Denote by Dp - {y e Rn~2 | \\\y\\ < p). We take the section transversal to v through the point (a(A),0)

Dynamic Bifurcation Theory: Finite Dimensional Case

147

as follows: S = {(xuy) =

W1'2}

\0<Xl
(0,P)xDp.

The Poincare map P : E —> E is given by P(z)

= (pi(z),$(«„z)),

z = (Xl,y)e

S,

where Pi : E —> (0, />) is a map, and $(£ z , 2) G £>p is given by rt

$(t,z)=etB*z+

Jo

e{t-T)BxGz(x,$)dT,

(5.85)

which is the solution of the equation i^=Bxy

+ G3(x,y),

11/(0) = z, and tz is the time that the orbit through z returns to intersect E. Because the x-plane is invariant for v(z,X), the line segment (0,p) x {0} C S is invariant for the Poincare map P. Hence we have *(*z,*)U=(*1,o)=0

(5.86)

Pi(xi,0) =r(2n,xi) (by(5-83^

= l-c(xuX)a(X)-KX)xl+o(\Xl\>)

(5-87)

= rri + c(xi,X)a(X)xi + b(X)x\ + o(|a;i|3).

For the fixed point (a(A),0) G S of P(z), from (5.85)-(5.87) we get

DP(a, 0) =

/dPl(a,0) dx\ d${tz,z) \ dxx

^

\

d$(tz,z) dy I 2

z={afi)

_ / l + ca + 3&a + o(|a|,|a| 2 ) v

* \ / z=(a,0)

Since tz = 1 at z — (a,0), and a(A) is positive solution of (5.84) we have a2(A) = ~\c-a + o(\a\) = - ^ a ( A ) + o(|a|).

148

Bifurcation Theory and Applications

Hence the eigenvalue of DP(a, 0) are as follows 1 - 47ra(A) + o(|a|),

e^ (A) ,

3 < j < n.

By assumption, Re{3j(\0) ^ 0 (3 < j < n), therefore DP(a, 0) is hyperbolic for all | A - Ao| > 0 small. STEP 4. Finally, we prove Assertion (1) for m = 2. We only need to prove the sufficiency because the proof of necessarity is similar to the case where m = 1. Let b(X0) > 0 and A-o be regular, i.e.

a(A)=ff(A-Ao) + o(|A-A 0 |), fl

a £ 0.

n

When vi £ Co (Ci x I,M. ) and ||ui — v\\C3,i sufficiently small, there exists an eigen-parameter Ai of v\ which is close to Ao such that near Ai, DF1(0,X)=

(ax{\) -(3 0 \ (3 ai(A) 0 ,

V 0

0

Bj

where a1(A) = a 1 (A-A 1 ) + o(|A-A 1 |), and B\ is close to B\. Meanwhile, the bifurcation number &i(Ai) of v\ is also close to 6(A0), therefore 6i(Ai) > 0. By the above assertions, v\ bifurcates to a unique branch of periodic orbits T\ for «i(A) < 0, which are hyperbolic for all |A — Ao| > 0 small. Thus the bifurcated branches T\ and F^ are on the same side of their bifurcation points. It is known that the hyperbolic periodic orbits of vector fields in C r (fi,R") (r > 1) are locally structurally stable. Because the parameterized vector field v G CQ'^Q X /,R") is C 1 on A, any two vector fields v(-,\x) and v(-, A2) are locally topologically equivalent at T\1 and F^2 for •^li ^2 < -^0 (°r Ao < Ai, A2). On the other hand, if v\ is close to v in CQ'^Q x /,M"), for some fixed parameter p, vi(-,p) is close to v(-,p) in C3(Q.,Rn), Therefore there exists a neighborhood U C CQ'\Q X /,K") of v such that for any v\ 6 U, v(-,p) and vi(-,p) are locally topologically equivalent at Tp and F^, which implies that the bifurcation points Ao and Aj of v and v\ have the same topological structure. D The proof of this theorem is complete.

Dynamic Bifurcation Theory: Finite Dimensional Case

149

It is easy to achieve directly from Theorem 5.12 the following global stability theorem for bifurcation. Theorem 5.13 A parameterized vectorfieldV\ € C o ' (fix I, R") is stable for bifurcation if and only if (1) all the eigen-parameters of v are simple and regular, and (2) each of the bifurcation numbers of v at eigen-parameters is nonzero. Moreover, the set of all vector fields with stable bifurcation is open and dense in C^'1 {fix I,W1). Remark 5.6 We note that in C Q ' 1 ^ X /, R n ) there are no vector fields which are stable for bifurcation except the vector fields which have no eigenparameters in /. In C0>:l(fi x /, M") the set of all vector fields with stable bifurcation are those whose eigen-parameters are regular with multiplicity m = 1. 5.5

Notes

5.1 The result on forced pendulum is introduced here for the first time, and the Kaldor model for business cycles is taken from [Gabisch and Lorenz, 1987], with new investing and saving functions introduced here. 5.2 The attractor bifurcation theorem, Theorems 5.2 and 5.3, are the core of the bifurcation theory developed in this book. The results in this section are taken from [Ma and Wang, 2004e]. 5.3 Theorem 5.10 was proved in [Ma and Wang, 2005a], and was used in [Ma and Wang, 2005e; Ma and Wang, 2005b]. 5.4 The results in this section are new, and are introduced here for the first time.

Chapter 6

Dynamic Bifurcation Theory: Infinite Dimensional Case This chapter develops dynamic bifurcation for infinite dimensional dynamical systems modeled by partial differential equations. The main topics to be addressed include

(1) attractor bifurcations for infinite dimensional dynamical systems, (2) attractor bifurcation for perturbed systems, and (3) bifurcations at simple eigenvalues or at eigenvalues with multiplicity two.

For both topics (1) and (2), we generalize attractor bifurcation theorems for finite dimensional systems in Chapter 4 to nonlinear operators (both first-order and second-order in time) equations, which model nonlinear partial differential equations from sciences and engineering, e.g. given in Chapters 7-10. In particular, the attractor bifurcation for perturbed system is applied to the Taylor problem in Chapter 10. Topic (3) combines the attractor bifurcation methodology with detailed analysis on the effect of higher order nonlinear terms to bifurcation. Among other things, for instance, sufficient conditions are obtained for the existence of saddle-node bifurcation and bifurcation to periodic solutions from real eigenvalues. These conditions are easy to verify for many bifurcation problems for partial differential equations. 151

152

Bifurcation Theory and Applications

6.1

Attractor Bifurcation

6.1.1

Equations with first-order in time

Let H and Hi be two Hilbert spaces, and Hi <—• H be a dense inclusion. We consider the following nonlinear evolution equation

idi=^u

+

G(u,X),

(6i)

[u{0) = H are parameterized linear completely continuous fields continuously depending o n A s I 1 , which satisfy

{

L\ = —A + B\ A : Hi —> iJ

a sectorial operator, a linear homeomorphism,

(6-2)

BA : -ffi —• # parameterized linear compact operators. It is easy to see from Chapter 2 that L\ generates an analytic semi-group {e~tLx}t>o. Then we can define fractional power operators L" for any 0 < a < 1 with domain Ha = D(L") such that Hai C Ha2 if ai > a.^, and H0 = H. Furthermore, we assume that the nonlinear terms G(-,A) : Ha —> i? for some 1 > a > 0 are a family of parameterized Cr bounded operators (r > 1) depending continuously on the parameter A € M1, such that G(u,X)=o(\\u\\Ha),

VAeM 1 .

(6.3)

Let the eigenvalues (counting the multiplicity) of L\ be given by /3i(A),/32(A),---,/?fc(A)GC, where C is the complex space. Suppose that

{

<0

ifA
ifA = A0) =0 >0 ifA>A 0 , RefriXo) < 0, Let the eigenspace of L\ at Ao be

(6.4) Vl
(6.4)

V m +1 < j .

(6.5)

Eo= | J {J{u£Hi\(LXo-[3i(\o))ku l
= 0}.

Dynamic Bifurcation Theory: Infinite Dimensional Case

153

By (6.4), we know that dimEo = m. We are now in position to state the attractor bifurcation theorem for the infinite dimensional system (6.1). Theorem 6.1 Assume that the conditions (6.2)-(6.5) hold true, and u = 0 is a locally asymptotically stable equilibrium point of (6.1) at X — Ao. Then the following assertions hold true. (1) Equation (6.1) bifurcates from (u, A) = (0, Ao) attractors SA for A > Ao, with m — 1 < dim E^ < m, which is connected as m > 1; (2) The attractor Y.\ is a limit of a sequence of m-dimensional annulus Ak with Ak+i C Ak; especially if T,\ is a finite simplicial complex, then Y,\ has the homotopy type of the (m — 1)-dimensional sphere Sm~1; (3) For any u\ € £A, u\ can be expressed as u\ = v\+o(\\v\\\Hl),

v\£E0;

(4) If the number of the equilibrium points of (6.1) in Y,\ is finite, then we have the index formula }_^

md[-(Lx

+ G),Ui] = \

(5) If u = 0 is globally asymptotically stable for (6.1) at X = Ao, then for any bounded open setU C H with S^ C U, there is an £ > 0 such that as Ao < A < Ao + e, the attractor T,\ attracts U\T in H, where T is the stable manifold of u = 0 with co-dimension m. In particular, if (6.1) has a global attractor for all X near Ao, then the £ here can be chosen independently ofU. Proof. By the canonical reduction, the bifurcation equation of (6.1) is in the following form OjCC

— = JmXx + g{x,hx(x),X),

ieRm,

(6.6)

where Jm\ is the Jordan matrix associated with eigenvalues /% (1 < i < m), and g(x, hx(x), X) = o(\x\). Since u = 0 is asymptotically stable for (6.1), x = 0 is asymptotically stable for (6.6). Then this theorem follows from Theorems 5.2—5.3. This proof is complete. •

154

Bifurcation Theory and Applications

Remark 6.1 If m = 1 in (6.4), i.e. /?i(A) is a simple eignvalue of L\, and G : Hi —> H is analytic, then Theorem 5.5 is valid for the infinite dimensional system (6.1).

6.1.2

Equations with second-order in time

This subsection is devoted to the bifrucation problem of a class of nonlinear evolution equations with the second-order in time and with a damping term. Consider ' cPu

du

^+2a-=Lxu

r

_,

,,

+ G(u,X),

(6.7)

ut(0) = ^ , where the constant a > 0 is the damping coefficient, L\ and G are as defined in the previous subsection. Here in this subsection, we always consider the case where L\\ H\ —> H is symmetric, and assume that G(-, A) : H1/2 -> H is Cr (r > 1) bounded.

(6.8)

We also assume that for any ( 0. Theorem 6.2 Let (6.2)-(6.5) and (6.8) hold true. Then we have the following assertions. (1) IfX < Ao, then u = 0 is a locally asymptotically stable equilibrium point of (6.7). (2) If u = 0 is a locally asymptotically stable for (6.7) at A = Ao, then the assertions of Theorem 6.1 hold true for (6.7). Proof.

We proceed in several steps as follows.

STEP l. REDUTCTION TO EQUATIONS WITH FIRST-ORDER IN TIME. It

is easy to see that (6.7) is equivalent to ' du — = -a.u + v, at < ^=Lxu + a2u-av + G(u,X), at (u(0),v(0)) = (<po,ipo),

(6-9)

Dynamic Bifurcation Theory: Infinite Dimensional Case

155

where <po =
H = #1/2 x H,

H1/2,

equipped respectively with inner products ((ui,v1),(u2,v2))n1 ((U1,V1),(U2,V2))H

Let L\:Hi-^H

= (ui,«2)ffi

= (ui,u2)Hl/2

+(VI,V2)H1/2,

+ (v1,v2)H-

and G(-, A) : Hi -> H be two maps defined by Lx = -A + B, G(u,v,X) = (0,G(u,\)),

where (u, v) 6 "H\, and

-.

A{u

.

fal -I\ fu\

>V)={Aal){v)

=

( au-v \

{Au + av)>

ZM={a>i°+Bx§Q

= {a>u0+Bxu)-

Thus, (6.9) is rewritten as ^=Lxw

(6.10)

+ G(w,X),

where w = (u, v). We infer then from (6.8) that f G : H -» W is Cr bounded, { .

(6.11)

STEP 2. PROOF OF ASSERTION (1). Since L\ is symmetric, it has a sequence of eigenvectors {efc(A)} C H\, which constitutes an orthogonal basis of H. Moreover, by taking proper norms for Ha, we can make {efc(A)} common orthogonal basis of Ha for any a £ R. Then it is easy to see that eigenvalues of L\ are given by Pk(X)

= -a± ^Jcfi+(3k{\),

fc

= l,2,"-.

(6.12)

By (6.4) and (6.5), we find t h a t for A < A0) R e p f c ( A ) < 0 , fc = l , 2 , - - - .

(6.13)

156

Bifurcation Theory and Applications

Furthermore, it is easy to check that the semigroup T\(t) of linear operators generated by L\ is given by

Tx{t) e

{£)<>*#) *!(«) J U J '

^)-

(6 14)

'

for any (
*i(t) = coshtCl'2 = l(e~
+ e
$2(i) = sinhtC]!2 = \ ( e < / 2 - e - < 2 ) . Namely, for any (
oo

we have

k=l 1

k=N+l

AT

oo

fc=l fc=JV+l

where the positive integer iV is given by 2

a+A

f >0

\
if it < N,

akin.

The direct caclculation shows that there is a constant K > 0 such that for any t > 0, ||rA(t)||=

sup e-* t [||$ 1 (*) v > + ^ 1 / 2 $ 2 ( t ) V ' | | i / 2 llvlll/2 + IIV'l|0 = l

+ n4 / 2 $ 2 (% + $1(^111/2] (6.15)

KKe-*, where by (6.13) /9 = mini Repfcl > 0, k

157

Dynamic Bifurcation Theory: Infinite Dimensional Case

and the norm

fc=l

for any a € R l . Thus, by Theorem 3.12, Assertion (1) is proved. 3. PROOF OF ASSERTION (2). It suffices to verify the condition (2.35) in the center manifold theorem, Theorem 2.14. Under the basis {e/j(A)}, the equation (6.10) can be decomposed into the following form STEP

—r^ = —axi + yt at < ^ = Pi(\)xi + a2xi - ayi + G{{u, t) at

if 1 < i < m, if 1 < i < m,

(6.16)

^ = Lxw + PG(u,X), ^ at where m

U = 22xiei

+Wl,

i=l m

i=l e Ex,

w = {wi,w2) Ex = {(w1,w2)

and Cx =

LX\EX

eH

I (wj,ei)H

= O,j = l,2;i = l,---

,m},

'• Ex —» -^A is defined by \L*X + a21 -alj

\w2j

H = span{e m+1 , e m + 2 , • • •} in H. Hence the eigenvalues of Cx are pk(^) (k = m + 1, m + 2, • • •) given by (6.12), and the semigroup Sx(t) generated by Cx is Sx(t) = Tx(t)\Ex where Tx{t) is defined by (6.14).

: Ex -> EX,

158

Bifurcation Theory and Applications

As in the proof of (6.15), we infer from (6.5) that

\\Sx{t)\\\ P=

W>0,

m i n |Rep fe (A)|, k>m+l

for any |A — Ao| > 0 small. Hence the conditions of Theorem 2.14 are verified. • The proof is complete. Now we consider the case where m = 1 in (6.4). Let

(G(u,X) = G1(u,X)+o(\\u\\k), \Gi(x«o,A) = W^xGtfaX),

' '

where k > 1, and let H=-a

(a > 0),

(6.18)

where e £ £i(Ao) is the eigenvector corresponding to /?i(A) at A = AoTheorem 6.3 Assume m = 1 in (6.4), and assume (6.17) and (6.18). Then there exists a neighborhood U C Hi x -#1/2 of (u,ut) = (0,0) such that the system (6.7) bifurcates from (u,ut,X) = (0,0, Ao) exactly two equilibrium points (ui\,0), (it2A)0) € U. In addition, U is decomposed into two open sets U\ and XJ\:

such that

(0,0)edUlndU2x, (Ui,o)eUi i = i,2, and for any (
lim \\ut(t,(p,i>)\\ = 0 ,

t—*oo

where u(^,^,^) w £/ie solution of (6.7).

Dynamic Bifurcation Theory: Infinite Dimensional Case

159

Proof. First it is easy to see from (6.16) that the bifurcation equation of (6.7) is given by

{

dx — = -ax + y,

(6.19)

J

J = fa(\)x + a2x - ay+ < G{xex + h{x, A), A), ex >H . By (6.17) and (6.18) we have (G(xei + h(x, A), A), ex)H = -oiX\x\k-xx

+ o(\x\k).

We make the change of variable x = x, y = y + ax. Then at A = Ao, the bifurcation equation (6.19) becomes

{

dx (6.20)

T-

-^ = -2ay-a\x\k-1x + o(\x\k), where a > 0 is given by (6.18). By Theorem 5.5, it suffices to prove that (x,y) = (0,0) is locally asymptotically stable for (6.20). To this end, let /i

=y,

h = -a\x\k~1x + o(\x\k), 91=0, 32 = -2ay. Then ox ay divg = —2a, /i
160

6.2 6.2.1

Bifurcation Theory and Applications

Bifurcation from Simple Eigenvalues Structure of dynamic bifurcation

Hereafter we always assume conditions (6.2) and (6.3). Let the nonlinear operator G(-, A) : Hi —> H in (6.1) have the Taylor expansion near u = 0 as follows G{u, A) = Gk(u, A) + o(||u||fc),

k > 2 an integer,

(6.21)

where Gk : Hi x • • • x Hi —> H is a k multilinear mapping, and we set Gk(u,X)

=

G1(u,...,u,X).

Let pj(X) G C be the eigenvalues (counting the multiplicity) of L\. Assume that /3i(A) is real, and

{

<0

ifA
(6.22) = 0

if A = Ao,

> 0

if A > Ao,

(6.22)

Re/^Ao) < 0, Vj>2. (6.23) Let ei(A) and e£(A) be the eigenvectors of L\ and L\ respectively corresponding to /?i(A), and L Ao ei=0,

Ll0e\=Q,

<ei,e\>H=l.

Let a=H, (6.24) where Gk is the ^-multilinear operator defined by (6.21). Then we have the following bifurcation theorems. Theorem 6.4 Assume (6.21)-(6.23), and k = odd in (6.21). Then the following assertions hold true. (1) If a > 0, then (6.1) bifurcates from (0, Ao) exactly two saddle points ^i(^)>^2(A) G Hi with Morse index one on X < Ao, and (6.1) has no bifurcation on Ao < A. (2) Ifa<0, then (6.1) bifurcates from (0,Ao) exactly two singular points vi(X) and i>2(A), which are attractors on Ao < A, and (6.1) has no bifurcation on X < XQ.

Dynamic Bifurcation Theory: Infinite Dimensional Case

161

(3) If a < 0 and \0 < X, there is an open set U C H with u = 0 £ U, which can be decomposed into two open sets U^ and f/^

Ctf n Etf = 0,

U = u\+V\,

such that F = dUi D dU^ is the stable manifold ofu = 0 with codimension one in H, Vi(X) £ Uf(i = 1,2), and lim \\u(t,
t—>oo

= 0, if
(i = 1,2),

where u(t,ip) is the solution of (6.1). (4) The bifurcated singular points vi{\) and U2(A) in the above cases can be expressed in the following form

t/li2(A) = ±\(3i{X)/a \Wk-»ex{\)

+ o(|/?i/a |VC=-D).

Theorem 6.5 Assume (6.21)- (6.23), k =even, and a ^ 0. T/ien the following assertions hold true. (1) (6.1) bifurcates from (0, Ao) a unique saddle point v(\) with Morse index one on X < Ao, and a unique attractor v(X) £ Hi on Ao < A. (2) If Ao < A, there is an open set U C H of u = 0, and U is divided into two open sets U* and [/£ by the stable manifold T of u = 0 with codimension one in H:

U = u\ + ux2,

u? n C/2A = 0,

r = dU? n dU$,

such that v(X) G J7j\ and lim \\u(t,
Ve

U?,

t—>oo

where u(t,ip) is the solution of (6.1). (3) The bifurcated singular points v(X) of (6.1) can be expressed as v(X) = -(/3 1 (A)/a) 1 /('=-i) e i + 0 (| / 3 1 / a |i/(fe-i) ) . If we replace the condition (6.23) by |Re/3j(A0)>0

i f l < j < n + l,

\Re/3j(A 0 ) < 0

if j>n

+ l,

(6.25)

then the bifurcated singular points of (6.1) are saddle points, which are characterized in the following theorem.

162

Bifurcation Theory and Applications

Theorem 6.6 Assume the conditions (6.21), (6.22),(6.25) and a ^ 0 in (6.24). Then the bifurcated singular points of (6.1) from (0, Ao) have Morse index n + 1 on A < Ao, and have Morse index n on Ao < A. Moreover, u = 0 has Morse index n on X < Xo and has Morse index n + 1 on XQ < X. Remark 6.2

In general, if we replace (6.24) by

(G{xex + h(x, Ao), Ao), e\)H = axk + o{\x\k),

ieR,

where h(x, A) is the center manifold function, then for a ^ 0 and k > 1, Theorems 6.4-6.6 are valid. Remark 6.3 The topological structure of the dynamic bifurcation of (6.1) is schematically shown in the center manifold in Figures 6.1-6.3.

(a)

(b)

Fig. 6.1 Topological structure of dynamic bifurcation of (6.1) when k = odd and a > 0: (a) A < Ao; (b) A > Ao. Here the horizontal line represents the center manifold.

Proof of Theorems 6.4-6.6. By the canonical reduction in Chapter 3, the bifurcation equation of (6.1) is as follows: — = A(A)x +
(6.26)

where h is the center manifold function satisfying h(x,X) = o(\x\) V A G K .

(6.27)

By (6.21) and (6.27), (6.26) can be rewritten, near A = Ao, as ^=(31(X)x at

+ axxk+o(\x\k),

(6.28)

163

Dynamic Bifurcation Theory: Infinite Dimensional Case

v

u=0

u=0

l

V

2

i I

1

,

1

(a) Fig. 6.2

—«

,

(b)

Topological structure of dynamic bifurcation of (6.1) when k = odd and a < 0.

» v^

«-

—•«

«

u=0

—«

»

u=0 t

(a)

u=0

<— v^

i

(b)

(c)

Fig. 6.3 Topological structure of dynamic bifurcation of (6.1) when k = even and a ^ 0.

where ax = {Gk(e1(X),X),e*1(X))H^a,

if A - Ao.

(6.29)

It is then easy to see that Theorems 6.4 and 6.5 follows from (6.28) and (6.29). The proof of Theorem 6.6 is trivial; we omit the details. The proof is complete. 6.2.2

Saddle-node

bifurcation

Now we consider a class of bifurcations as shown in Figure 4.1, called saddlenode bifurcations. First, we recall a new version of Theorem 4.2 as follows. Theorem 6.7 Assume that the conditions (6.2)-(6.5) hold true and the eigenvalues j3j{\) ^ 0 of L\ for all A < Ao. If equation (6.1) bifurcates from (0, AQ) on A < AQ a branch SA of singular points having nonzero

164

Bifurcation Theory and Applications

index near Ao, which is bounded in H x (—00, Ao), then there exists a point (u*, A*) e Hi x K1 with X* < Ao and u* ^ 0 satisfying ind(-(Lx

+ G),u*) = O,

atX = X",

(6.30)

and there are at least two branches F^ of singular points of (6.1) bifurcated from (u*,\*) on A > A*, i.e. ux->u*, for any ux£T$

(l<j<J,J>

if\->\*+0, 2).

Definition 6.1 We say that (6.1) has a saddle-node bifurcation from (u*,A*) on A > A* (resp. on A < A*) if u* satisfies (6.30), and (6.1) has at least two branches of singular points T* (1 < j < J, J > 2) of singular points of (6.1) bifurcated from (u*, A*) on A > A* (resp. on A < A*). The following theorem is motivated toward to applications in hydrodynamic bifurcation and stability. Theorem 6.8 true, and

Assume that the conditions (6.21)-(6.24) with a^O hold Vue#i,AeR\

j H=0, 2

\ H>C\\u\\

1/2,

(6 31)

'

for some Ai < Ao and a constant C > 0, where Ao is as in (6.22). If (6.1) has uniformly bounded attractors for bounded X, and all eigenvalues flj(X) T^O of L\ for A < Ao, then the following assertions hold true. (1) If k =even in (6.21), then (6.1) has a saddle-node bifurcation from (u*,X*) with Ai < A* < Ao, and the connected component C\ containing (u*, A*) of the following set

Tx = {(u,X)eH1 xR1 I Lxu + G(u,X) = 0,u^0} is nonempty at Ao < A < Ao + e for some z > 0. (2) If k =odd in (6.21) and a > 0, then (6.1) has at least one saddlenode bifurcation from (u*,X*) with X\ < A < Ao, and if all singular points on Tx are regular at Ao, then the connected components Cx of Tx containing all singular bifurcation points on X < Ao has at least two singular points of (6.1) at each X with XQ < X < XQ+S for some £ > 0.

165

Dynamic Bifurcation Theory: Infinite Dimensional Case

Proof. Based on (6.30) and (6.31), (6.1) has no nontrivial singular points at A = Ai. By assumption and Theorems 6.4 and 6.5, the bifurcated branches of (6.1) from (0, Ao) on A < Ao is nonempty and bounded in iJxR 1 . Hence, the existence of saddle-node bifurcation follows from Theorem 6.7. It is then routine to prove the rest par t of the theorem by calculatin g the indices of the singular points of (6.1) near Ao. The proof is complete. D 6.3 6.3.1

Bifurcation from Eigenvalues with Multiplicity Two An index formula

In order to investigate dynamic bifurcations of (6.1) from eigenvalues with multiplicity two, it is necessary to discuss the index of the following vector field at x = 0. u =

/anxl + a12xix2 + a 2 2 z| \ \bnxl + bi2XiX2+b22X%)'

{

'

'

We assume that the vector field (6.32) is 2nd order nondegenerat e at x = 0, which implies that a^x + b^ ^ 0. Without loss of generality, we assume that on ^ 0. Let A = a\2 - 4ana 22 , and if A > 0, let -au

+ y/E

- a i 2 - y/E 2an & = buot? + b12ai + 622,

i = l,2.

The following index theorem will be useful in studying dynamic bifurcation of (6.1) hereafter. Theorem 6.9 Let the vector field (6.32) be 2nd order nondegenerate at i = 0, and a n ^ 0. Then

{

0,

if&<0or

2,

ifauPt > 0 and an/32 < 0,

-2,

ifanPi

Pxp2 > 0,

< 0 and anp2 > 0,

(6.33)

166

Bifurcation Theory and Applications

We proceed in several steps as follows.

Proof.

1. When A = a\2 — 4ana 2 2 < 0> t n e following quadratic form is either positively or negatively definite: STEP

anx\ + a12xix2 + a22x\ > 0 (or < 0), V i e K 2 , i ^ 0 , depending on the sign of a n . Hence the following system of equations

f anxj + auxiX2 + a22x\ = -£ 2 (or = £2), \ bnx\ + b12xix2 + b22x\ = 0, has no solution for any e ^ 0, which implies that ind(u, 0) = 0, as A < 0. STEP 2. In the case where A > 0, the vector field u given in (6.32) can be rewritten as u =

fan(Xl - aix a )(xi - a 2 a; 2 )\ V 6111? + 612^12:2 + 622^2 )

(6.34)

Since u is 2nd order nondegenerate at x = 0, /3j • /32 ^ 0. By (6.34), u = (0, ±£2)*, with £ ^ 0, is equivalent to Pix\ = ±e2

xx = otix2,

(i = 1,2).

(6.35)

If /?i • /?2 > 0, then one of the systems in (6.35), for either +e 2 or —e2, has no solution, which means that the index of u at x = 0 is zero. STEP 3. When /?i • /?2 < 0, it is easy to see that a i ^ a 2 and A > 0. The vector field u — (ui,u2y given in (6.34) takes the following form:

ui=au(xi-a1x2)(xi-a2x2), < u2=

)2 [/?i(si

- a2Z2)2 + /32(xi - aix 2 ) 2

+ "/(xx - axx^ixi - a2x2)}, where 7 = — (2bnaia2 + 6120:1 + 6120:2 + 2622). Let /?i > 0,

/?2 < 0

if a n > 0,

/?i < 0,

/32 > 0

if a n < 0.

(6.36)

167

Dynamic Bifurcation Theory: Infinite Dimensional Case

Then the solutions y = (2/1,2/2) of (6.35) are given by ,

ifan>0, if an < 0,

\otiV2 I a2yf 1/2

)

( ±Pi1/2e

ifa n >0,

t ±/?2" ' e

if an < 0 .

6 37 ( (6.38) - )

Let zi=xi—aix2,

(6.38)

z2 = xi — a2x2.

Then the Jacobian matrix of u is given by

(

dui dui\ /dzi dzi\

~dz~l ~dz~2 I I ~dx[ ~dx~2 \ 9zi 9z 2 / \9xi 9x2 /

It is easy to see that

* fefe -<^(!::;)—>»• \9xi 5x2 / Hence we infer from (6.36) and (6.38) that / anZ2 anzi \ detZ?u(x) = (ai - a 2 )det 2(32zi + -yz2 20iz2 + jzi . \ (ax - a 2 )

2

(ai - «2) /

On the other hand, by (6.37) we deduce that ' . . f0 z

i = 2/f -

Q

i2/2 = S

(6.39)

2

1/0

I ±(a2-a0/32-1/2£ ± ± f ± (ai - a2)/3r1/2e ± zf = yf - a2yf = 1 [0 Therefore, by (6.39) and (6.40) we arrive at

deti ?tt (^) = {2a"/3l(ai-aa)"1(f)a:ba

if an > 0, ifan<0, if an > 0,

(6.40)

if an < 0, ifOll>0

'

(6.41)

168

Bifurcation Theory and Applications

By the Brouwer degree theory, we know that ind(u, 0) = deg(u, Br, xo),xo = (0,e2) £ Br

(6.42)

where Br = {x e R2\ \x\ < r}, and r > 0 sufficiently small. It follows from (6.41) and (6.42) that ind(u, 0) =sign det ~Du(y+) + sign det Du(y~) =2, for an/?! > 0 and an/3 2 < 0. We can obtain in the same fashion that ind(u, 0) = - 2 , for an/3i < 0 and onft > 0. Thus, the formula (6.33) is proved. The proof of the theorem is complete. • Remark 6.4

If an = 0 and bn ^ 0, we let A = b\2 - 4&11&12.

If A > 0, we define

51

-612 - \ M = —26^—'

-6ia + \[h &2 =

26 n ' A = ana? + ai2o:i + a22, i = 1,2. Then, the formula (6.33) is written as

{

0

if A < 0 or frfa > 0,

2

if bnPi < 0 and bnJ32 > 0,

- 2 if 6u/3i > 0 and bnp2 < 0. Remark 6.5 The index formula (6.33) shows that a two dimensional vector field, which is 2nd order nondegenerate at x = 0, takes only values {0, ±2} as its indices at x = 0. In fact, let u be an m-dimensional vector

Dynamic Bifurcation Theory: Infinite Dimensional Case

169

field, which is A;-order nondegenerate at x = 0, defined by /

a1

V

«=

xjl

• ••

xjm\

~ji

. . . ™jm

:

E \ji+-+jm=k

nm a

, X

ji-jmXl

m

(6.43)

/

then its index at x = 0 is given by

{

0

if m = odd,fc= even,

even

if m = even,fc= even,

odd if k= odd,Vm> 1. Moreover, the index of (6.43) at x = 0 takes values in the following range. 0, ±2, ••• , ± f c m - 1 m 1

! ±l,---,±k ~

6.3.2

if k = even, m = even, if k = odd,

Main theorems

Under the conditions (6.4) and (6.5), the integers m and r are the algebraic and geometric multiplicities of the eigenvalue /3i(Ao) of LA at A = AO. Here, we assume that m — r — 2, and the operator L\ + G(-, A) is second-order nondegenerate at (u, A) = (0,Ao), i.e. fc = 2 and Gk — G2Let aii(A) = (G 2 (ei(A) iei (A),A) )e *(A))tf, fl22(A)

= (G 2 (e 2 (A),e 2 (A),A),e*(A))H,

a12(A) = (G 2 (e 1 (A),e 2 (A),A)+G 1 (e 2 (A) ) e 1 (A),A),et) i f , 6ii(A) = (G 2 (e 1 (A) ) e 1 (A) ) A),e^(A)) ff , 622(A) = (G 2 (e 2 (A),e 2 (A),A),e^(A)) i/ , 612(A) = (G 2 ( e i (A), e2(A), A) + G 1 (e 2 (A), ei(A), A), e*2(X))H,

where G 2 is given by (6.21), and ej(A),e^(A) (i,j = 1,2) are respectively the eigenvectors of L\ and L*x near Ao: Lxei{\)

= A(A)e,(A), LAe*(A) = /?,(A)e*(A), i,j = 1,2.

170

Bifurcation Theory and Applications

Thus, we obtain a vector field

„ m _ fanWxl

+ a12(X)x1x2 + a22(A)a^\

(6.44)

By assumption, u0 is second order nondegenerate at x = 0 near Ao. According to Theorem 4.4, under conditions (6.4) and (6.5) with m = r = 2, if the two vectors (an, ai 2 , o22) and (bu, 6j2,622) are linearly independent near Ao, then there are at most three and at leat one bifurcated singular points of (6.1) on each side of A = Ao. By Theorem 6.9, the index of UQ given by (6.44) at x = 0 is either 0, or 2 or —2. Now we state the main dynamic bifurcations of (6.1) in each situation. We start with the case where ind(uo(Ao),O) = —2. Theorem 6.10 Let the conditions (6.4) and (6.5) with m — r = fc = 2 hold true, L\ + G(-,X) be second order nondegenerate at (u, A) = (O,Ao), and ind(uo(X0),0) = —2 for uo defined in (6.44)- Then (6.1) bifurcates exactly 3 saddle points with Morse index 1 from (0, Ao) on each side of A = A0. For other two cases, we need to introduce a notation. A set S{6) c R2 is called a sectorial region with angle 6 G [0, 2TT] , if S(9) is enclosed by two curves 71,72 starting with x = 0 and an arc F, and the angle between the two tangent lines L\ and I/2 of 71 and 72 at x = 0 is 0; see Figure 6.4. Let Sr(6) be the sectorial domain with angle 6 and radius r > 0 given by Sr(9) = {x€R2\

\x\
x=0

Fig. 6.4

and x G S{9)}.

171

Dynamic Bifurcation Theory: Infinite Dimensional Case

Theorem 6.11 Assume (6.4) and (6.5) with m = r = 2, and ft(A) = ft(A) near Ao. Let L\ + G(-, A) be 2nd order nondegenerate at (0, Ao), and UQ(\) be given by (6.44)- Then the following assertions hold true. (1) If ind(uo(Xo),0) = 2, then (6.1) bifurcates an attractor Ax with &imA\ < 1 from (0,Ao) on Ao < A, and A\ attracts a sectorial region Dr{6) in H with angle 9 € (TT, 2TT], and radius r > 0, where Dr{6) = {u = x + v£H\x

= xiei + x2e2 e Sr(9), \\u\\H < r}.

(2) The attractor A\ contains minimal attractors, which are singular points, as shown in Figure 6.5 (a) - (c). (3) If mrf(uo(Ao),O) = 0 and (6.1) bifurcates from (0, Ao) three singular points on Ao < A, then one of them is an attractor, which attracts a sectorial region Dr(9) with 0 < 9 < n, as shown in Figure 6.6 (a) and (b). Remark 6.6 If ft (A) ^ ft. (A) near A = Ao and A ^ Ao, then Theorem 6.11 may not be valid. Consider, for instance, the following equation dx — =Axx + G(x),

(6.45)

where x = {xi,x2)t, ai = 2 + V^,a2 = 2 - \/3, and Ax

~ { 0 a2\) '

Q(X\ _ fanxl + ai2xix2 + a22x\\ \bnx\ + b12x1x2 + b22xl)' Assume that G is second order nondegenerate at x = 0 and ind(G, 0) = 2. Let P : M.2 —> K2 be a linear transformation y = P x =

(6.46)

(PII pv) VP21P22J

such that

PG(P-Iy) = G{y) = M ~ 4 y i 2 / 2 ~ y") . \

2/12/2

)

172

Bifurcation Theory and Applications

Here we can choose the coefficients to make G as given here. Then, (6.45) is transformed into

{

-£• = 4Ay! + Xy2 + y\-

f

4yiy2 - y\,

(6.47)

dy , -ir 2 = -Xy1+y1y2. It is easy to see that y = (0, A) is a unique bifurcated singular point of (6.47) from (0,A0) = (0,0), and (6.47) has no attractor on Ao < A near (y, A) = (0, Ao), which has the topological structure as shown in Figure 6.7.

6.3.3

Proof of main theorems

First order approximation By the center manifold theorem, the dynamic bifurcation of (6.1) is equivalently reduced to that of the following equations

f - ^ = )8i(A)a:i + (G{xiei + x2e2 + h(x, A),e?(A))H,

I f

(6.48)

[ - ^ = fo{\)x2 + (G(xiei + x2e2 + h(x, A), e*2(X))H, where h(x, A) is the center manifold function satisfying h(x,X) = o(\x\) for x e R2. Thus, near (a;, A) = (O,Ao), (6.48) can be written as ^ = JxX + F(x,X)+o(\x\2), at

(6.49)

where

JxX

-{

/A (A)

0

0 /MA)JW~U(W'

PC \\ _ {anM^i

*

[X A)

\fx1\_((31(X)x1\

' ~\bu(X)xl

+ ai2(A)a;ia;2 + a22(X)x%\

+ bu(X)Xlx2

+ 6 22 (A)^ ) '

and atj,bij are as in (6.44). Since F is 2nd order nondegenerate at (x,X) = (0, Ao), the vector field on the right hand side of (6.49) is a perturbation of J\ + F near x = 0. Hence, by the stability of nondegenerate singular points and attractors (Theorem 5.4), it suffices to prove Theorems 6.10 and 6.11 for the following

Dynamic Bifurcation Theory: Infinite Dimensional Case

173

(a)

\PI

* (]<:

a

•

(b)

(c)

Fig. 6.5 (a) If (6.1) bifurcates one singularity p, the attractor Ax = {p}; (b) If (6.1) bifurcates two singularities pi and p2, then A\ = y U {pi,P2}, where 7 is the orbit connecting pi and p2\ and (c) If (6.1) bifurcates three singularities po, pi and p3, then .AA = 71 U 72 U {PI,P2IP3}, where 7i are the orbits connecting po and pi.

system, which is the first order approximation of (6.48) dx — = Jxx + F(x,\).

(6.50)

174

Bifurcation Theory and Applications

(a) Fig. 6.6

(b) (a) A = Ao, (b) Ao < A with {p} being an attractor.

Fig. 6.7

Proof of Theorem 6.10

The proof can be achieved by Theorem 3.10 and the following lemma. Lemma 6.1 Ifind(F{-,\0),0) = - 2 , then (6.50) bifurcates from (x,X) = (0, Ao) exactly three saddle points with Morse index one on each side of A = A0. Proof. By Theorem 6.9, as ind(.F, 0) = —2, the two vectors (an, ai2,022) and (6n, &12,622) are linearly independent. Therefore it follows from Theorem 4.4 that (6.50) has at most three bifurcated singular points from (0, Ao). We shall prove that (6.50) has just three bifurcated singular points on each side of A = AQ.

Dynamic Bifurcation Theory: Infinite Dimensional Case

175

It is known that ind (Jx +F,0)=

sign [ft(A) • p2(\)} = 1, if A ± Ao,

fc

Y,ind (JA + F, Pi ) + ind (JA + F,0) = ind (F(-,Ao),0) = -2, where p» (1 < i < k) are the bifurcated singular points of (6.50) from (0, Ao). Hence, if A ^ Ao, then k

53ind(JA+^Pi) = - 3 .

(6.51)

If the number k < 3 in (6.51), then one of the bifurcated singular points, say pi, of (6.50) satisfies that |ind(J A + J P,p 1 )|>2.

(6.52)

By the Brouwer degree theory, if D(Jx + F)(Pl)^0, we have |ind(J A +F,pi)| < 1 . Therefore, it follows from (6.52) that the Jacobian matrix of J\ + F at p\ is zero: D{ Jx + F)(Pl) = JX+ ( ^ ^ ] V dxi

= 0.

(6.53)

)

Let pi — (21,^2)1 then we infer from (6.53) that ft + 2 a n z i +C112Z2 = 0 , <

al2zi + 2a22z2 = 0, f32 + 2&22-Z2 + &12-21 = 0,

(6.54)

bi2z2 + 2bnzi = 0, which, together with J>pi +i ? (p i , A) = 0 , imply that px = 0, a contradiction to pi 7^ 0. Thus, we have shown that k = 3. Prom (6.51) and Theorem 4.4 we have md(Jx + F,Pi) = -l, 1 = 1,2,3,

176

Bifurcation Theory and Applications

which implies that pt (1 < i < 3) are saddle points with Morse index one. This proof is complete. • Poincare formula In order to prove Theorem 6.11, we need the following lemma, which establishes the Poincare formula; see [Chow and Hale, 1982]. L e m m a 6.2 Let v be a two dimensional Cr(r > 0) vector field with v(0) = 0. Then

(6.55)

ind(v,0) = l + Ue-h),

where e is the number of elliptic regions, and h number of hyperbolic regions. Here the elliptic, hyperbolic and parabolic regions E, H and P in a neighborhood U C M.2 of x = 0 are defined as follows; see Figure 6.8: E = {xeU\

S(t)x

and S(-t)x

H = {x e U\ S(t)x, -S(-t)x

-+ 0 as t -» oo},

$. U for some

P = {x £ U\ either S(t)x -> Q(t -> oo), S(-t)x or S(~t)x

-> 0, S{t)x <£ U, or S(t)x, S(-t)x

t>to>O}, $ U(t > t0), € U,Vt > 0}.

Fig. 6.8

Proof of Theorem 6.11 The proof is achieved in a few lemmas hereafter. Here, we always assume that (3{X) = /3i(A) = /?2(A) for A near Ao. First, by the homotopy invariance of indices, for A near Ao, md(F(; A), 0) = ind(F(; Ao), 0).

(6.56)

Dynamic Bifurcation Theory: Infinite Dimensional Case

177

Lemma 6.3 Let incLF(-, A),0) = 0 or 2. Then for X near Ao, the vector fields F(x,X) have k straight orbit lines with 1 < k < 3: aiXi+piX2=0,

a?+j3?^0,

i = l,--- ) fc,

(6.57)

where Oi = (Xi/[3i orOi = —fii/cti are the solutions of the following algebraic equation:

{

a22cr3 + (ai2 - b22)a2 + {an - h2)a - hi = 0, or

(6.58)

hicr3 + (&12 - an)a2 + (b22 - o,i2)cr - a2i = 0. Proof. When F(x,X) are second-order nondegenerate at x = 0 near Ao, °ii + ^li 7^ 0. We assume that an ^ 0. By the homogeneity of F(x, A), a straight line Z2 = ox\ is an orbit line of F(x, A) if and only if FaQc.A) Fi^.A) _ 6na;f + 612^12:2 + ^22^2 anx^ + ai2a;ia;2 + 022^2 _ ^li + bi2
Ifind(F(-, A0),0) = 2, then we have

(1) F(x, A) has no hyperbolic regions at x = 0, (2) F(x, A) has exactly two elliptic regions E\ and E2, (3) F(x,X) has no parabolic regions if k = 1, which is the number of solutions of (6.58), and has exactly two parabolic regions Pi and P2, if k>2, (4) the elliptic and parabolic regions E and P are sectorial regions E = S{0i),P = S(92) with 0 < 6i,62 < n,91+e2 = ir, and the edges of Sr(9i) and Sr(62) are the straight orbit lines of F(x,X); see Figure 6.9(a)-(c). Proof. Based on Lemma 6.3, we take an orthogonal coordinate transformation y = Ax with a straight orbit line of F(x, A) as the yi-axis. Under

178

Bifurcation Theory and Applications

this transformation, the vector field F(x, A) is changed into the following form F(y,X)=(~a^^y)yl

+

(6.59)

^ )

Since ind(.F(-, A),0) = 2, b\ j^= 0. Take another coordinate transformation as follows x\ = £>i?/i +£>22/2, x'2 = 2/2Then, by Theorem 6.9 the vector field in (6.59) is transformed into the following form, where for brevity, we omit the primes: F(x, A) = (%) = (a^ ~ «if a)(*i + « 2 x 2 ) \

V

\F2J

bxxx2

)

(6.60)

where a • b > 0, a\, a2 > 0. It is known that a coordinate system transformation preserves the topological structure of F. It is easy to see that (6.60) has the topological structure as shown in Figure 6.9(a) - (c) for o, b > 0 in (6.60). To derive the topological structure in Figure 6.9(a)-(c) of (6.60). Let Di, £>2, D3 and D\ be the 4 open quadrants in R2, and the two straight lines x\—ot\X2 = 0,x2 + a^xi = 0 also divide the plane R2 into four regions G R2 I xi - a.\xi > 0,xi +a2x2

Q\ = {(xi,x2)

2


> 0}, > 0},

2

Q3 = {(xi,x2)

G R I xi - axx2 > 0, xi + a2x2 < 0},

Qi = {(xi,x2)

G R2 I xi - axx2 < 0, xi + a2x2 < 0}.

It is easy to see that ' Fi>0

in Qi and Q4,

Fx<0

in Q2 and Q3,

<

F2 > 0 in Dx and D 4 ,

(o.bl)

Fi < 0 in D2 and D 3 . The properties (6.61) ensure that (6.60) has only two elliptic regions E\ and E2, with £1 C R2+ = {(11,12)1x2 > 0} and E2 c R i = {(xi,x2)|a;2 < 0};

see Figure 6.10.

Dynamic Bifurcation Theory: Infinite Dimensional Case

X

*2

(a)

179

2

(b)

(c)

Fig. 6.9 Toplogical structure of (6.60): (a) The number of straight orbit lines k = 1, (b) fc = 2, and (c) k = 3.

Thus, by Lemma 6.2, i*1 has no hyperbolic regions, and Assertions (1) and (2) are proved. By (6.58), it follows from (6.60) that the straight orbit lines Li(i = 0,1,2) of F(x, A) are given by LQ : X2 = 0,

L\\

X2 = crixi,

L2 : x2 = (T2X1,

and if o\,
£1,

c*i

(72 =

1

H£2,

a2

for some real numbers 0 < E\ < 1/ai and 0 < e2 < l/a2. Hence we have LiCQiUQ3

(i = 0,12).

(6.62)

180

Bifurcation Theory and Applications

. , - "

• • • - , . .

"

xr-a2x2

Fig. 6.10

Therefore, Assertions (3) and (4) follow from (6.62) and the radial symmetry of F(x, A). The proof is complete. • Lemma 6.5 / / ind(F, 0) = 2, then (6.50) bifurcates from (x, A) = (0, Ao) an attractor A\ on XQ < A, which attracts a sectorial region Sr(9) with n < 6 < 2TT. Actually, Sr(0) C (Ex U E2 U Pi) n Br, where E\,E2 are the elliptic regions of F, P\ is the parabolic region where all orbits of F reach x = 0, Br = {x S R 2 ||:r| < r}, 6 = 2TT — 6Q, and 9Q the angle of the parabolic region. Proof. We know that under an orthogonal coordinate system transformation, the linear operator

r _ (PM

0 \

is invariant. Therefore, without loss of generality, we take the vector F as given by (6.59). By Theorem 6.9, F(x, A) can be written as F

-\F2)

=

{

bx2(Xl-ax2)

)'

^ 6 - 63 )

where a • b > 0, a\ + a | ^ 0. We proceed with the case where a, b > 0 and a > 0. The other case can be proved in the same fashion. By Theorem 6.9 we know that cci > a > a2, which implies that thelineszi— o.iX2 = 0(i = 1,2),X2 = 0, andxi— ax2 = 0 are alternatively positioned in M2.

Dynamic Bifurcation Theory: Infinite Dimensional Case

181

Based on the definition of elliptic and parabolic regions, by Lemma 6.4 we obtain that lim Sx(t)x = 0, V x e £ i U £ 2 U Pu

t—>oo

(6.64)

where S\(t) is the operator semigroup generated by F(x,X). On the other hand, we obtain from (6.63) that for any x € E\ U E2 UPi, there is a to(x) > 0 such that Sx(t)x G D = {x G R2 I X! - to(x).

(6.65)

It is clear that Pi C D c Ei U E2 U D. Let L>(r) = {x £ D I |a;| < r}, £>(ri,r 2 ) = {s G D I 0 < n < |x| < r 2 } . Let T\(£) be the operator semigroup generated by J\ + F(-, A). It is known that for A > Ao all orbits of J\x are straight lines emitting outward from x = 0. Therefore, by (6.65) we deduce that Tx(t)x GD,

V t > 0,

x€dD,xj^0.

(6.66)

Now, we shall prove that for any A — Ao > 0 sufficiently small there are ri,r2,r3 > 0 with ri < r 2 < r 3 such that TA(*)x G 0 ( r i , r 2 ) , V i e D(r3),t > tx,

(6.67)

for some tx >0. We know that for A > Ao, the singular point x = 0 of Jx + F has an unstable manifold Mu with dimM" = 2. We take r% > 0 such that the ball Bri C Mu. Then, by (6.66) we obtain that Tx(t)x e D(ri,r2), V xeD(n),t>tx.

(6.68)

If (6.67) is not valid, then by (6.66) and (6.68) there exist An -> Ao + 0,t n —> 00 and {a:n} C D(r 3 ) such that I T An («„)!„ | > r 2 , V n > l .

(6.69)

Let xn —» xo G -D(^3)- Then by th4e stability of extended orbits (Lemma 5.1) and (6.69) there is an exgtended orbit 7 of F(-, Ao) with starting point XQ € D(r3) which does not reach to x — 0. This is a contradiction to (6.64).

182

Bifurcation Theory and Applications

It follows from (6.66) and (6.67) that D{ri,r2) is an absorbing set in a neighborhood U of D(ri, r 2 ). Hence, by the existence theorem of attractors (Lemma 2.2), for A > Ao, the set A\=w(D(n,r2),\), with 0 ^ Ax, is an attractor of (6.50), which attracts D(r3). Applying Lemma 5.1 again we infer from (6.64) that lim max |x| = 0.

A—>A0

z£Ax

Thus A\ is a bifurcated attractor of (6.50) from (0, Ao). We can deduce from (6.65) that A\ attracts a sectorial region Sr(6) c E1UE2UP1, with 6 = 2TT — #0, where #0 is the angle of the parabolic region P%. The proof is complete. • Lemma 6.6 The attract A\ has dimension dim A\ < 1, and Ax contains minimal attractors consisting of singular points. Proof. It is clear that .4^ contains all singular points of (6.50). We shall prove that Ax does not contain extended orbits homeomorphic to S1. By Lemma 6.3, all singular points of (6.50) must be in the straight orbit lines L of F(x, A), and L are invariant sets of (6.50) which consist of orbits and singular points. Use the method as in the proof of Lemma 6.4, for any straight orbit line L we can take an orthogonal coordinate system transformation with L as its x\—axis. Thus, the vector field F(x, A) take the form of (6.63), and the singular point xo = (x®, x^) of Jx + F on L is given by xo = {x°1,xo2) = (-l3(\)/a,O), and the Jacobian matrix of Jx + F at XQ is given by

D +F

^ ^=(~T\i-hm)-

<6-7°)

Hence, on each straight orbit line there is only one singular point XQ of Jx + F, and there are two orbits 71 and 72 in L reaching to XQ. Moreover one of 71 and 72 connects from x = 0 to x$. It follows that the attractor Ax containing all singular points has no closed extended orbits. By the Poincare-Bendixon theorem we obtain that dim.4,\ < 1.

Dynamic Bifurcation Theory: Infinite Dimensional Case

183

When J\+ F has three singular points zt (1 < i < 3), by Theorem 4.4 they are regular, and ind (Jx + F, zi) = - 1 , ind (JA + F, z2) = ind (Jx + F, z3) = 1.

(6.71)

It follows from (6.70) and (6.71) that z\ is a saddle point, z2 and z3 are attractors. In this case the attractor Ax has the structure as shown in Figure 6.5(c). When J\ + F has two singular points z\ and Z2, md(Jx + F,Zl)=0,

md(Jx + F,z2) = l,

which implies, by Theorem 4.4 and (6.70), that z2 is an attractor, and z\ has exactly two hyperbolic regions. Thus, Ax has the topological structure as shown in Figure 6.5(b). When Jx + F has only one singular point z, then ind(JA + F,z) = l, which implies by (6.70) that z is an attractor, and Ax = {z} has the topological structure as shown in Figure 6.5(a). The proof is complete. • Note that Lemmas 6.4-6.6 are still valid for (6.49), Assertions (1) and (2) in Theorem 6.11 follows from Lemmas 6.5 and 6.6. Assertion (3) of Theorem 6.11 is an immediately consequence of the following lemma. Lemma 6.7 Ifind(F,Q) = 0, and (6.50) bifurcates three singular points from (0, Ao) on \Q < A, then one of them is an attractor which attracts a sectorial region Dr(6) with 0 < 6 < IT. Proof. By Theorem 4.4 the three bifurcated singular points pi (1 < i < 3) are nondegenerate, and ind (Jx+F,Pl)

= ind (Jx+F,p2)

= - 1 , ind (JA +F,p3) = 1.

Then as in the proof of Lemma 6.6, we can deduce that p3 is an attractor. Since pi and p2 are in the other two straight orbit lines which enclose the parabolic region P, the singular point p3 € P and attracts a domain P n Br = Dr(0) for some r > 0. The proof of the lemma is complete. •

184

Bifurcation Theory and Applications

6.3.4

Case where

k > 3

Now we consider the case where the operator L\ + G(-,X) is kth-order nondegenerate at (0, Ao) with k > 3. Let

"o=(U>-^f-j^-;;^\

(6.72)

where a

h-Jk(X)=
,ejlc,X),e*(X) >H,

» = 1,2,

and Gk : # i x • • • x H\ —» H is thefc—multilinearoperator defined in (6.21). The following theorem is the generalization of Theorem 6.11 to the case where k > 3. Theorem 6.12 Assume (6.4) and (6.5) with m = r = 2, and /?i(A) = /?2(A) near Xo. Let L\ + G(-, X) be kth order nondegenerate at (0, Ao), and uo(X) be given by (6.72). If ind(uo(Xo),0) > 1, then (6.1) bifurcates an attractor A\ with dim.4.\ < 1 from (O,Ao) on Ao < A, and A\ attracts a sectorial region Dr(6) in H with angle 9 £ (0,2TT], and radius r > 0. The proof of Theorem 6.12 is based on the fact that if ind(uo(Ao), 0) > 2, by the Poincare formula (Lemma 6.2), there is at least one pair of elliptic regions E\, £7 and a parabolic region P, which may be empty, of «o(A) at x = 0, as shown in Figure 6.11, such that lim S\{t)x = 0,

t—>oo

Vz G Ex U E2 U P,

where S\(t) is the operator semigroup generated by ito(A). Then Theorem 6.12 can be proved in the same fashion as the proofs of Lemmas 6.5 and 6.6; we omit the details. 6.3.5

Bifurcation to periodic solutions

In this subsection, we consider the case where m = 2, r = 1 and k = odd > 3 in (6.4) and (6.21). We also assume that ^(Ao)^0,

V j > m + 1.

(6.73)

Since m = 2 and r = 1, the two eigenvectors vx and u2 of L\ at A = Ao

Dynamic Bifurcation Theory: Infinite Dimensional Case

p

e

185

^ \ \

Fig. 6.11

enjoy the following properties; see Section 3.3.2: LXovi=0, L*Xov2=0,

Lx0v2=av!, LXov{=av*2,

J>0,

i=j,

[ = 0,

i+j.

a + 0,

Let a £ R be the number denned by a = (Gk{vuXo),vZ)H,

(6.74)

where Gk is as in (6.21). Then, we have the following bifurcation theorem of periodic orbits from the real eigenvalues with m = 2 and r = 1. Theorem 6.13 Assume the conditions (6.6) and (6.73) with m = 2,r = 1 and k = odd > 3. Let a be given by (6.74). If a • a < 0, then (6.1) bifurcates from (u, \0) = (0, Ao) a periodic orbit.

Proof. STEP 1. By the center manifold theorem, it suffices to consider the bifurcation of the following equations f ^ \

dx

{~i=

= /31(X)x1 + ax2 + (G(x + h(x, A), A),v\(X))H,

/32(X)x2 + (G(x + h(x,X),\),vl(X))H,

<6-75)

186

Bifurcation Theory and Applications

where x = xiu^A) + x2v2(\), h(x, A) is the center manifold function, Lxv2(X) = f32{\)v2{\) + at;!(A), L*xv*2(\) = (32(\)v*2(\), L*xvl{\)=(31(\)vl+av*2{\), v*(\))H=6ij, {Vi(X),

a / 0,

where Sij is the Kronecker symbol. By (6.21) and (6.74), equation (6.75) at A = Ao reads as

f-W-(™).

(^6)

where F1(x)=ax2+0(\x1\k,\x2\k), F2{x) = a4 + 0(\Xl\k+1, \x2\k, |o: 2 | fc -Vi|, • • • , N • l^l"- 1 ). Since a • a < 0 and k = odd > 3, we have ind(F,0) = l.

(6.77)

STEP 2. We now prove that the number of elliptic regions of F at x — 0 is zero, i.e. e = 0. Assume otherwise, then there exist an orbit 7 of (6.76) connected to x = 0, i.e.

lim S(t)x = 0, Vz G 7,

t—*oo

where S'(i) is the operator semigroup generated by (6.76). Let 7 can be expressed near x = 0 as ^2 = /(zi), (Zi,^) £7-

It follows from (6.76) that for any ( i i , ^ ) S 7 ax2 _ axk + Odxtl^1, l n H / ^ ) ! * - 1 , \xl\2\f{x1)\k-'i, • • • , ^ l ^ 1 ! / ] ) dxi o/CiO + OdnlM^xi)!*) Thus we obtain af{xl)f'{x1) + 0(\Xl\k, |/| f c )/' = mf + 0(|xi|*+1, I x i l ^ i / H ,

(6-78)

187

Dynamic Bifurcation Theory: Infinite Dimensional Case

which implies that f{x) = f3xm + o{\x\m),

2 < m = ^~-


(6.79)

Therefore, from (6.78) and (6.79) we get a = am/32, where a • a < 0, m > 2 and /? ^ 0. It is a contradiction. Hence e = 0. STEP 3. By (6.77) and the Poincare formula (6.55), h = 0. Therefore i = 0 must be a degenerate singular point, and is either (a) a stable focus, or (b) an unstable focus or (c) a singular point having infinite periodic orbits in its neighborhood. The case (c) implies a bifurcation to periodic orbits for (6.75). For the case (a), x = 0 is an asymptotically stable singular point of (6.76). Then by Theorem 5.2, the equation (6.75) bifurcates from (x, A) = (0, Ao) an i^-attractor SA on A > AoFor the case (b), x = 0 is an asymptotically stable singular point of the vector field —F(x), therefore the vector field

_(ft(A)

0

\x_F(x)

bifurcates from (0, Ao) an 5>1-attractor EA on A > Ao, which implies that (6.75) bifurcates from (0,Ao) on A < Ao an S1- repelor EA, which is an invariant set. STEP 4. Now, we need to prove that the S^-invariant set EA contains no singular points. Consider the following equations /?i(A)x 1 +aa: 2 +0(|a:i|M a :2|*)=0,

(31(\)x2+axk1+O(\x1\k+1,\x2\k-i\x1\i)=O. Hence

X2 = aaxi'1

-/3i(\)a-1x1+0(\x1\k,\p1\k), - ft(A)/%(A) + 0(\Xl\k, |*i|fc|/?i|) = 0.

(6.80)

By aa < 0 and /?i(A)/?2(A) > 0, (6.80) has no solution near (x, A) = (0, Ao). Hence, there is no singular points in SA, which means EA must contain a periodic orbit. The proof is complete. •

188

6.4

Bifurcation Theory and Applications

Stability for Perturbed Systems

In this section, we shall present two theorems on the bifurcated attractor s of systems with perturbation , which will be used in studying the bifurcation phenomena for the Taylor problem; see Chapter 10. One theorem is on the general case and the other on the case with simple eigenvalues.

6.4.1

General case

Consider the perturbe d equation of (6.1) given by -

= (L>+Si)v + G(v,\),

(6.81)

where L\ and G ar e as in (6.1), and Sf^ € Ba is a perturbatio n operato r of L\, depending continuously on A € R, such that (6.82)

\\Sio\\B,<e.

Here Ba = C(Ha, H) is the space of all linear bounded operator s from Ha to H, i.e. Ba = £(Ha, H) = {B : Ea - H linear bounded}. We always assume that 0 < a < 1. By the spectra l theory of the linear complete continuous fields [Kato, 1995], under conditions (6.5) and (6.6) as s > 0 sufficiently small, there exists a paramete r AQ approximating to Ao such that the eigenvalues {/S^(A)} of L\ + S^ satisfy

{

< 0

for A < A£ (6.82)

= 0

for A = A^

>0

for A > Ag

ifl
i f j > m i + l, >Rej3?(A§)<0 where 1 < mi < m, and m is as in (6.4). The main theorem in this section is the following attracto r stability theorem for the perturbe d equation (6.81). Theorem 6.14 Let the conditions in Theorem 6.1 hold true. Then there aree > 0 and 5 > 0, such that as 5 | e Ba satisfies (6.82) and 0 < A — AQ < 5, the following assertions hold true.

Dynamic Bifurcation Theory: Infinite Dimensional Case

189

(1) There exists a neighborhood U C H of u = 0 such that the equation (6,81) has an attractor S | C U such that dimS^<m,

0 i S|,

and Ti\ attracts an open and dense set UQ of U. (2) Each element u\ e E^ can be expressed as v\ € Eo, < lim vx = 0,

(6.84)

A->A 0

lim |K||/K||=0, A—*AQ £—.0

where Eo is as in Theorem 6.1. (3) If u = 0 is globally asymptotically stable for the unperturbed equation (6.1), and (6.81) has a global attractor for any A € M., then S | attracts any bounded open set of H. Proof. We shall apply Theorem 5.4 to prove this theorem, and remark that Theorem 5.4 is still valid if the domain of each vector field depends on the field. We know that the eigenvalues {/3f.(\)} of L\+Sl depend continuously on the operator 5 | ; see Ch. IV - 3.5 of [Kato, 1995]. Therefore, by condition (6.5) there are numbers £ > 0 and 6 > 0, as S^ e Ba satisfies (6.82) and | A — Ao| < S, the eigenvalues of L\ + S^ satisfy RePj(\) < - 7 ,

V j > m + 1 for some 7 > 0.

(6.85)

Let L\ = Lx + S{. By (6.4), (6.5) and (6.85), the spaces Hx and H can be decomposed respectively into

JJi = ££©££, and Hi = Eie@E^, H = Et®E$£,

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Bifurcation Theory and Applications

such that dim Ei = dimi?*£ = m, and for i = 1,2,

£$ = L\Er.E?-+E*, The operators £^ and £Ae have eigenvalues {/?i(A),--- ,/?m+i(A)} and {/3f (A), • • • ,/?m+i(A)}, while the operators £A and £A£ have eigenvalues {#j(A) | j > m + 2} and {/3?(A) | j > m + 2} respectively. Thus, by the center manifold theorems in Section 2.2.4, there are C 1 functions h : O —» ^(fl), ^ : OE — JB2A£(^), with O c E± and C7£ C ^ ^ being neighborhoods of a; = 0, such that the attractors of (6.1) and (6.81) in the locally invariant manifolds

Mx = {(x,h(x))\xeOcE?}, £ M x = {(x,h£(x))\xeOecE?e}, are homeomorphic to the attractors of the following ordinary differential equations respectively dx — = £$x + PG{x + h(x), A),

XGOC

£j\

(6.86)

OJXJ

^

= C\ex£ + PeG(x + he(x), A),

x £ e O £ C Et,

(6.87)

where P : Hi —» E± and P£ : H\ —> ^ £ are the canonical projections. It is known that

•hm ££ = £*, , lim£^=^,

(6.88)

lim h£ — h.

In addition, by Theorem 6.1, (6.1) bifurcates from (u, A) = (0,Ao) an attractor EA C MA for A > Ao. Hence it is easy to see that as Ao — 5 < A < Ao the system (6.86) has a unique trivial attractor x = 0 in O, and as Ao < A < Ao + S, (6.86) bifurcates from (x, A) = (0, Ao) an attractor, which is denoted by SA C O, and SA attracts O \ {0}. By the attractor stability theorem, Theorem 5.4, we infer from (6.88) that as £ > 0 is sufficiently small, (6.87) has an attractor EA in Oe for all

Dynamic Bifurcation Theory: Infinite Dimensional Case

191

| A — Ao | < S, such that E^ —» Ex

a s e - t 0.

Obviously d i m £ | < m. By (6.83) we deduce that 0 £ S | for all A > Ao. Thus, Assertion (1) is verified. By Assertion (3) of Theorem 6.1, Assertion (2) follows from (6.88) and (6.89). Finally, Assertion (3) follows D from Theorem 5.4. The proof is complete. Remark 6.7 In the above proof, the center mainfold theorem is used by considering perturbation of (6.1) with 5 | ; see [Temam, 1997] and [Henry, 1981]. 6.4.2

Perturbation at simple eigenvalues

We now consider the case where m = 1 in (6.4), i.e. the first eigenvalue /?i(A) of L\ is simple. Let VQ £ Hi be the eigenvector of L\ at A = Ao: LXov0 = 0,

||VD||

= 1.

We assume that L\ is symmetric, m = 1 in (6.4) and (6.5), G(u) is bilinear, (G(u,v),v)=0, \/U,VGHO.

(6.89)

Here as G is bilinear, one can write G as G(-, •), which is linear for each argument. We remark here that the condition (6.89) is necessary for the uniqueness of a pair of equilibrium points in the attractor S^ in the floowing theorem; otherwise, E | may contain more than two points. Theorem 6.15 Let the conditions in Theorem 6.14 and the conditions (6.89) hold true. Then we have the following assertions. (1) The attractor Yj\ of (6.81) obtained in Theorem 6.14 consists of exactly two equilibrium points of (6.81), i.e. T,\ = {w^u^} such that

{

u^ = ai{\,e)vQ+wi{\,e), "2 = -a2(A,£)uo + w2(X,e), Wi{\,e) = o(|ai(A,e)|) G Hi,Oi € E, i = 1,2,

(6.90)

192

Bifurcation Theory and Applications

where at > 0, and a(X, e) —> 0 as e —» 0, A —> Ao. (2) There is a neighborhood U C H of u — 0, and U can be decomposed into two open sets U± and U^:

u = u\ + ux2,

t/A n t/2A = 0, o £ du? ndu£

with uA £Uf (i = 1,2), such that

lim ||t>(£,v?)-uA|| = 0 ,

as(peU?

(i = 1,2),

t—»oo

where v(t,(p) is the solution of (6.81) with v(0,ip) = tp. Furthermore, if (6.81) has a global attractor for each X eR1, then U can be taken as any bounded open set of H. Proof. It suffices to prove that there are exactly two nonzero singular points of the operator U\ + G in H\ near u = 0 as e > 0 and A — Ao > 0 sufficiently small. We shall proceed by using the Lyapunov-Schmidt method. The following equation L\u + G(u, A) = Lxu + Sexu + G(u, A) = 0

(6.91)

can be reduced to the following one-dimensional equation f3f(\)x + P?G(xv£0 + y£(x, A), A) = 0,

x e R,

which can be rewritten as (3£1(\)x

+ P 1 G ( x v 0 + y ( x , \ ) , \ ) + g ( x , e ) = 0,

x £ l

(6.92)

where

jg(x,e)=o(\x\),

(6.93)

and y(x, A) e E% is the solution of the equation £$y + P2G(xv0+y,\) = 0.

(6.94)

Since the linear operator L\ is symmetric, the eigenvalues {/?fc(A) | k = 1,2,... } of L\ are real and complete, and the eigenvectors {vk} (v\ = VQ)

193

Dynamic Bifurcation Theory: Infinite Dimensional Case

constitute an orthogonal basis of H. Then we have y = ^2fL2yiv3> an( ^ (6.94) becomes oo

0j(X)yj

+ {G(xvo

+ J2ykvk,X),vj)H

= O,

(6.95)

j = 2,3,---.

fc=2

By (6.89) we find OO

OO

OO

2

(G(xv0 + ^2 VkVk, X),VJ)H = <*jX + ^2ckxyk fc=2

fc=2

+^

(6.96)

cikykyk

i,k=2

where a^, c^, c^ are constants, and atj = { G ( V O , \ ) , V J ) H ,

j = 2,3,---.

It follows from (6.5), (6.95) and (6.96) that yJ = \l3j\-1aja?+o{xt),

j = 2,3,---.

(6.97)

On the other hand, we see that OO

PiG(xvo + y,X) -{G(xvQ +

^2ykvk),v0)H fc=2

=(by (6.89)) OO

=

-^2aixyj 3=2

OO

+ Yl c^ykyk-

(6.98)

i,k=2

By (6.97) and (6.98), (6.92) reads P\{\)x - ax3 + g(x,e) + o{\x\3) = 0,

(6.99)

where

a = f> J r 1 a J 2 >0, 3=2

as G(VQ, X) ^ 0. Because G is analytic, so is g(x,e). By (6.93) we find g(x, e) = b(e)x2 + o{x2), where b(e} —> 0 as e —> 0.

(6.100)

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Bifurcation Theory and Applications

Then it follows from (6.83), (6.99) and (6.100) that there are only two nonzero solutions of (6.99) near x = 0 as A > Ao and e > 0 sufficiently small; they are given by

K.)^>l.) +W W ,

'

(6.101)

2(7

Thus, the proof of the theorem is complete.

•

Remark 6.8 The coefficients at{\, e) (i = 1,2) in (6.90) are the numbers given by (6.101); namely <*1 =

^ ACT

~,

"1 =

^

lf b

^

= KS) > 0>

2,(7

ai =

^ , a2 = \ if b = b(e) < 0. la la It is clear that if b ^ 0, e.g. b > 0, then the attractor «£ = ~~a2Vo + W2 is bifurcated from (0,A0), and the other one, u\ — OL\VQ +W\ is not, i.e. lim u\ = - wo + o | - )

T^

0.

In fact, the number b(e) in (6.101) can be expressed as &(e)= * , where UQ* is the first eigenvector of Z,|*. By the saddle-node bifurcation theorem, Theorem 6.8, b{e) ^ 0 implies that (6.81) has a saddle-node bifurcation on A < AQ as shown in Figure 6.12.

6.5

Notes Theorem 6.1 is first proved in [Ma and Wang, 2004e], and Theorems 6.2 and 6.3 are new here. Sections 6.2 and 6.3 are based on [Ma and Wang, 2004a; Ma and Wang, 2004b], and Section 6.4 is taken from [Ma and Wang, 2005d].

Dynamic Bifurcation Theory: Infinite Dimensional Case

:'

Fig. 6.12 cation.

^ ^

195

\

When 6(e) ^ 0, the attractors (6.90) are originated from a saddle-node bifur-

Chapter 7

Bifurcations for Nonlinear Elliptic Equations 7.1 7.1.1

Preliminaries Sobolev spaces

Let Cl C M.n be an open set, and Ck(Q) (k > 1) the space of all fc-th differentiable functions on $7, endowed with the norm ||u||Cfc = V^ sup\Dau\, |a||
where a — (ai, • • • ,an), at > 0 (1 < i < n) are integers, \a\ = J3"=i a«i and

Du=— dxi'

D uy

92u

dxidxj'

Let

c£(n) = {«e cfe(n) | suppu c n}, where supp u is compact support set of u defined by suppu = closure of {x G Q | u(x) ^ 0}. Let Lp(£l), 1 < p < oo (or L°°(fi)) be the space of real functions denned on Q with the p-th power integrable (or essentially bounded real functions), endowed with the norm

\HLP = \J \u\pdx\ , 197

198

Bifurcation Theory and Applications

(or, for p = oo, II U ||L°°

= ess.sup a

|W(X)|).

For 1 < p < oo and nonnegative integer k, the Sobolev spaces are defined as follows Wk'p(Q) = {ue L p (fi) | Dau G L p (ft), |a| < fc}, with the norm

\\Dau\\Lr.

\\u\\w,.P = £

\<x\
As p = 2, we write HkQ.) = Wfc'2(fi) which are Hilbert spaces with the scalar product

(u,v)Hh = V

/

Dau-Davdx.

Jn

\a\
We define WofclP(ft) = the closure of C£°(fi) in Wrfc'p(fi),

For integers /c > 0 and 0 < a < 1, we define the spaces = {u G Cfc(fi) | [Z)"«]a < oo, |/?| = fc}

Ck'a(n) with the norm

||u||cfc.° = \\u\\ck + Yl [DPu]a, |/3|=fc

Ha= sup

K*)-y,

Embedding Theorems. Lei fi C R n 6e a bounded domain and 1 < p < oo, then

Theorem 7.1

^'P((1)CL«(S1)

VI < ? < — ^ ,

W0fe'p(O) C Lq(Q)

VI < g < oo,

P

1 01

W£' (Q.) C C" ' ^) \/kp>n,

n>fcp,

n = kp,

m+a =

k--.

199

Bifurcations for Nonlinear Elliptic Equations

Moreover, the inclusions are continuous, namely TIT)

\\u\\Li < C\\u\\wk-r

ifl
||w||c'm,a < C||u||^fc,P

ifm + a = k

IX —

T-, Kj) 77.

,

kp > n.

where C = C(n, ft, q). Theorem 7.2 have

Let Cl C M.n be Lipschitz, not necessarily bounded, then we

Wk'p(Q.) c L"(Q)

q = - ~ - , n > kp, n — kp m+a = k--, kp > n,

Wk'p(n) C Cm'a(Q) and the inclusions are continuous. Compactness Theorems.

Theorem 7.3 Let Q c M.n be a bounded domain, then the following embeddings are compact W^(n)^L"(Q),

Q<^~,

W^p(n)^L"(n),

q
W*'p(£l) ^ C°
n>p, n=p,

a < 1- -,

p > n.

Interpolation Theorems. Theorem 7.4 follows

For p < q
where e > 0 is an arbitrary real number. Theorem 7.5 Let Q C K" be a Lipschitz domain and e0 > 0 be given, then, for any e (0 < e < EQ) we have

H > < C[e\u\m,p + e-»\\u\\LP},

\ukP= E \l \Dau\pdXP, |a|=fc

Un

J

200

Bifurcation Theory and Applications

where C = C(m,p,£ 0 ,ft), 0 < j <m-l,

n= ^ j .

Trace Theorems. Theorem 7.6 Let Q, C Kn be a domain with class Cm+1. u € W£'p(n), p > 1, we have

Then for any

Dau |an= 0, a.e., V|a| < m - 1. 7.1.2

Regularity

estimates

We consider the linear elliptic equation given by

{

n

n

(7.1) - ^2 a^DijU + ^2 biDiU + cu = f(x), x E Q, u\da = V,

(7.2)

where 0 C Mn is a bounded domain, and n

^2 aijtej > m2 (A > 0), Var e U, £ € M". (7.2) The following theorems provide the Schauder global estimates and the Lp~estimates for the equation (7.1). Global Schauder Estimates. Theorem 7.7 Let Q C Rn be of class C2'a, and aijr bit c, f G C°>a(Q.),