The Science of Hysteresis : 3-volume set

Example Consider the system

L (~)x y(t) rl(t)

-

fit, x,y(t)),

=

(I)(r/(t)),

=

(F[rl(to)]x)(t),

(2.83)

with

L(p) = p~ + alp ~-1 + . . . q- a~, where x(t) and y(t) are inputs and outputs of the Preisach nonlinearity and the functionf(t, x, y) is supposed to be continuous and satisfiesflt, x, y) _-fit + T, x, y) for some T > 0. We split the set of all T-periodic solutions (x(t), r/(t)) = (x(t + T), tl(t + T)) (t>~to) of system (2.83) into disjoint equivalence classes, assigning two periodic solutions to the same class if they have the same first component and their second components ~lj(t) satisfy (I)(t/1 (t)) = q)0/2(t)), i.e. the variable states tlj(t) define the same periodic output of the Preisach nonlinearity. We denote the equivalence class of periodic solutions (x(t), rl(t)) by Ix(t), y(t)] where y(t) = (I)(r/(t)) for all its representatives. Classes [x(t), y(t)] are natural for several reasons. Firstly, the differential equation in system (2.83) contains inputs and outputs rather than

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184

variable states r/(t) of the hysteresis nonlinearity. This situation is typical since states of the hysteresis nonlinearity are basically considered as values that one can neither observe nor control. Secondly, the second components of all periodic solutions (x(t), q(t)) that belong to the same class Ix(t), y(t)] are related by very simple explicit formulas, which we discuss below. Let for all t, x e R

[fit, x, y)l<<.qlxl + b,

y e [0, 1],

(2.84)

(here [0, 1] is the range of output values of the Preisach nonlinearity). Suppose that all the zeros of the polynomial L(p) are different from the numbers nwoi (n 9 7/)where w0 = 21t/T. Set

(

k - max IL(nwoi)1-1,

kl - T -1/2 IL(O)1-2 + 2

neZ

~ )1/2 IL(nwoi)1-2 . n--1 (2.85)

In fact, these formulas define the norms k = IIH IIC2----~C2 and kl = IIH IIL2-,C of the solution operator H = H(T) sending a function u = u(t) to the solution

x

Hu of the T-periodic problem for the linear Eqn. L ( d ) x = u(t) in the

spaces L2 = L2[0, T] and C - C[0, T]. To keep things simple, we shall always assume in this subsection that the measure It of the Preisach nonlinearity has a bounded density h = h(~, t) with respect to the Lebesgue measure. Define the scalar function ~(r) = It({(~,//)'~ < - r , r
r>~0,

(2.86)

and denote by Ru the least nonnegative solution of the Eqn. ~(r) = 0 if any exists. If ~(r) > 0 for all r>~0 then we set Ru = c~. The value Ru plays an important role in all the statements below. PROPOSITION2.3.16. Let estimate (2.84) hold and

k l b V/-T 1 - qk < Ra.

qk < 1,

(2.87)

Then there exists a one-parameter continuum of classes [x,~(t), y2(t)] (0~<2~1) of T-periodic solutions to system (2.83), whose classes are different for different 2. Assume additionally that the density of the measure It satisfies h(~, fl)<.v(fl- ~),

~ < fl,

(2.88)

v(s) ds < cxz.

(2.89)

where v is integrable on the positive semiaxis, i.e. v~-

f

OO

0

185

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

Moreover, assume that the function fit, x, y) satisfies the Lipschitz condition Lf(t, Xl, yl) - f i t , x2, y2)l~qlxl - x21 if- qllyl - y2l

(2.90)

for all t, xj ~ JR, O<~yj<~l and 1 - qk ql < 2v~kl ~/T"

(2.91)

Then the set of all T-periodic solutions of system (2.83) is the join of the classes [x,~(t), y,~(t)] (0~<2~<1) above and there are numbers cj such that - 221,

IIx2~ (.) - x~ 2 ( . ) I I c ~ c l l , ~ l

IIY21(') - Y22(') IIc~c2121 - 221,

2j e [0, 1]. (2.92)

For example, consider Eqn. (2.9) of Section 2.1, i.e. x" + x = sin(~/2t) + p(2y(t) - 1),

(2.93)

where x and y are the input and output of the Preisach nonlinearity, p > 0. Here T - ~/2~z, estimate (2.84) holds with q = 0, b - 1+ p and the quantities (2.85) equal

k-

1 ~ v/2~z+ sin(V~n) ~. 0.829.

1,

kl - ~

1 - cos(~/2~z)

Therefore the first of the estimates (2.87) holds. Assume that the density h = h(c~, fl) of the measure/~ is defined by h(~, fl) = 1/(2r 2)

if

- r~<~ < fl<<.r;

h(~, fl) = 0

otherwise.

Then R~ = r and the second of the estimates (2.87) has the form r

> ~2~z 2 + ~/2~z sin(~/27r)

1+ p

4(1 - cos(~/~z))

1.747.

(2.94)

By Proposition 2.3.16 this estimate implies that Eqn. (2.93) has a oneparameter continuum of classes [x,~(t), y2 (t)] of periodic solutions with the period ~/2~z. Furthermore, the Lipschitz estimate (2.90) holds with ql = 2p and from

h(c~, fi)<~l/(2r 2)

if

0 < f l - c~<~2r,

h(c~, fi) = 0

otherwise

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186

it follows that Voo = 1/r, and therefore estimate (2.91) takes the form 4/t/2~ 2 q- ~/-2~sin (w/2~) P > V 4(1 -- c--0-os(---~~)) ~ 6.9876.

r

(2.95)

The second part of Proposition 2.3.16 implies that if estimates (2.94) and (2.95) are valid, then the set of all v/2~-periodic solutions of Eqn. (2.93) is the join of the classes [x2(t), y,l(t)] (0~<2~<1). In fact, these classes are related here by the simple formulas

x,~(t) = xo(t) + 2p6(2),

y~(t) = yo(t) + 6(2),

2 e [0, 1],

where the continuous function 6 = 6(2) strictly increases and 6(0) = 0. Note that estimate (2.94) implies r>1.747. One can use operator norms different from (2.85) to conclude that all Tperiodic solutions of system (2.83) constitute a one-parameter set of classes [x~(t), y,l(t)]. For example, that conclusion and estimates (2.92) are valid if the relation k2(q + 2vooql) < 1 holds with k2 - IIHIIc--.c. This relation and relation (2.91) of Proposition 2.3.16 do not imply each other and give different estimates for the coefficients q and ql of the Lipschitz condition (2.90). We remark that the norm k IIHIIL2__+L2 equals the spectral radius of the operator H = H(T) and that if qk < 1 (i.e., the first relation of (2.87) holds) then estimate (2.84) alone implies that system (2.83) has at least one T-periodic solution, which may be eventually unique. =

Structure of the Set of Periodic Variable States and Outputs The following proposition, describing the set of all possible periodic variable states and outputs of the Preisach nonlinearity for a given periodic input, explains why the one-parameter continua of equivalence classes of periodic solutions appear and play the main role in the proofs of Proposition 2.3.16 above and further statements of this subsection on such continua. Consider a continuous input, which is periodic on the semiaxis t>~to, i.e. x(t) - x(t + T) (t>~to). For some admissible initial states r/0 = r/0(~, fl), the variable state r/(t) = (F[r/0]x)(t) and the output y(t) = O(r/(t)) of the Preisach nonlinearity are also periodic on the semiaxis t>,to. To describe the class of all such variable states and outputs for a given input x(t), let us define the numbers Xm = min x(t), XM = max x(t) and the sets

G(xm, XM) = {(0r fl) : 0r < Xm~XM < fl}, Gc(xm, XM) = 1-I\G(xm, XM) on the half-plane FI = {(a, fl) : a < fl}. From the relay definition it follows that if (a, ]/) e Gc(xm, XM) then the relay output (R~,/~[t0, .]x)(t) is periodic

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

187

on the semiaxis t~to for exactly one of the two initial relay states 0 a n d 1, for the other initial state the o u t p u t is either not periodic or not defined. A u n i q u e (for a given x(t)) periodic relay o u t p u t is d e n o t e d by r/per(t, (x, fl) a n d we set

s s (J,~x)(t) - I ] r/per(t, (X, fl)d#(a:,//) + ,~#(G(Xm, XM)), d JGc(Xm,XM)

t~to,

(2.96) w h e r e 0 ~ 2 ~ 1. For each ,l, this operator is defined on the class of all continuous periodic functions x : [to, co) ~ R, acts in this class and sends a function x = x(t) with a period T to the function J , tx with the same period. Denote by (9 the class of all functions g : ~+ --> ~ such that Ig(yl) g(y2)l~lYl - y2[ (yl, y2~0) and define the function @(e) =

sup # {(~, fl) ~ 1-I : I~ + fl - g(fl - ~)lKe}, g=g(y)~q)

e~0,

(2.97)

w h i c h is nondecreasing, nonnegative a n d satisfies ~(e) ---> 0 as e ---> 0, since we a s s u m e that the m e a s u r e # has a b o u n d e d density. PROPOSITION2.3.17. The following statements hold:

(i) For any periodic input x(t) =_ x(t + T) (t~to) the formula y(t) = (J2x)(t) with 2 ranging over the segment 0 ~ 2 ~ 1 defines the class of all outputs of the Preisach nonlinearity that are periodic on t ~ to. (ii) For any periodic input x(t) =_ x(t + T) (t~to) the formula r/(t, a, fl) = {

r/,r/per(t fl)(X''(fl) (x, ififeithero~< Xm~XM fl > O~Xm<

[~,~ XM~fl

> O~or both,

(2.98) where r/~(~, fl) is an arbitrary measurable (with respect to the measure #) function with the values O, 1, defines the class of all periodic variable states r/(t) = r/(t, ~, fl) of the Preisach nonlinearity. Their periods equal the period of the input , i.e. r/( t) = r/( t + T) ( t ~ to ) , and the periodic output y(t) = ~(r/(t)) for the variable state (2.98) is defined by y(t) = (J2x)(t) with

2#(G(xm, XM)) -- #({(~, fl) 6 G(xm, XM): r/~(~, fl) -- 1}).

(2.99)

(iii) Each variable state r/(t) = (F[.]x)(t) (t~to) equals one of the variable states (2.98) on the semiaxis t~to + T. (iv) For every pair of periodic inputs xj = xj(t) (t~to) with a period T and every ,l ~ [0, 1] II,J2Xl - J2x211C[to,to+T]~26(llXl - x211C[to,to+T]).

(2.100)

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CHAPTER 2 Differential Equations with Hysteresis

From conclusion (iv) it follows that operator (2.96) is uniformly continuous in the space C[to, to + T] for each 2 (we identify periodic inputs and outputs with their restrictions to the period). If relations (2.88) and (2.89) hold, then I~,(E)I~
IIJ x X 1

- - ,.,']'2X2

]]C[to,to+T]<~2v~ ]Ix1 -

x2

[[C[to,to+T].

(2.101)

By Proposition 2.3.17 for every class [x(t), y(t)] of T-periodic solutions of system (2.83) the relation y(t) = (J,~x)(t) with some 2 E [0, 1] is valid. Under the conditions of Proposition 2.3.16 such classes [x(t), (J,~x)(t)] exist for all 0K2~<1 and are pairwise disjoint. In fact, the dependence of the periodic problem on the parameter 2 rather than any special behavior of the function fit, x, y) is the reason for the existence of a one-parameter continuum of classes of periodic solutions. Simple assumptions like estimates of the measure # density and Lipschitz continuity of the nonlinearity fit, x, y) used in Proposition 2.3.16 may guarantee that the set of all classes [x(t), y(t)] of T-periodic solutions is the image of the segment 0~2K 1 under a one-to-one continuous mapping. The easiest way to construct examples of periodic solution sets with any number of connected components is to consider open-loop systems consisting of the differential Eqn. L(d/dt)x --- fit, x) without y, and the relations r/(t) = (F[rl(to)]x)(t), y(t) = @(x(t)) that define variable states and outputs of the hysteresis nonlinearity. Here every periodic solution xj(t) of the differential equation (if any exists) defines a one-parameter set of classes [xj(t), (J,~xj)(t)] (0K2~1) of periodic solutions for the system.

Classes of Periodic S o l u t i o n s Now we consider individual T-periodic solutions (x(t), rl(t)) of the system (2.83) from any class [x(t), y(t)]. Simple relations between their second components r/(t) follow from Proposition 2.3.17. Recall that, by definition, (x(t), rl(t)) belongs to the class [x(t), y(t)] if and only if r/(t) = (F[tl(to)]x)(t), y(t) = (I)0/(t)) and r/(t) = r/(t + T) for all t>~to. Statements (i) and (ii) of Proposition 2.3.17 imply that for every periodic solution (x(t), rl(t)) E Ix(t), y(t)] the component r/(t) = r/(t, a, t) coincides on the domain (~, t) E Gc(xm, XM) for all t~to with the function ~/per(t, ~x, t) uniquely defined by the input x(t). Thus, the components ~/(t) = r/(t, a, t) of the representatives (x(t), rl(t)) of the class Ix(t), y(t)] differ from each other on the domain G(xm, XM) only, where these components do not change with time. That means that r/(t, :r fi) - ~/. (a, t) on G(xm, XM) for each individual solution (x(t), r/(t)). If #(G(xm, XM)) > 0, then one obtains all representatives of the class [x(t), y(t)] inserting here characteristic functions r/. (~, t) of all measurable subsets of G(xm, XM) that

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

189

satisfy a unique restriction, namely (2.99) should hold with the 2 ~ [0, 1] defined by the equality

y(t) - f f r/per(t, a, fl)d#(a, fl)+ 2#(G(xm, XM)), d JGc(Xm,XM) which is nothing other than as y - J, tx. Since there is a continuum of such functions r/. = r/~(a, ]/), each class Ix(t), y(t)] consists of a continuum of periodic solutions (x(t), r/(t)) if l~(G(xm, XM)) > 0. For example, under the conditions of Proposition 2.3.16, system (2.83) has a one-parameter continuum of classes [x,t(t), Y,t(t)] of T-periodic solutions and each of those classes is a continuum parameterized by the functional parameter r/~ Formula (2.96) implies that J2x - J0x for all 2 and the Preisach nonlinearity assigns a unique periodic output to a given periodic input x - x(t) if and only if #(G(xm, XM)) -- 0. In contrast to the situation considered above, it means that the a class [x(t), y(t)] of periodic solutions is not included in the one-parameter set of such classes [x,~(t), y2(t)] if #(G(xm, XM)) = O. Moreover, formula (2.99) implies that the class [x(t), y(t)] contains a unique element (x(t), r/per(t)) in this case, since we consider functions r / = r/(~, fl) that are, with respect to the measure #, equal almost everywhere as the same state of the Preisach nonlinearity. Thus, periodic solutions (x(t), r/(t)) such that #(G(xm, XM)) -- 0 are basically isolated (if the nonlinearity f itself is not so complicated that it 'produces' nonisolated solutions). For example, the relation #(G(xm, X M ) ) - - 0 holds if Xm < - R ~ and XM > Ra.

Guiding Functions It is instructive to modify the statements of Subsection 2.3.3 that guarantee the existence of at least one T-periodic solution for systems with the Preisach nonlinearity in order to formulate sufficient conditions for the existence of the continuum of such solutions. Under the hypotheses of each of the Propositions 2.3.5, 2.3.6, 2.3.7 the periodic solutions admit an a priori estimate llz(t)licKp of the z-component. Such an estimate implies the estimate llx(t)llc~P0 for the corresponding periodic inputs x = (c, z) of the Preisach nonlinearity, for example with P0 - plcl. If we assume that P0 < R~, in addition to the conditions of any of above propositions, then the set of periodic solutions will include the one-parameter continu u m of disjoint classes [z2(t), y,t(t)], 0K2K1. Here and henceforth we denote by [z(t), y(t)] the class of all periodic solutions (z(t), r/(t)) satisfying y(t) = (I)(r/(t)) for systems (2.104) below, similar to the notation Ix(t), y(t)] used for system (2.83). We consider in more detail how this approach works in another application based on the method of guiding functions.

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190

A scalar-valued, continuously differentiable function V(z) is called a guiding function for the system of ODEs z' = F(t, z) with z ~ Ra if the gradient VV(z) of this function satisfies for some p > 0 (VV(z), F(t, z)) < 0

if

Izl > p, t ~ E.

(2.102)

Let the continuous function F(t, z) be T-periodic with respect to t. The simplest fact illustrating the role of guiding functions in the study periodic problems is that estimate (2.102) and the relation IV(z)I --> oe as IzI --* c~ imply the existence of at least one T-periodic solution of system z' = F(t, z). An important idea that leads to less restrictive existence conditions is to use several guiding functions. From (2.102) it follows that outside some ball in Ra the vector fields VV(z) and -F(to, z) are linearly homotopic and do not have zeros for every to. Therefore if Vj(z) (j = 1 . . . . . k) are guiding functions for the system z' = F(t, z), then the index at infinity is defined for each gradient vector field VVj(z) and all these indices coincide. The common value of these indices is called the topological index of the finite set of guiding functions 111(z) . . . . . Vk (z). Such a set is called complete if IWl(Z)i-}-..--}-IWk(z)l--+

oo

as

[zl--*

oo.

(2.103)

There is a classical theorem which states that if there exists a complete finite set of guiding functions with a nonzero topological index, then the system has at least one T-periodic solution. If one uses a straightforward modification of the definition of a guiding function, then the same statement holds for systems with hysteresis nonlinearities. So, we consider the system

dz/dt y(t) ~l(t) x(t)

= = = =

F(t, z(t), y(t)), (I)0/(t)), (F[rl(to)]x)(t), (c, z(t))

Z E [~d,

(2.104)

with the Preisach nonlinearity, where F(t, z, y) =- F(t + T, z, y) is a continuous function. Assume that for j = 1 . . . . . k

(VVj(z), F(t, z, y)) < 0

if

Izl > pj, 0KyK1, t ~ ~,

(2.105)

the relation (2.103) holds and the common value of the index at infinity of the gradient vector fields VVj(z) is nonzero. Then the system (2.104) has at least one T-periodic solution (see, e.g. [61]). Combining this statement with Proposition 2.3.17 in accordance with the basic approach of this subsection, we arrive at the following conditions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

191

for the existence of a continuum of periodic solutions. We use the notations mj and Mj for the smallest and the largest values of the guiding function Vj(z) on the ball Izl~Pj and the notation c j - {z ~ ~d " m j < . V j ( z ) < . M j } .

PROPOSITION2.3.18. Let scalar-valued continuously differentiable functions V] (z), . . . . Vk(z) satisfy (2.103) and (2.105) and the index at infinity of the gradient vector fields VVj(z) in ~cl be nonzero. Let a ball {z E ~ 9 Izl~p.} contain the set G1 D ... D Gk and let R~ > p.Ici. Then the set of T-periodic solutions of system (2.104) includes a one-parameter continuum of disjoint classes [z2 (t), y2 (t)] (0~2~1). We remark that relation (2.103) implies that the set G1D... NGk is bounded. The conditions of Proposition 2.3.18 imply the inclusion z(t) E G1Vi... DGk, t>~to for the z-component of each periodic solution of system (2.104) and therefore the a priori estimate Ilz(t)Ilc~p. for all such solutions. If IV(z)] --~ cx~ as ]z] --> cx~ (i.e., k = 1) then the index at infinity of the vector field VV(z) is either I or -1. There are many other simple conditions which guarantee that the topological index of the complete set of guiding functions Vl(Z) . . . . . Vk(z) is different from zero. For example, if one of these guiding functions is even, then the topological index is odd. The proofs of Propositions 2.3.16 and 2.3.18 are based on the same idea. Using the operator (2.96), one constructs an operator equation depending on 2, whose solutions define the classes Ix(t), y(t)] of T-periodic solutions of the system (2.83) (the classes [z(t), y(t)] for system (2.104), respectively), then proves the existence of a solution for each 2 E [0, 1] and shows that the solutions are different for different 2. For example, one can use the operator equation

~(t) =fit, (H~)(t), (,.72H~)(t))

(2.106)

in the space L2[0, T] to prove Proposition 2.3.16. From the definition of the linear operator H = H(T) and Proposition 2.3.17 it follows that each solution ~ = ~(t) of (2.106) defines the class Ix(t), y(t)] of T-periodic solutions of the system (2.83) with x(t) -- (H~)(t), y(t) = (J,~x)(t). The existence of a solution ~,~ of (2.106) for each 2 E [0, 1] follows from (2.84) and the first of the estimates (2.87) by the Schauder principle. The second of estimates (2.87) and the a priori estimate ]lxllcKklb~/T/(1 - qk) of all T-periodic solutions of (2.83) imply that all the classes [x,l(t), y,~(t)] of such solutions defined by the solutions ~,~ of (2.106) are disjoint. If the additional assumptions of the second part of Proposition 2.3.16 hold, then Eqn. (2.106) satisfies the conditions of the contraction mapping principle and therefore has a unique

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CHAPTER 2 Differential Equations with Hysteresis

solution r for each ;~ ~ [0, 1]; moreover, the map 2 --, r is Lipschitz continuous. For the proof of Proposition 2.3.18, one can use the operator equation

z(t) = z(O) +

~0t F(s, z(s), (J2x)(s))

ds

with

x(t) = (c, z(t))

in the space C T ( [ 0 , T]; •a) of continuous functions z - z(t) satisfying z(0) = z(T) with the uniform norm. The proof of the existence of a solution z2 of this operator equation for each ;~ ~ [0, 1] follows standard lines of the guiding functions method. For details of the proofs of Propositions 2.3.16 and 2.3.18 and for the proof of Proposition 2.3.17 we refer to [162].

A u t o n o m o u s Systems Our approach also works in periodic problems for autonomous systems, and here it leads to statements on the existence of continua of cycles. For the same reasons as in problems on forced periodic oscillations, the continua of cycles have a similar type of one-parameter structure as described above if the a priori estimate lixllc < R~ holds for the cycles. We give one particular statement. Consider the system

L (-~)x y(t) ~l(t)

= = =

f(x, y(t)), (I)(r/(t)), (F[~l(to)]x)(t)

(2.107)

with the Preisach nonlinearity. Its solutions of the form (x(t), ~l(t)) =(x0, r/0) are called stationary, since for such solutions y(t) = Yo with Y0 = 9 (r/0). We are interested in conditions that guarantee the existence of nonstationary periodic solutions (x(t), rl(t)) -- (x(t + T), ~l(t + T)) of any period T > 0. Since system (2.107) is autonomous, those periods are u n k n o w n a priori. Furthermore, one should take into account that the set of nonstationary periodic solutions contains, together with any solution (x(t), r/(t)), all its time shifts (x~(t), rl~(t)) = (x(t + ~), r/(t + ~)) with 0 < ~ < T. Passing to the equivalence classes [x(t), y(t)] = {(x(t), ~/(t)) : y(t) = (I)(r/(t))} of periodic solutions of the system (2.107), we see that the time shifts of solutions from a class Ix(t), y(t) ] by the time r form the class [x(t + z), y(t + ~)]. In order to simplify the description of shift invariant structures of periodic solution sets, we introduce closed curves 7 = {(x(t), y(t)) ~ [~2 . 0Kt K T} on the i n p u t output plane of the Preisach nonlinearity and say that the curve 7 represents the class of all periodic solutions from the join Uo<.~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

193

The curve 7 is called degenerate if (x(t), y(t)) =_ (xo, Yo) and nondegenerate otherwise. We consider stationary solutions in more detail. For 0<~A~
-

#({(a,/~)

9 ~ < fl < ,o}) + ,~ ~ ( { ( ~ , / ~ )

9 a < ,o < fl}).

(2.108)

This function is continuous with respect to the set of arguments A, p. As we know, the Preisach nonlinearity assigns stationary outputs to stationary inputs for any admissible initial states. Moreover, from conclusion (i) of Proposition 2.3.17 it follows that the values of all the outputs y(t) -- Yo corresponding to the input x(t) - xo form the segment {Y0 = ~b,t(x0) : 0<~A~
(2.109)

If L(0) ~- 0 and the continuous functionf(x, y) is bounded, then Eqn. (2.109) has at least one solution p - p*(2) for each 2 e [0, 1] with Ip*(2)l~pl = sup ~(x, y)llL(0)1-1, and therefore the system (2.107) has stationary solutions. In particular, if there is a unique solution p = p* (A) of (2.109) for each 2 and #({(a,//) 9 ~ < p < fl}) > 0 whenever p e [ - P l , Pl], then the points representing all the classes of stationary solutions of system (2.107) in the plane (x, y) form the continuous curve F -- {(x0, Y0) - (p*(2), q~,t(p*(A))) 9 0~<~.~<1} without self-intersections. N o w we are ready to formulate a statement on the existence of nonstationary solutions. Let fix, y) be a bounded continuously differentiable function satisfying the global Lipschitz condition

~(Xl, Yl) --f(x2, y2)l<~qlxl -- x21-+- qllyl -- Y21,

xj, yj 6 ~.

(2.110)

We assume that the relations (2.88), (2.89) are satisfied for the Preisach nonlinearity. It is easy to see that if there are no zeros of the polynomial L(p) on the imaginary axis and the coefficients q, ql in (2.110) are small enough, then the system (2.107) does not have nonstationary periodic solutions. Let us assume that the polynomial L(p) has a pair of the conjugate imaginary zeros 4-woi, and the so-called nonresonance condition L(nwoi) g= 0

for

n - 0, 4-2, 4-3 . . . .

(2.111)

holds. Define on the positive semiaxis r > 0, the function d(r) -

/0

sin t fir sin t, 1) dt -

/0

sin t f ( - r sin t, O) dt.

(2.112)

194

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.19. Let the polynomials L(wi) and ~mL(wi) of the variable w have a pair of common zeros • (wo > O) of the same odd multiplicity. Let the nonresonance condition (2.111) hold. Suppose that estimates (2.88) and (2.89) are valid, that the bounded continuously differentiable function fix, y) satisfies the Lipschitz condition (2.110) and function (2.112) satisfies Id(r)l~c > Ofor r~r,. Then there are numbers q0 > 0, Wl e (w0/2, w0) and w2 > w0, such that if q + vooql < q0, the following statements are valid:

(i) The set of all stationary solutions of the system (2.107) is represented on the plane (x, y) by a continuous curve F, which may degenerate to a unique point. (ii) If there is a point (x*, y*) o f f such that

ef

d(r.) ~xx(X*, y*) < 0,

(2.113)

then the system (2.107) has at least one class of nonstationary periodic solutions represented by a nondegenerate closed curve ~; = {(x, y) e R 2 9 x = x(t), y = y(t), t e [0, T)} with a period satisfying 2~/w2~T~2rc/w]. (iii) If estimate (2.113) is valid for all points of the curve F, then there exist numbers r], ro > 0 such that the x-components of all periodic solutions of the system (2.107) with periods 2~/w2<~TK2rc/wl satisfy IIx(t) - YClIL2[O,T]~rl,

]Ix(t)IIc[o,T]~ro,

(2.114)

where Ycdenotes the average value of the function x(t) over its period. If in addition Ra > ro, then the set of all nonstationary periodic solutions of the system (2.107) includes a one-parameter continuum of disjoint classes represented at the plane (x, y) by nondegenerate closed curves ?,~ where 2 ranges over the segment 0K2~l. These curves are different for different 2 and the periods of all periodic solutions represented by them satisfy 2rc/w2 ~ T K2rC/Wl. By conclusion (iii), the x-components of all periodic solutions represented by the curves 7,l satisfy the estimates (2.114) and consequently are separated from the one-dimensional subspace of constant functions and from infinity in L2, C and other functional spaces, which implies, in particular, that such periodic solutions are separated from stationary ones. Additional assumptions about the functionf and the measure # of the Preisach hysteresis nonlinearity guarantee that the one-parameter set of curves ?,t represents all periodic solutions of the system (2.107) and these curves depend continuously on 2. Formulas for the numbers q0, w], w2 (they are defined by the polynomial L(p)) and estimates (2.114) can be given explicitly. We do not present

195

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

these cumbersome explicit expressions here, 3 but consider, as an example, the equation x'" + x" + x' -t- x =fix, y(t)). (2.115) The polynomial L(p) = p3 + p2 + p + I has the pair of simple roots +i,

L(wi) = (wi + 1)(1

3mL(wi) = w(1

- w2),

- w2),

w0-1

and 1/2 < Wl < 1 < w2. For example, if the Lipschitz estimate (2.110) holds with q = 0.305 and v~q1~<0.295, the Eqn. p = tip, qbx(p)) has a unique solution p = p~ for some 2 ~ [0, 1] and for all sufficiently large r

d(r) -~x

<

o,

[d(r) l~c > O,

then Eqn. (2.115) with the Preisach nonlinearity has nonstationary T-periodic solutions with 5.575~
L ( d ) x - g(x) + qly(t). Then

d(r) = 2ql + 4

f rcl2 J0

sin t godd (r sin t) dt,

where godd (x) -- (g(x) -- g(--x))/2 is the odd part of g. If godd (x) ~ go as x --, +oe, then d(r) ~ do = 2ql + 4g0 as r --~ +cx~. If do ~ 0, then the condition Id(r)l>~c > 0 of Proposition 2.3.19 is satisfied for all sufficiently large r.

2.3.6 AVERAGING PRINCIPLE General Description The averaging principle, see [146] and bibliography therein, is a powerful tool in nonlinear dynamics. Here we apply averaging theory to differential equations with hysteresis operators on the right-hand side. We remind the reader, that the standard averaging principle is applicable to the ordinary differential equation of the form z ' = ef(t, z) 3Further details, statements and proofs can be found in [163].

(2.116)

196

CHAPTER 2 Differential Equations with Hysteresis

where the function f is periodic or almost periodic in t. The idea is to introduce the function

F(z)-

lim Z1 IJoT f(t, z) dt

T~oo

(2.117)

and then to consider the equation

z' = eF(z)

(2.118)

as an approximation to (2.116). It has been proved by N. Bogoljubov, that the averaged Eqn. (2.118) is a good approximation to (2.116) over intervals of the size O(1/e), which is the characteristic time for Eqn. (2.116). Now we consider the system with hysteresis given by

~(t) = el(t, z, y(t)), y(t) = (W[rlo]x)(t),

z ~ ~N,

(2.119)

x(t) = g(t, z(t)).

(2.120)

Here y(t) is the output of the hysteresis operator W with the initial state t/0, and e > 0 is a small parameter. The functions f and g are almost periodic in t. We note that the system (2.119), (2.120) is not a particular case of (2.13). We explain how these Eqns. (2.119), (2.120) 'come to life' in the analysis of Eqn. (2.13) below.

Averaged Hysteresis Operators First we need some analysis of hysteresis nonlinearities for inputs that are almost periodic functions depending on a slowly changing parameter s ~ ~+. Formally, the function u(t, s) is given in such a way that for each fixed s ~ R+ it is almost periodic as a function of t ~ ~ and the map s --+ u(.; s) is uniformly continuous on any bounded interval [0, T]. For each s ~ ~+ we define the sequence of inputs

VSk(t) = u(t, min{t/k, s}),

t~>0,

to the operator W[~/0] and denote z~ (t) = W[r/0]v~(t). Since v~(.) converges to an almost periodic function, the same is true for the output z~ (t); by y~ (t) we denote the almost periodic limit of z~ (t). Note that y~ (t) is defined for all t ~ R. If, for each s ~ ~+, the sequence y~ (t) converges to a function ys(t) as k --+ oo uniformly in t ~ 0~, then ys(t) is almost periodic and we say that the map y(s) = ys(') from ~+ to the space of almost periodic functions is the output of the averaged hysteresis operator W'[r/0]. We return to the system (2.119), (2.120) The solutions are traced on a finite interval of the 'slow' time ~ = et. The functions fit, x, z) and g(t, x)

197

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

are assumed to be almost periodic in t and uniformly continuous in all the variables. We denote 1 T + ~ -T /0 fit, z(s), W*[~lo]X s (t)) dt, (I)z(s) = T ~lim

x s(t) = g(t, z(s)).

The averaging m e t h o d replaces (2.119) by the equation

~(t) = ~(bz(t),

(2.121)

whose solutions converge to the corresponding solutions of (2.119) as e --+ +0, if the averaged operator W* exists. Existence of the averaged operator W* has been proven for m a n y classical models, including the Preisach nonlinearity and hysterons. We note especially Vladimirov's results concerning the multidimensional hysteresis operator [111] and their applications to IT problems.

Averaging of Preisach Nonlinearity Let us give a simple example of an explicit description of the limit nonlinearity. Let P be a Preisach nonlinearity and let function u(t, s) be T(s)periodic. We suppose also that the set f~ coincides with the half plane It is sufficient to consider the case w h e n the initial state ~/0 is the characteristic function of a set S, which looks like the shaded set in Fig. 2.35(a). That is, S is located to the left of the line a - ] / a n d below a continuous linear curve L = Ls, with a non-positive slope. We denote the set of states ~/describing the sets S as above by (~.. Additionally, we assume that the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. We will discuss this type of initial condition in more detail in Section 2.4.1. N o w we will give a correspondence between a given function u(t, s) and a varying set S(s). For each s>~0 introduce u_ (s) = min u(t; s) t

and

u+(s) = max u(t; s).

(2.122)

t

For simplicity we consider only the case w h e n both these functions are piecewise monotone. We define S(0) as the part of S which is located to the left from the vertical line :~ = u_ (0), that is,

s(0)see Fig. 2.35(b).

< u_(0)}.

CHAPTER 2 Differential Equations with Hysteresis

198

FIGURE 2.35 The sets S, S(0). Suppose that S(cr) is defined for 0~cr~s0 and that the functions (2.122) are monotone at the interval so Ks. Let us consider four cases: Case Case Case Case

1. 2. 3. 4.

u_(s)~u_(so), u_(s)~u_(so), u_(s)Ku_(so), u_(s)Ku_(so),

and and and and

u+(s)Ku_(so). u+(s)~u_(so). (u_(s), u+(s)) r S(so). (u_(s), u+(s)) ~ S(so).

Figure 2.36 provides the 'picture definitions' for the corresponding set S(s), which is the union of the hatched and shaded regions. The bold line is given by the parametric Eqn. ~ = u_ (rT),//-- u+ (rr). For instance in Case 1, S(s) is the union of the set S(so) and the set {(a, ]/) "u_(so)~o~ < u_(s), and a < / / < u+(a)}. The limit Preisach nonlinearity P*[~/0] can be described by

(P*[~lo]xS)(t) = (P[rl(s)]xS)(t), where x s (t) = u(s, t) and r/(s) is the characteristic function of the set S(s).

Numerical Example Here we demonstrate how the techniques described above work in a particular example. Consider our standard scalar differential-operator equation of the form x"+x = ey(t), y(t) -- (I)0/(t)) 2, (2.123) r/(t) = (F[r/0])x(t).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

FIGURE 2.36

199

Set S(s).

w h e r e O(r/(t)) is a Preisach n o n l i n e a r i t y w i t h a m e a s u r e #. We s u p p o s e that p is a L e b e s g u e m e a s u r e on the triangle A ( - 1 , 1) = {-1K0cKflK + 1} a n d zero a n y w h e r e else. U s i n g the s u b s t i t u t i o n x = r cos(0 + t),

x' = - r sin(0 + t)

the Eqn. (2.123) can be r e w r i t t e n in the f o r m of the s y s t e m

dr/dt dO/dt y(t)

= =

r/(t)

=

-

- e s i n ( t 4- O)y(t), ecos(t 4- O)y(t)/r, O(r/(t)) 2, (F[~/0])(r(t) cos(0(t) + t)).

C o n s i d e r the initial c o n d i t i o n s r(O) = ro e (0, 1],

0(0) = O,

rt(O) = ~o.

(2.124)

CHAPTER 2 Differential Equations with Hysteresis

200

FIGURE 2.37 Set S(s) for (2.124) is hatched; all possible sets described by characteristic function r/belong to the union of the filled areas.

For these initial conditions, the system (2.124) has a unique solution. The initial state of the Preisach nonlinearity ~/0 is chosen to be u n i o n of the s h a d e d and hatched areas on Fig. 2.37. Using the averaging m e t h o d from the previous subsection and the properties of the Lebesgue measure, we can write a differential equation for the average ~ of r

f2~+o

1 d~/dt= _ m2~ Jo

e Sin(S -}- 0)(I)((1P[r/0])(~ cos(0 if- S))) 2 ds

1 fo ~ e sin(s)(c q- -~~.2 (cos(s) - 1)2) 2 ds = - 2---~ ~2 1 f2~ 2re e sin(s)(c + 2~z - ~ ( 1 - cos(s))2) 2 ds

4~2(C if_ ~2) 3~ Here c > 0 and ~ is decreasing with time, that is the case of the evolution of a Preisach nonlinearity s h o w n in Fig. 2.35(a). The solution x - r sin(t + 0) has symmetric m i n i m a and m a x i m a which means that c is the area in the hatched p o l y g o n in Fig. 2.37. Thus, c - 1 - ~2 and the equation above can

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

201

be written as d~ dt

4~ 2 =

3rt

~(0) - r0

and has the solution 3~zr0 ?(t) = 3rr + 4erot"

Similarly, one can write d o w n and solve the equation for the average of 0. From the averaging principle d0 1 f020 ~ cos(s + 0)(I)((F[r/0])(r cos(0 + s))) 2 ds d--~ = 2~z~ = 2 1~ fo ~ e cos(s)(c q- -~~2 (cos(s) -- 1)2) 2 ds

1 ~2~r 2~z~

~2 e cos(s)(c + 2r 2 - ~-(1

- cos(s))2) 2 ds

= e?(c + 72). N o w the equation for dO dt

= ~

3~ro 3~z + 4roet '

o(o) - o

has the solution 0(t) - -3/4(zt log(3r0 - zr log(3rc + 4er0t)). In Fig. 2.38 we c o m p a r e the results of the averaging m e t h o d with straightforward numerical integration. We consider two values of p a r a m e t e r e: E = 0.01 a n d e = 0.001. Initial conditions are the same in both cases, r0 = 1 a n d we c o m p a r e the solutions on the time interval [0, 1000].

A p p l i c a t i o n to Q u a l i t a t i v e A n a l y s i s of S y s t e m s with Hysteresis We will d e m o n s t r a t e h o w it could w o r k by a particular example. We consider the scalar differential-operator equation of the form x" + x -

qb(t) + ey(t),

y(t) - (I)0/(t)),

(2.125)

r/(t) - (F[r/0])x(t),

(2.126)

CHAPTER 2

202

Differential E q u a t i o n s w i t h H y s t e r e s i s

in ! ! ....

0.2

0.5 0.1

o

!

C 0.1

-0.5

0.2

-10

|

--

4'0

i

6'0

i

i

900

(a) c = 0.01, t r [0, 100] 1

i

0.5

f

nir,f

|

1

10(

920 940 960 980 (C) c=O.O01, t E [900, 1000]

1000

-

0.5 :

i

0

2

-0.5 -1

920 940 960 980 (b) ~ -- 0.01, t E [900, 1000]

i 0

i

20

I

40

60

-0.5

I i

i~

.

,

80

-1 90O

100

(C) c=O.O01, t E [0, 100]

FIGURE 2.38 Dashed oscillating-numerical solution; solid oscillating-solution of averaging equations. w i t h the Preisach nonlinearity. We will s u p p o s e that the c o r r e s p o n d i n g m e a s u r e # is finite, positive, absolutely continuous w i t h b o u n d e d density, a n d defined in the half plane a < ft. Suppose that the external force ~b(t) is a s u m of periodic functions qbi(t), i = 1 . . . . . n, w i t h periods Ti, for w h i c h n

2~

Z ki-~ii + kn+l ~ 0

(2.127)

i=1

for a n y integers kin, m = 1 . . . . . n + 1. As w e know, this equation for a n y initial conditions x(0) = x0, x'(0) = x 0' has a u n i q u e solution x(t , x(O) , x'(O) , e). We denote b y ui(t), i = 1 . . . . . n, the periodic solution (it is unique) of the equation u" + u = qbi(t ), and we set n

u(t) = Z i=1

ui(t).

(2.128)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

203

By m e a n s of the usual substitution x = u(t) + r cos(0 + t),

x' = u' (t) - r sin(0 + t)

the Eqn. (2.125) can be rewritten in the form of the system dr d--t = - ~ sin(t + O)y(t), dO d---t-= e cos(t + O)y(t)/r, y(t) = ~(r/(t)),

~/(t) = (F[~/0])x(t), x(t) = u(t) + r(t)cos(0(t) + t).

For any initial conditions r(0)=r0>0,

0(0)=00,

~/(0)=~/0

this system has a u n i q u e solution (r(t; ro, 0o, tl o, e), O(t; ro, 0o, ~1o, e), r/(t; r0, 00, ~/0, e)). We denote by f~(r), r > 0 the following d o m a i n in the half plane ct < fl: f~(r) =

~ <

m a x ui(t) + r or e > i=1

m i n ui(t) - r

.

i=1

For r > O, (e, fl) ~ f~(r) and 0 < Oi < 2re, i = 1 . . . . . n + 1, we set F(r; ~, fl, 01 . . . . On+l) =

lim

to---+ - - ~

(R[z(e, fl); ~, fl)x) (0)

n ( x(t) = ~

ui

t -- to -t- - ~ T i

-t- r cos(t - to -t- On+l).

i=1

The limit of the right-hand side always exists. For fixed values of r > 0 a n d (e, fl) e f~(r) it is convenient to consider this function to be defined on the (n + 1)-dimensional torus H n+l. For r > 0 we set P(r) = ~

(r)

d/.t(0c,fl)fH

n+l

F(r; ct, [3, 6)) sin On+l d |

Since in any d o m a i n 0 < r < R, P(r) satisfies a Lipschitz condition, each Cauchy problem d? m

= -ePff),

dt has a unique solution ?(t; ?0, e).

r(-0) = r0

204

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.4. The function P(r) is strictly positive for r > O. COROLLARY2.3.10. Let R > O. Then to each ~ > 0 there correspond numbers l eo > 0 and L > 0 such that for 0 < e < e0, and Ix0[ + Ix0] < R Ix(t; x0, x~) - u(t)] + Ix'(t; x0, x~) - u'(t)] < 6,

t > L/e.

2.3.7 BIFURCATIONANALYSIS In Section 2.2 we observed a rich bifurcation behavior of Eqn. (2.9) depending on the parameter b. Here we survey some statements about bifurcation points. We restrict ourselves to one type of bifurcations, namely Hopf bifurcations in autonomous systems, discussing mainly sufficient conditions for some parameter value to be a bifurcation point and the structures of solution sets, to some extent, and omitting the proofs.

H o p f Bifurcation Let some autonomous system depending on a scalar parameter b have an equilibrium z.(b) for each parameter value from some interval (b_, b+) and let those equilibria form a smooth curve in the phase space Z of the system. We say that Hopf bifurcation occurs and a parameter value b0 is a Hopf bifurcation point if for any sufficiently small r > 0 one can find a parameter value br E (bo - r, bo + r) such that the system with b -- br has a cycle which lies in the ball of radius r centered at the equilibrium z.(br) (this type of weak definition of Hopf bifurcation was introduced in [164], where nonsmooth problems were studied for the first time). Freely speaking, we consider bifurcation of a small cycle from an equilibrium point of the autonomous system. The simplest example is the equation x" + bx' + x = fix, x', b)

(2.129)

with f satisfying flZl, Z2, b) = o(]Zll q-IZ21) as Zl, Z2 --+ 0 uniformly with respect to b. Here the origin is an equilibrium of the equivalent planar firstorder system for every b; the value b = 0 is a unique Hopf bifurcation point. Figure 2.39 presents a typical structure of the set of cycles in the product of the phase space ~2 of the equivalent system and the axis of parameter values b in the case of a sufficiently smooth f. The cycles form a smooth two-dimensional surface ~ (a 'cup') passing through the origin (note that the picture is local both in b and z = (Zl, z2), i.e. we consider cycles lying in a small vicinity of the zero equilibrium and b small in absolute value only). Appropriate nondegeneracy conditions guarantee that ~ is tangent

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

205

I Z2 s ss

E1 %

FIGURE 2.39 Hopf bifurcation.

to the plane b = 0 and lies on one side of it. In Fig. 2.39, s lies to the right of this plane, which means that the system has no small cycles for b < 0 and has such cycles for small b > 0; for b = b' the cycle is a cross section of with the plane b = b'. Under the conditions of the classical Hopf Theorem (see e.g. [165]) a similar picture is observed for a u t o n o m o u s systems

d z / d t = F(z, b),

z c ~a,

(2.130)

of ODE of any dimension in some vicinity of the equilibrium z,(b). Let us recall more precisely the algorithm to determine bifurcation points by linearization of (2.130). Since the change of variables z ~ z + z, (b) places the equilibrium at the origin, let us assume without loss of generality that z, (b) ~ 0, i.e. system (2.130) has zero equilibrium for all b. In this situation, we call b0 a Hopf bifurcation point for system (2.130) with a limit period To > 0 if for any sufficiently small r > 0 the system has a periodic solution zr(t) of a minimal period Tr > 0 for some b = br and max Izr(t)l ~ O,

br ~ bo,

Tr ~ TO

as

r ~ 0.

t~

We remark that periods Tr are not k n o w n a priori and that we use an auxiliary parameter r. The reason w h y the parameter b is not good for the parameterization of the set of small cycles becomes clear if one considers linear equations, for example x" + bx' + x = 0, i.e. Eqn. (2.129) w i t h f ~ 0.

206

CHAPTER 2 Differential Equations with Hysteresis

Here all the cycles exist for one parameter value b0 = 0, which is a unique Hopf bifurcation point (the surface ~ in Fig. 2.39 becomes a piece of the plane b = 0). Suppose F is differentiable with respect to z at the point z = 0 uniformly with respect to the parameter b, i.e. F(z, b) = A(b)z + g(z, b) where suPb ]g(z, b)l/JzJ ~ 0 as izi --* 0, and let the matrix-valued function A(b) and the vector-valued function g(z, b) be continuous. Assume that the matrix A(bo) has a pair of simple imaginary eigenvalues 4 +iwo (w0 > 0) for some b0, which implies that A(b) has a unique pair of simple eigenvalues rr(b) 4- iw(b) close to 4-iwo for b close to b0. Let the derivative rr'(b0) of the real part of these eigenvalues at the point b = b0 exist and the so-called transversality condition cr'(b0) ~= 0 (2.131) and nonresonance condition

iwok ~ Sp(A(b0))

for all

k ~ 77, k ~= 4-1

(2.132)

hold, where Sp(A) is the spectrum of A. Then b0 is a Hopf bifurcation point for system (2.130) with the limit period 2rc/wo (for example, this follows from the results of [164]). If in addition F is sufficiently smooth, then according to the Hopf Theorem the small cycles form a two-dimensional smooth surface in the product of the phase space ~c/of (2.130) and the axis b. Moreover, information about the Taylor expansion of F at the origin allows us to study further important problems including stability of the cycles, but we do not consider those problems here.

Bifurcation in Systems with Preisach Hysteresis Consider the autonomous system

dz/dt y(t) rl(t) x(t)

= = = =

F(z(t),y(t),b), (I)(r/(t)), (F[rl(to)]x)(t), (c, z(t))

z ~ ~a (2.133)

with the Preisach nonlinearity, where the first equation depends on the scalar parameter b; the function F is supposed to be continuous. We use notation similar to that of Subsection 2.3.5. Solutions of the form (z(t), r/(t)) = (z0, r/0) are called stationary, the point (z0, ~/0) in the product ~a x (~. is called an equilibrium of the system. By a cycle we mean the trajectory of any nonstationary periodic solution (z(t), rl(t)) =_ (z(t + T), rl(t + T)) 4The existence of imaginary eigenvalues of A(bO) is a necessary condition for b0 to be a Hopf bifurcation point.

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

207

in ~a x (~., with any T > 0; actually, the cycle is a c o m m o n trajectory for all the time shifted periodic solutions (z~(t), ~/~(t)) = (z(t + ~), rl(t + ~)), 0~ (c, z0). Since the Eqn. y0 = ~0/0) with a given Y0 has a solution r/0 in this class of admissible states if and only if #({(~, f l ) : ~ < fl < (c, z o ) } ) ~ y o ~ # ( { ( ~ , fl): ~
(2.134)

we conclude that system (2.133) has an equilibrium if and only if the Eqn. F(zo, Y0, b) = 0 has a solution (z0, Y0) satisfying (2.134). Similarly to the definition for ODEs above, we call b0 a H o p f bifurcation point for system (2.133) with a limit period To if for any sufficiently small r > 0 the system has a nonstationary periodic solution (zr(t), rlr(t)) with a period Tr > 0 for some b = br and max I z r ( t ) - z o l ~ 0, max/9 l(r/r(t ), r/0) ~ 0, br -+ bo, Tr ~ TO t t

as

r --, 0,

where (z0, ~/0) is an equilibrium for b = b0 and/91 (., .) is the metric (2.20) in the state space @. of the Preisach nonlinearity. To formulate the simplest statement on H o p f bifurcation for the system (2.133), suppose that there are z0, Y0 satisfying (2.134) such that F(z0, y0, b) - 0

for all b.

Therefore system (2.133) has an equilibrium at the same point (z0, r/0) ~a x @. for all b. A s s u m e that the function F can be linearized at the point (z0, Y0), more precisely F(z, y, b) - A ( b ) ( z - zo) + a(b)(y - Yo) 4- g(z, y, b),

(2.135)

with the matrix A = A ( b ) and the vector a - a(b) d e p e n d i n g continuously on b and lim maxlb-b01~R Ig(z, y, b)l - 0 (2.136) z--,zo, y-*yo Iz - z01 + lY - Y01 for some R > 0. In this situation, H o p f bifurcation points for system (2.133) are d e t e r m i n e d by the matrix A (b) in the same w a y as for ODE. PROPOSITION 2.3.20. Let relations (2.135), (2.136) be valid and strict estimates #({(a, fl) 9 a < fl < (c, z0)}) < Y0
CHAPTER 2 Differential Equations with Hysteresis

208

hold in place of (2.134). Let the measure # have a bounded density with respect to the Lebesgue measure. Suppose that the matrix A(bo) has a pair of simple eigenvalues • that the transversality condition (2.131)and the nonresonance condition (2.132) hold. Then bo is a Hopf bifurcation point for the system (2.133) with a limit period 2~/wo. As an example, consider the equation x" + bx' + x = 2y(t) - 1 where x(t) and y(t) are the input and output of the Preisach nonlinearity. If #({(a, f l ) : a < f l < 0 } ) < l / 2 < # ( { ( a , fl):a 0 for some b = br and max [zr(t)- Zr(Z)[--+ cx~, br ~ bo, Tr ~ TO as t,'~

r ~ cx~.

(2.138)

In other words, the system has cycles with arbitrarily large amplitudes of the z-component and periods close to To for some parameter values arbitrarily close to b0. Suppose that the function F can be represented as

F(z, y, b) = A(b)z + g(z, y, b),

(2.139)

lim max Ig(z, y, b)] = 0 Izl-~c~ Ib-bol<~R,lYI~R~ ]Z[

(2.140)

where

for some R > 0 and every R1 > 0. Then the derivative A(b) of F with respect to z at infinity plays the same role in problems on Hopf bifurcation of large cycles as the derivative at the equilibrium point in local problems above.

209

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

PROPOSITION2.3.21. Suppose that relations (2.139), (2.140) hold, the matrix A (bo) has a pair of simple eigenvalues +iwo, the transversality condition (2.131) and the nonresonance condition (2.132) are valid. Then bo is an asymptotic Hop]: bifurcation point for system (2.133) with a limit period 2n/wo. This proposition is a consequence of a more general statement of [166]. Proposition 2.3.20 and some further statements on Hopf bifurcation from an equilibrium can be found in [167].

Bifurcation with Respect to Internal Parameter As in Subsection 2.3.5, let [z(t), y(t)] denote the class of periodic solutions (z(t), rl(t)) = (z(t + T), rl(t + T)) (t~to) such that y(t) = (I)(r/(t)) and let us say that the closed curve 7 - {(z(t), y(t)) 9 0~t < T} in the space Ea+l of pairs (z, y) represents the periodic solutions of the join Uo<~
dz/dt y(t) rl(t) x(t)

= = = =

F(z(t), y(t)), (I)(r/(t)), (F[ri(to)]x)(t), (c, z(t))

ZE[~ d, (2.141)

that does not depend explicitly on parameters. Nevertheless, as we already know from considerations in Subsection 2.3.5, problems on cycles of this system contain the parameter 2, which we call 'internal'. This results in the fact that each closed curve 7 c ~a+l representing a class of cycles (if any exists) such that the amplitude of their z-component is smaller than the characteristic size of the Preisach nonlinearity is generically included in the one-parameter continual set of such curves 7,~; a typical example is presented by the conclusion (iii) of Proposition 2.3.19. It turns out that the curves 7 representing small cycles of the system (2.141) can accumulate at its equilibrium and form the structure similar to that observed for Hopf bifurcation problems for ODEs with a scalar parameter. Moreover, sufficient conditions for the existence of those structures for system (2.141) with hysteresis are close to the classical assumptions of the Hopf Theorem. Let us consider one example. Let the Eqn. F(z, 2) = 0 define a continuous branch z = ~p(2) of the implicit function on some open interval A*. Denote by F the curve (z, y) = (~p(2), 2), 2 E A* in [~d+l and assume that for each point (z0, Y0) ~ F relations (2.137) hold, which implies that this point represents the class [z0, Y0] of equilibria of system (2.141). We shall suggest sufficient conditions that

CHAPTER 2 Differential Equations with Hysteresis

210

guarantee the existence of a one-parameter set of closed curves ~ with vanishing diameters that accumulate at one of the points Ado = (z0, Y0) 6 F. The density of the m e a s u r e / t of the Preisach nonlinearity will be everywhere assumed to be bounded. Let us suppose that the function F(z, y) is continuously differentiable with respect to z and Lipschitz continuous with respect to y in some neighborhood of the curve F. As usual, set A(2) = 8F/8z(q)(2), 2),

2 e A*.

Suppose that at some point .Ado = {q~(20), 2o} e F the spectrum Sp(A(20)) of the d x d matrix A(20) = 0F/8z(q~(20), 2o) contains a pair of simple eigenvalues +iwo (w0 > 0) and the nonresonance condition iwok r Sp(A(20)) holds for all integer k 4= +1. Consider the pair of the simple eigenvalues cr(2) + iw(2) of A(2) satisfying cr(20) 4- iw(20) = +iwo, where the real continuous functions a(2), w(2) are defined in some neighborhood of 2o. PROPOSITION2.3.22. Let the function r~(2) take values of both sign in each neighborhood of 2 = 20. Then for every small r > 0 system (2.141) has a class [Zr(t), yr(t) ] of nonstationary periodic solutions of a period Tr, all the closed curves ~ r = { ( Z , y) 6 [ ~ d + l . z - - Zr(t), y -- yr(t), t ~ [0, Tr)} are different for different r and max ]zr(t)- q~(,~0)] ~ 0, t

max lyr(t)- ,~0] ~ 0, t

Tr --* TO = 2~/wo

(2.142) as r --+ 0, i.e. the curves ~r shrink to the point .Ado and periods approach To. Particularly, the conditions of this proposition are satisfied if the function r~(2) is differentiable at the point 2 = 20 and the transversality condition r~'(20) ~= 0 holds, which one can consider as the main case. The simplest example of system (2.141) satisfying the conditions of Proposition 2.3.22 is x" + (y(t) - K)x' + x = g(x, y), where x and y are the input and output of the Preisach nonlinearity and g(x, y) = o(]xl + lYl) as x,y --+ 0. If u ( { ( ~ , / ~ ) : a < / ~ < 0}) < ~: < u ( { ( a , / ~ ) : a < / ~ , a < 0})

then this second-order equation has the continuum of disjoint classes [Xr(t), yr(t)] (0 < r < r0) of nonstationary periodic solutions with periods Tr such that maxt Ixr(t)l --+ 0, maxt lYr(t) - tel ~ 0 and Tr --+ 2~z as r --+ 0. If the function F(z, y) is sufficiently smooth in a neighborhood of the point M0, then the functions r~(2), w(2) are continuously differentiable and F(q~(2) + ~, 2 + v) = A(2)~ + a(2)v + g(2, ~, v)

(2.143)

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

211

for 2 close to 20 and ( ~ ~a, v ~ R small in absolute value. Let the matrixvalued function A(2) and the vector-valued function a(2) be Lipschitz continuous and let the residual term g satisfy uniformly in 2 ]g(2, ~1, Vl)]~ o(p),

(2.144)

]g(21, ~1,121) - g(22, ~2,122)]~ o(P)]~,l-22]nus(P) (]~1-~2] if- ]121-122])

(2.145)

whenever ]~jl, ]vj]~fi with e(p) = o(1), p ~ 0. In this situation, Proposition 2.3.22 is supplemented by the following statement. PROPOSITION2.3.23. Let relations (2.143)-(2.145) hold and r~'(20) ~= 0. Let the density of the measure t2 of the Preisach nonlinearity be Lipschitz continuous with respect to both the arguments ~, ft. Then there is an rl > 0 such that the curves ~/r -- {(Z, y) E ~ d + l zr(t), y -- yr(t), t ~ [0, TF)} representing cycles of (2.141) in Ra+l and satisfying (2.142) as well as their periods Tr depend Lipschitz continuously on r for 0 < r < rl. Moreover, there is a neighborhood U c Ra+l of the point ;~do ~ F such that each closed curve ~ lying in U and representing a class of cycles of (2.141) with a period sufficiently close to To belongs to the set of curves ?r, 0 < r < rl, i.e. there are no other such curves in U. .

Z

--

We stress the difference between the conclusion of this proposition and that of the statements in the begining of this section on Hopf bifurcations in systems without hysteresis. System (2.141) with the Preisach nonlinearity has a continuum of small cycles in the phase space (this continuum is the join of the classes of cycles represented by the curves 7r with 0 < r < rl), while system (2.130) has typically at most one small cycle in the phase space for each value of the parameter, the surface like that shown in Fig. 2.39 in the product ~a+l of the phase space and the parameter axis consists of cycles existing for different parameter values. Another remarkable difference is seen if we consider the asymptotics of the cycles. Small cycles of system (2.130) form a smooth surface 5 in the space [~d+l of pairs (z, b). The curves ~/r representing cycles of system (2.141) in the space R~a+l of pairs (z, y) form a structure close to a cone, as is shown in Fig. 2.40. More precisely, the curves ?r are asymptotically close to the ellipses gr = Ado + r(z,, y,) + r(C, 0) for small r > 0, where Ado + r(z,, y,) is a ray tangent to the curve F at the point ,Ad0 and C is an appropriate cycle of the linear system z' = A(20)z (the cycles rC form a plane in the phase space ~d of z' = A(20)z). The Hausdorf distance between 7r and gr is o(r) as r ~ 0; Fig. 2.40 shows the curves ~r and the 5We consider sufficiently smooth nonlinearities F(., .).

212

CHAPTER 2 Differential Equations with Hysteresis

~Z2

/ z1

FIGURE2.40 Curves 7r. two-dimensional cone surface formed by the ellipses gr with the vertex at the point .Ado. Some further details can be found in [117].

D i m e n s i o n of the Set of Cycles of System with a Parameter Let us consider again small cycles of the system (2.133) in the case of Hopf bifurcation from an equilibrium (z0, ~/0) at the Hopf bifurcation point b = b0. Here the Problem includes the 'external' parameter b and the 'internal' parameter 2. So, one might expect that the set of closed curves 7 = {(z(t), y(t), b) ~ [ ~ d + 2 . t E •} representing small cycles of (2.133) in the space R a+2 of triples (z, y, b) in a vicinity of the point.M0 - (z0, Y0, b0) with Y0 = (I)0/0) is generically two-parametric and that these curves accumulate at the points of some curve $ passing through the point.M0 and representing equilibria of (2.133). Namely, if F(zo, Y0, b0) = 0 with Y0 = cI)(r/0) satisfying (2.137), the matrix A(.,., .) = cgF/cgz(., .~ .) has a pair of simple eigenvalues • iwo (w0 > 0) at the point AF0 and proper smoothness and nondegeneracy conditions are satisfied, then the equality F(z, y, b) = 0 defines a two-dimensional surface | in some vicinity of the point d~0 in ~a+2, each point (z~, y~ b~) ~ | defines the stationary solution (z(t), y(t)) = (z~, y~) of (2.133) for b = b~ and the curve 8 c | is defined by the relations Sp(A(z, y, b)) ~ iw, Iw - w0J < ~, w > 0 with a sufficiently small ~. Yet, we do not know exact statements of this type. In problems on Hopf bifurcation from infinity for system (2.133) one should distinguish between the cases R~ = cx~ and Ra < cx~. For example, suppose that the conditions of Proposition 2.3.21 are satisfied. If R~ -- cx~

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

213

then the set of closed curves 7 - {(z(t), y(t), b) e ~d+2. t e R~}representing large cycles of (2.133) for b close to the asymptotic Hopf bifurcation point b0 is generically two-parametric; if R~ < cx~ then this set is generically (for 'nonexotic' nonlinearities F) one-parametric. One can see it more precisely from operator equations equivalent to the problem. We outline the approach roughly, not presenting any particular operator equation, which can be constructed in different ways in the form W r (z, y, b, T) = 0. Here r is an auxiliary parameter; the u n k n o w n s are the functions z = z(t), y = y(t), the parameter value b for which the class Uo<.~ r0 and every 0~<2~<1, the relations m a x I Z r , 2 ( t ) - Zr,2('c)l---> cx~, br, 2 ---> bo, Tr,2 ~ t,'c

TO = 27r/wo

as

r ~ cx~

similar to (2.138) hold uniformly with respect to 2 e [0, 1], and Zr, 2 -Zr,,2', Yr,2 -- Yr',2' if and only if r = r', 2 = 2 I. This implies that there is the two-parameter set of c u r v e s 7r,2 - {(Zr,2(t), Yr,2(t), br,2) 6 [~d+2 . t e [0, Tr,,~)} (r > r0, 0K2~<1) which are different for different (r, ;~) and that the diameter of 7r,,~ goes to infinity as r increases. Moreover, it turns out that the m a x i m u m max{d(b, 2, b,,V) " 0~2, 2' <~1} of the Hausdorf distance d(., .) between the c u r v e s 7r,2 and 7r,2' with the same r vanishes as r increases, i.e. the set U0~<,~<~l7r,;~ shrinks to the individual curve 7r,0 as r ~ cx~. If the vector c does not belong to the invariant plane of the matrix A(bo) corresponding to the pair of its simple eigenvalues +iwo, then the relations mint (c, z(t) ) ~ - c x t maxt (c, z(t) ) ~ ooasb ~ bo hold a priori for the cycles bifurcating from infinity. In case R~ < cx~ this implies (J,~x)(t) = (Jox)(t) with x(t) = (c, z(t)) for all 0~2~<1 and all the large cycles. Therefore in place of (2.146) one arrives at the system Wr(z, y, b, T) = 0, y = J0x, x = (c, z) with only the parameter r. One proves that for all r > r0 this system has solutions (Zr, yr, br, TF) satisfying (2.138). In this case, large cycles of system (2.133) form the one-parameter set of classes represented by the curves 7r (r > r0) with the diameters going to infinity as r increases. The auxiliary parameter r can be chosen in different ways. For example, a ' g o o d ' choice (which allows one to construct topologically nondegenerate systems (2.146)) is r -- Ilzllc. With this choice, the above statements

CHAPTER 2 Differential Equations with Hysteresis

214

guarantee the existence of cycles for all sufficiently large amplitudes of the z-component. Statements of this type on Hopf bifurcations from infinity for systems with the Prandtl-Ishlinskii hysteresis nonlinearities (a class of nonlinearities close to a particular class of the Preisach models) in case Ra - ~ are presented in [116]. Case Ra < cx~ for systems with the stop hysteresis nonlinearity was studied in [168], including stability analysis of large cycles; a similar approach can be applied here.

2.3.8 CHAOTIC BEHAVIOR Definition of Chaos Letf : S ~ S be a continuous mapping in a subset S of an Euclidean space Re. Important attributes of chaotic behavior include sensitive dependence on initial conditions, an abundance of periodic trajectories and an irregular mixing effect describable informally by the existence of a finite number of separated subsets U0. . . . . Urn-1 of Re which can be visited by trajectories of some fixed iteratef k off in any prescribed order. Let/d = {U0 . . . . . Urn-l}, m > I be a family of disjoint subsets of S and let us denote the set of onesided sequences co = o90, o91. . . . " ogi 9 {0, 1 . . . . . m - 1} by f~R m. Sequences in f~R m will be used to prescribe the order in which sets Ui are to be visited. m-1 For x 9 Ui=0 Ui we denote by I(x) the number i satisfying x 9 Ui. A mapping f is called (/d, k)-chaotic (k is a positive integer) if there exists a compact f-invariant set Z C ~Ji Ui with the following properties: (p l) for any o9 9 f2R there exists x 9 Z with fik (x) = (fk')i 9 Uo~i for i~1; (p2) for any p-periodic sequence o9 9 f~R m, there exists a pk-periodic point x 9 Z with fik(x) 9 U~i; (p3) for each ~/> 0 there exists an uncountable subset Z(r/) of S, such that the simultaneous relationships lim supi__,~

I(fik(xl)) -- l(fik(x2))] ~1,

lim infi~ c~

~ik(xl) -- fik(x2)l < rI

h o l d for all Xl, x2 6 I ( r ] ) , Xl # x2.

The above-defined properties of chaotic behavior are similar to those in the Smale transverse homoclinic trajectory theorem with an important difference being that we do not require the existence of an invariant Cantor

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS TABLE 2.2 i

V~1) (i)

V~2) (i)

0 1 2 3 4 5 6 7 8

2.595 2.597 2.606 2.641 2.808 3.045 2.677 2.622 2.602

1.286 1.278 1.258 1.189 0.9016 1.150 1.599 1.366 1.304

Va(1) (i) 0.001328 0.001194 0.001099 0.001086 0.001535 -0.002766 0.001789 0.001615 0.001465

215

Parallelograms Va(2) (i) -0.002690 -0.002419 -0.002190 -0.002059 -0.002426 0.004905 -0.002620 -0.003255 -0.002989

v~1) (i) 0.003545 0.004168 0.004869 0.005637 0.007217 0.0003166 0.001140 0.002219 0.002922

v~2) (i) 0.009351 0.01097 0.01284 0.01472 0.01352 -0.009191 0.006788 0.007069 0.008053

set. Instead, the definition includes property (p2), which is usually a corollary of uniqueness, and (p3), which is a form of sensitivity and irregular mixing as in the Li-Yorke definition of chaos [169], with the subset/701) corresponding to the Li-Yorke scrambled subset So. A similar definition was used previously in [170]. Note also that the one-sided left shift cr+ is a factor of the restrictionfkls (See [171], p. 68).

Chaos in an ODE with Preisach Nonlinearity Consider Eqn. (2.9) with b = I and an appropriate Poincare map F1 : FI 1-I generated by it. Let v = (v (1), v (2)) denote the element of FI. A part of the co-limit set of F that was constructed with help of the computer is given in Fig. 2.21, (p). Introduce the parallelogram sets R(i) ~ l-I, i = 0 . . . . . 8 given by the center points vc(i), and the halves of the side vectors va(i) and vb(i):

R(i) -- {vc(i) 4- ~va(i) 4- flvb(i) : --1 < ~, fl < 1}.

(2.147)

Values of vc(i), va(i) and vb(i) are given in Table 2.2. Let/d denote the family of connected components of the set ~8=0 R(i). 2.3.6. The family bl has nine components, i.e. R(i) N R(j) = O, i ~ j, i , j = O . . . . . 8.

LEMMA

PROOF. This is clear from the definition of set R(i). See also Fig. 2.41.

THEOREM2.3.5. Mapping F1 is (ld, 16)-chaotic. Proof of this theorem is given in Section 2.4.4.

[]

CHAPTER 2

216

Differential Equations with Hysteresis

I

I

'

I

'

I

'

1.35 1.8 1.3 1.6 1.25 1.4

-

,

I

,

2.59

I

,

2.6

I

,

2.61

I

,

2.62

I

2.63

1.2 -

1-

0.8 2.5

2.6

F I G U R E 2.41

2.7

2.8

2.9

3

Sets R(i), i = 0 . . . . . 8 in c o o r d i n a t e s t m o d V ~ r , z (2).

2.5 -

~,,~

-

1.4

r :Y~'G

1.5

v-,,,, #?.. , : , . ~ , - 4 d ~ ".

:V,

..

~" \'~

2.6

2.7

2.8

2.9

0.5

1 F I G U R E 2.42

2

3

4

Sets R(i), i = 0 . . . . . 8 a n d a c h a o t i c a t t r a c t o r .

5

,

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

217

Note also the following. PROPOSITION2.3.24. The map F1 has a fixed point v, ~ R(0) and for any positive

integer p greater than eight there is a periodic point v ~ R(O) which has the minimal period p, and which is not asymptotically stable. Moreover, Eqn. (2.9) has periodic solutions of minimal periods v/2rc 9 (3kl + 29k2), where kl, k2~>0, kl + k2 > 0. In particular there exist solutions of all periods x/-2rcn, with n > 55. Proof of the proposition closely follows ideas in [172,173].

Interpretation In this subsection we discuss another interpretation of our main result from the previous section. It is convenient to begin with some definitions in terms of Eqn. (2.13). We will always suppose that the functionf(t, z, u) is periodic in t with minimal period T > 0. Thus any periodic solution of that equation must have a period which is a multiple of T. Let no be a positive integer and zo(t), - ~ < t < ~ , (2.148) be a function with the minimal period noT. This function will play the role of an approximate n0T-periodic solution of Eqn. (2.13). We will always deal with the situation when the function zo(t) is piecewise continuous, and the limits z(t - 0) = lim z(t)

rSt

are well defined at the points of discontinuity. In our setting the wording 'the function zo(t) can be considered as an approximate n0T-periodic solution of Eqn. (2.13)' means that, firstly, the jump Iz0(t) - zo(t - 0)l at any discontinuity point t is rather small, and, secondly, the function almost satisfies the equation between successive points of discontinuity (the quantitative bounds for the corresponding discrepancies will be specified below). Suppose that for a positive integer nl < no, the number Iz0(0) - z0(nlT - 0)1 is small. Then we can treat the niT-periodic function Zl(t), - o e < t < oe, which is defined by zl(t) = zo(t), 0 <~t < niT, as an approximate niTperiodic solution of Eqn. (2.13). Since no is the least period of zo(t) and

CHAPTER 2 Differential Equations with Hysteresis

218

z

Zl.

$_;

,..G/ Zo

x

t

!

FIGURE 2.43 Functions z0 (solid) and Zl (dotted). Here z0 has period 2T, z I has period T; thus z0, Zl coincide on 2nT<.t < (2n + 1)T, n = 0, 4-1, 4-2. . . . in this case. nl < no, w e define the distance r b e t w e e n the c o r r e s p o n d i n g functions

zo(t), zl (t) b y r = sup Izo(t) - zl(t)l > 0.

(2.149)

t

Figure 2.43 illustrates the situation for the case no = 2, nl - 1: the functions z0 a n d Zl are s h o w n by a solid a n d a dotted line, respectively. We can use as a p p r o x i m a t e patterns for possible b o u n d e d o u t p u t profiles, not only the functions zo(t) a n d zl(t) themselves, b u t also all possible 'shuffiings of these functions'. To explain w h a t this m e a n s w e need additional notation a n d definitions. Let f~ be the totality of all bi-infinite oo b i n a r y sequences o9 = {o9i}i=-oo w i t h ogi E {0, 1} for i = 0, 4-1, 4-2 . . . . . To each sequence co e f2 there corresponds the sequence of n u m b e r s ti to be defined b y to = 0, i-1

ti = T E no~] j=0 for i = 1, 2, 3 . . . . . a n d b y i

ti--T

E

no~j

j=-I

for i = - 1 , - 2 , - 3 . . . . . . That is, the real line R 1 is partitioned into the intervals Ii = [ti, ti+l) w h e r e the length of the interval Ii equals no~iT. Figure 2.44 illustrates this idea: here 09= . . . . 0 , 1 , 0 , 1 , 1 , 0 . . . . . a n d the element coo is in bold.

(2.150)

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

-T

0

T

219

2T

3T

4T

FIGURE 2.44 Partition of R 1 corresponding to o~ . . . . . 0, 1, 0, 1, 1, 0 . . . . ; w e show the intervals corresponding to coi = 0 by the solid lines and intervals I i, corresponding to coi = 1, by the dotted line, with co0 = 0 in bold.

Z l ZI.

9

M

-.~.. .~"

Z1"

Z1.

ir

".~

If ~

l

,

'0

i

r

CT

l

~3T

i

/

/

if

,

,

I

~4T

t

I

!

z0 FIGURE 2.45

Function zo~ for o~ . . . . .

0, 1, 0, 1, 1, 0 . . . .

C o r r e s p o n d i n g to a n y o9 ~ f~ t h e f u n c t i o n zoo(t) is d e f i n e d b y

zoo(t) - zo~i(t - ti),

for

t i ~ t < ti+l.

F i g u r e 2.45 s h o w s a f r a g m e n t of t h e f u n c t i o n zo~ for t h e case w h e n co satisfies (2.150). We will u s e t h e n o t a t i o n Z ~ = {zm(t) 9 o9 ~ f~}. PROPOSITION 2.3.25. A n y two functions Zm~l), zo)(2) 09(2) . Moreover, sup

IZco(1)--Z(o(2) I -- r(09 (1), to(2))~rmin > 0,

~_

Z ~ are different if 09(1)

09(1) 09(2) E ~'-~, ~(1) ~: 09(2).

-c~
inf r(09 (1), 09(2))/~rmin > O, co(l) ~_co(2)

09(1), 09(2) E ~

is strictly positive. We call r ( Z n ) t h e s e p a r a t i o n t h r e s h o l d for t h e set Z n .

(2.152)

CHAPTER 2 Differential Equations with Hysteresis

220

Z

zo)(t) + e / I ~" ~ " ~ ,

II l ~

,/

"

~ '",

,'

'" .... ',,

"

,' ,'

"',

'" ....

"~, 9 ,

'

"'"

r

, i

,

2~T.... "'/

',,

!

I

.

I

~

9 ~ aL

~" ~

'

,' I 9

~

IL

""

9 ',.: v

:'T--'" " "

t ,"

",

'

zto(t)- e

FIGURE 2.46 A function z(t) (solid dark curve) satisfying (2.155), with o~ . . . . . O, 1, O, 1, 1, 0 .... If the sequence o9 is p-periodic in the sense that ogi = 0, 4-1, +2 . . . . . then the function z~(t) is T~-periodic where

ogi+p

for i =

p-1

To~ - T Z

n~i"

(2.153)

i=0

The number (2.153) is the minimal period of the function zoo, provided that p is the minimal period of the sequence co. Thus, among the elements of the set Zfl there are periodic functions with arbitrary large minimal periods. However, the main bulk of the set Zfl consists of functions which do not follow any particular pattern. Let e be a positive number satisfying e < r(Zn)/2.

(2.154)

We will say that Eqn. (2.13) is (zo, Zl, e)-compatible if the following two conditions hold: (C1) For any function zoo(t), o9 ~ f2 there exists a solution z(t) of Eqn. (2.13) satisfying the uniform estimate max

-cx~
Iz(t) - z~(t)l < e.

(2.155)

(C2) Additionally, if the sequence o9 is p-periodic, then there exists a T~periodic solution z(t) of Eqn. (2.13) satisfying the uniform estimate (2.155), where To~ is given by (2.153). Figure 2.46 graphs a function z(t) that satisfies (2.155), where o9 satisfies (2.150).

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

'

i

i

!

'

i '

2

"

221

i

'

i

'

'

1'0

'

i

'

_

0-1 -2

,...

,

0

|

20

,

i

40

,

!

60

,

i~ , 80 100 120 140

-20

;

1;

'

20

FIGURE 2.47 A function zo(t) on the left and a function z l(t) on the right. In the scalar function z for small e, functions z(t) satisfying (2.155) follow closely the 'guidance' of a c o r r e s p o n d i n g function zoo(t). In the term i n o l o g y of hyperbolicity theory, a function z(.) is an e-shadow of zoo(-). Since e is strictly less than half of the separation threshold r(Zr any solution z(t) can be considered as an e-shadow of not more than for one guide zoo. Thus, the m e a n i n g of (z0, Zl, e)-compatibility is that for any pattern given b y a function zo~ ~ Zr we can find a b o u n d e d solution profile which is an e-shadow of zoo. Consider Eqn. (2.9) with b = 1. As zo(t) = (xo, x'o) we choose the numerical solution of (2.9) c o r r e s p o n d i n g to the pseudo-trajectory of F1 of the length nine, which is given as the u n i o n of nine numerical trajectories with initial conditions x(ti) = 0.5, x'(ti) v~2) (i) on the intervals [ti, ti+l), i - 0 . . . . . 8; V~1) , V~2) are given in Table 2.2 and ti is given b y -

ti =

-

v~1) (i) + 3 i - v ~ n , v~1) (i) + (3i + i - 4 ) . ~/2n, v~1) (i) + (3i + 2). ~/2n,

i - 0 . . . . . 4, i - 5 . . . . . 6, i = 7 . . . . . 8,

vf 1) (0) + 29. v ~ n ,

i - 9.

Thus no = 29. Let us choose z] (t) as zo(t) on the interval [to, to + 3- ~/2n), in which case n l -- 3. The pictures of the w a v e g u i d e s are given in Fig. 2.47. LEMMA2.3.7. The following relation holds: r(Zn) > 0.1. By this l e m m a the (z0, Zl, e)-compatibility is well defined w h e n e v e r e<0.05. The principal result of the previous section, T h e o r e m 2.3.5, m a y be rewritten n o w as follows.

222

CHAPTER 2 Differential Equations with Hysteresis

THEOREM2.3.6. The equation is (zo, Z l , O.O1)-compatible.

Why It is Interesting and Important The (z0, zl, s)-compatibility implies, in particular, that the totality of bounded solutions is spatially chaotic in a topological sense. We recall, as an aside, that the three main aspects of chaotic behavior are equally important: the measure theory aspect such as the existence of the Sinai-Ruelle-Bowen invariant measures [174]; the 'sensitivity' aspect, such as positivity of Lyapunov exponents [171]; and the topological aspect describing irregular mixing properties in terms of symbolic dynamics [171]. At an informal level the topological aspect of the spatially chaotic behavior can be described as follows. Suppose that we register the forms of the solutions within a certain precision ~, where ~ is greater than s, but significantly less than the separation threshold (see the relevant numerical information in the next section). Then a 'shadow solution' z(t), satisfying (2.155), is indistinguishable from its guide zo~(t), although different guides are quite distinguishable. Thus any chaotic shuffling zo~ of two basic patterns z0, Zl provides a possible steady profile (within the accuracy ~). The next level of formalization requires ideas of symbolic dynamics. Denote by n the least common multiple of the numbers no, nl. Also let e [0, noT) be a fixed number such that Iz0(z) - Zl(Z)l > 2s (such a ~ exists by (2.154)). Firstly, we characterize a function z(t), -cx~ < t < cx~,by the bi-infinite sequence of its values Zi - -

z(inT + z),

i = O, 4-1, 4-2 . . . . .

(2.156)

That is, we measure the values of z(t) 'stroboscopically' with the uniform interval nT between the measurements. Further, encode the function z(t) by a binary sequence o9 e f~, which is constructed as follows. Let

Io = {z:lz- z00:)l~e},

I1 -- {z:lz- Zl('01- ~s}.

The sets I0 and 11 do not intersect since Iz0(z) - Zl('01 > 2S. If for any i the value zi belongs either to I0 or to I1, then we define the code sequence o9 - co(z(.)) by setting o g i = 0 in the first instance, and ogi = 1 in the second. In other words, we indicate the measurement positions inT, i = 0, 4-1, 4-2 . . . . where the function z(t) goes through a window I0 or I1 (see Fig. 2.48 for one-dimensional interpretation). If the value zi does not belong to one of these two windows for some i, then our coding procedure fails, and we are not interested in such 'unencodable' functions z(t). Since I0 N I1 -- 0, not more than one code can correspond to a particular function z(t). N o w the compatibility condition (C1) implies that any random

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

223

Z

I I f

t

FIGURE 2.48 A function z(t) with the code .... 0, 1, 0, 1, 1, 0 . . . . . sequence o9 a p p e a r s as a code for some b o u n d e d profile. (Indeed, let us introduce a sequence o9* b y substituting n/no ones (n/nl zeros) instead of a n y one (any zero) entry of the sequence o9. Then, the function z~. (t) ~ Z~ has the required properties.) For a given function z(t) w e denote b y ;:(-) = anTZ(.) its (left) nT-shift to be defined b y ;:(t) = z(t + nT). Let us denote also b y a the s t a n d a r d (left) shift on f2 given b y a(og) = o91 -- ( . . . . . o9I 1' 090' r 091t . . . . ) !

w h e r e ogi = o9i+1. For instance, if o9 - . . . , 0, 1, 0, 1, 1, 0 . . . . , then ogl

...,0,1,0,1,1,0

.....

PROPOSITION 2.3.26. Suppose that Eqn. (2.13) is (zo, Zl, e)-compatible. Then to

each o9 ~ ~ there is a solution z~(x) of Eqn. (2.13), such that z~(.) has the code co, and this correspondence og~-+zo~is shift invariant: O'nTZo~(') = Zao~(') for og~f~. Proposition 2.3.26 can be p r o v e d in a s t a n d a r d w a y using the Z o r n L e m m a . In technical terms, the above proposition m e a n s that the restriction O'nT of the shift an0T to the set {z~ : o9 ~ f2} is semi-conjugate to the shift a in f2. Such semi-conjugacy is w i d e l y recognized as the m o s t i m p o r t a n t attribute of topologically chaotic behavior, see, for example, [171]. It is also i m p o r t a n t to note that (z0, Zl, e)-compatibility implies the existence of solutions w i t h a w i d e range of nontrivial regular patterns. For instance, one more basic characteristic of chaotic b e h a v i o r is an a b u n d a n c e of periodic solutions and, in particular, an exponential rate of increase of the m a x i m a l n u m b e r of pairwise e-separated 6 kT-periodic solutions w h e n k increases, see [ 171]. 6Two functions z(x), y(x) are E-separated if sup Iz(x) - y(x)l > e.

224

CHAPTER 2 Differential Equations with Hysteresis

PROPOSITION2.3.27. Suppose that Eqn. (2.13) is (z0, Zl, e)-compatible and denote by n the least common multiple of the numbers no, nl. Then it has at least 2 k periodic solutions of period kn which are pairwise e-separated. This assertion follows immediately from the definitions. Another example of functions with a regular pattern is given by the socalled recurrent functions [97], which exhibit fractal-type features. The recurrence property of a function z(t) with - c ~ < t < o0 means that for each pair of positive numbers (e, a) there exists A > 0 such that for each real r, any interval which is longer than A contains r such that the function t --, z(t + r) is e-close to the function t ~ z(t + a), t ~ [-T, T]. Recurrent nonperiodic functions are important today in the analysis of desynchronized systems, various new models in biology, etc. The notion of a recurrent sequence co is defined analogously. Recall, in particular, the classical Morse example of a recurrent sequence. We begin from the one-element sequence Sl = 0 and then substitute at each step the two-element sequence 0, 1 instead of 0 and two-element sequence 1, 0 instead of 1. Thus, Sl = 0 , 1 ; s2 = 0 , 1 , 1 , 0 ; $3 = 0 , 1 , 1 , 0 , 1 , 0 , 0 , 1 ; S4 = 0, 1, 1,0, 1,0,0, 1, 1,0,0, 1,0, 1, 1,0; etc.

After infinitely many iterations we obtain the famous forward Morse sequence M. Considering backward shifts of this sequence and their weak limits we arrive at the set A//of bi-infinite Morse sequences. PROPOSITION2.3.28. Suppose that Eqn. (2.13) is (zo, zl, e)-compatible and the sequence co is recurrent in the Morse sense (for instance, one of the set .M). Then there exists a recurrent, nonperiodic, solution satisfying (2.155). The proof follows the standard constructions [175]. 2.3.9 RECURRENT OSCILLATIONS

We will be interested in recurrent solutions of the general equations

x' =fit, x, y(t)), y(t) = (Fit0, y(to)]x) (t).

(2.157) (2.158)

Here x ~ ~a, and y belongs to a complete metric space 3; with the metric py. Also, for a fixed Y0, the operator F[t0, Y0] transforms functions

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

225

x 9 [to, oo) ~ ~a into functions y 9 [to, oo) ~ 3;. For a particular function x(.), the notation (Fit0, y0]x) (t) refers as usual to the value of the corresponding function y(.) at the m o m e n t t. The operators describing the hysteresis nonlinearities can be treated with a typically infinite dimensional set 3; of internal states y. We will use the definition of recurrent property introduced on the previous page. Periodic, quasiperiodic and almost-periodic functions are recurrent, however the class of recurrent functions is m u c h wider. The typical examples of recurrent, but not almost-periodic behavior are given by various modifications of the classical Morse construction [176]. Such functions are getting rather important today due to analysis of desynchronized systems [177], various new models in biology, etc. We will establish, u n d e r some natural technical restriction, that the system (2.157)-(2.158) has a recurrent solution, providing that the function f in the right-hand side of Eqn. (2.157) is recurrent in t, and that there exists at least one solution (x(t), y(t)) which is uniformly b o u n d e d at t --+ oo. Surprisingly, some of our results seem to be new even for Eqns. (2.157)(2.158) without hysteresis.

(W, C)- and s

Functions

Let us introduce some notation. 9 Let a function a of scalar a r g u m e n t be measurable at the interval [r, r~] and take values in a finite-dimensional Euclidean space. We denote II~ll&.~ =

I~(t)l dt.

(2.159)

9 If a function ~ is additionally absolutely continuous at [~, rr], denote

Ilallw~,~ = la(~)l + Ila'lls

(2.160)

9 For any continuous functions yl, Y2 : [z, rT] --+ 3;, we define

P:,cr(Yl, y2)

-- max P3;(yl(t), y2(t)). te[~,cr]

(2.161)

We denote by Wr,r aa the Banach space of absolutely continuous functions x 9 [r, cr] --+ R a with the n o r m (2.160). Analogously, denote by C~,~;3; the complete metric space of continuous functions y : [~, or] ~ 3; with the uniform metric (2.161).

CHAPTER 2 Differential Equations with Hysteresis

226

Denote the z-shift of a function y by 0(r)y:

(O(r)y) (t) = y(t + r). We say that a set ,5' c ~ is A-dense if any interval [t, t + A] contains at least one element from $; that is ,_q~[t, t + A] ~a 0, for all t ~ ( - ~ , ~ ) . Let a function x 9 R ~ R~a be absolutely continuous on any finite interval, whereas a function y : ~ ~ 3; is continuous. The pair (x, y) is said to be (W, C)-recurrent if for any positive T, ~ there exists A = A(T, e) such that the set of rT ~ ( - ~ , o0) satisfying

II0(~)x- O(~)xlIw_T,T + P-T,T(O(r)Y, O(cr)y) < e is A-dense for each z ~ (-cx~, cx~). Let a functionf be measurable in t and continuous in x, y (that is, it satisfies the Caratheodory condition). Let also f satisfy the following growing condition: TT Ixl,
T< ~

(2.162)

for any T, R > 0. The function f is s in t if for any positive numbers T, R, e there existsA = A(T, R, e) such that the set of cr ~ (-oo, o0) satisfying T

max Ll:(t + ,, x, y) - f i t + or, x, y)l dt < e T ]xI<<.R,Y~3;

(2.163)

is A-dense for each 9 ~ (-oo, oo).

(W, C)-Correct Hysteresis Nonlinearities We will use a standard approach to describe hysteresis nonlinearities, s e e [15] and the references therein. We suppose that the operator F[t0, Y0] can be understood as a description of an autonomous control dynamical system F. That is, the family of operators F[t0, Y0] must satisfy the following properties: 9 the Volterra property: X(S) =-- Xl(S), S E

[to, t],

implies

(F[t0, y0]x) (t) = (F[t0, y0]xl) (t);

9 the semigroup property: (F[tl, (F[t0, y0]x)(tl)]X)(t) - (F[t0, y0]x)(t), for to < tl < t;

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

227

9 the autonomy property: (Fit0, y0]x) (t) = (F[t0, y0]Xl) (t -- t o q- tl), where Xl(t) = x(t - to + tl) and to < tl < t. N o w we introduce some additional properties of the hysteresis operators F. We will suppose that the operators F[t0, Y0] are defined for all absolutely continuous functions x 9 [0, oe) --~ ~ct and that their outputs are continuous. Using the Volterra property, the operators Fit0, Y0] are also well defined at any finite interval [to, tl]. Recall, that a set $1 of measurable functions x 9 [0, t] ~ ~a is uniformly integrable if the relationship f lim / [x(t)[ dt = 0 2--,oo dlx(t)l>2 holds uniformly in x 6 51. A set $2 of Y-valued continuous functions is said to be equicontinuous if for any e > 0 there exists 6 > 0 such that p y ( y ( t l ) , y(t2)) < e,

It1 - t 2 ] < 6, y 6 $2,

holds. We say that a nonlinearity F is (W, C)-correct if for any uniformly bounded set $1 of absolutely continuous functions, such that the set of derivatives {x', x 6 $1} is uniformly integrable, the set $2 -- {l-'[0, yo]x : Yo ~ 3;, x

~

$1}

is equicontinuous, and if the relationships [IXn - X]lWto,t 4- P 3; (,YO(n) ' Yo) ---> 0

imply p y ((F[t0, y~n)]xn)(t), (F[t0, yo]x)(t)) ~ O.

for any - ~ < to < t < c~. As examples of (W, C)-correctness we mention the Prandtl, Besseling, Ishlinskii, Mizes, Tresca, etc. models of plasticity, Madelung, Preisach, Giltay, Mayergoys, etc. models in ferromagnetism and many, many others, see [7,14,15,157] and the bibliographies therein. Actually all hysteresis nonlinearities with continuous output signals known to us belong to the class of (W, C)-continuous nonlinearities.

228

CHAPTER 2 Differential Equations with Hysteresis

Main Result Let us come back to the system (2.157)-(2.158). The function f in the right-hand side of (2.157) will always satisfy the Caratheodory condition. We will understand the solutions of Eqns. (2.157)-(2.158) in the classical sense: solution is a pair (x, y) where x 9 [to, cx~) "---> RJ is a locally absolutely continuous function and y : [to, cx~) :--+ 32 is continuous; the relationships (2.157) should hold almost everywhere and the relationship (2.158) everywhere. (Recall [178] that the function x 9 [~, a] ~ ~d is absolutely continuous if x' exists almost everywhere, is integrable on [r, a], and x'(t)dt = x(tl) - x(t2) for r < tl < t2 < a.) The pair (x, y) where the functions x and y are defined over the whole interval (-cxz, oo) is said to be a solution of the system (2.157)-(2.158) if the restrictions of the functions x, y to the interval [to, ~ ) provide a solution of this system for each to 6 (-cx~, oo). The last definition is correct by the Volterra property.

ft~

THEOREM2.3.7. Let f be s

in t and let F be (W, C)-correct. Let (x,(t), y,(t)),

t>~O,

be a solution of the system (2.157)-(2.158) with the function x,(t) uniformly bounded. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). N o w we will give some general comments concerning the use of the preceding theorem. Under simple additional assumptions, that hold for all practically important hysteresis nonlinearities, any initial problem

x(to) = xo ~ Rd,

y(to) -- Yo ~ Y,

(2.164)

has at least one solution (x, y) defined at the maximal interval [to, T) where either T = ~ or Ix(t)l --* ~ as t -+ T. It is sufficient, for instance, to suppose that the hysteresis nonlinearity F is continuous from C to C (as in the case of Prandtl, Besseling, Tresca, Mises, Giltay, etc. nonlinearities) or is continuous from W to W (as is the case for the Madelung nonlinearity). The proof proceeds in the usual way using the Showder fixed-point theorem. We will say in this situation that the initial problems for Eqns. (2.157)-(2.158) have local solution. If the initial problems have local solutions, we say that the system (2.157)-(2.158) is dissipative if for any R > 0 there exists R1 > 0 such that lim sup Ix(t)l~R, t--, oo ix01~
2.3 EXAMPLES OF RIGOROUS CONCLUSIONS

229

where x(t) is a solution of Eqn. (2.164); the sufficient conditions of dissipativity are easy to suggest in terms of Lyapunov functions, etc. COROLLARY2.3.12. Let f be s in t and let F be (W, C)-continuous. Let the system (2.157)-(2.158) be dissipative. Then there exists a (W, C)-recurrent solution (x(t), y(t)) of the system (2.157)-(2.158). Note that for analysis of stability of recurrent solutions, the method suggested in [83] is applicable without trouble.

2.3.10 HYSTERESIS VERSUS TIME RELAXATION Generalities Time relaxation and hysteresis are common strongly nonlinear phenomena which occur in many industrial, physical and economic systems. Often these two types of phenomena manifest different stages of the same or similar processes. A number of fundamental hysteresis models can be considered as limit cases of time relaxation processes, or admit an approximation by a differential equation which is singular with respect to a particular parameter. However, the level of interaction between theories of systems with time relaxation and systems with hysteresis (and between the 'relaxation' and 'hysteresis' research communities) is quite low. The aim of this section is to discuss, at the very basic level, relationships between those two topics. An instructive singularly perturbed system is the following

x' =fit, x, y), ey' = g(x, y),

(2.165) (2.166)

where e > 0 is a small parameter. We concentrate first on the auxiliary equation

~y' = g(x(t), y), where x(t) is an arbitrary function. We denote the solution of this equation, with a given initial condition y(0) = Y0, by

y(t) = VE[t0, yo]x(t). Under suitable technical conditions it can be proved that for a generic function x(t) there exists a limit

y~(t) = V[to, yo]x(t) = lim V~[t0, yo]x(t). 6--+0

CHAPTER 2 Differential Equations with Hysteresis

230

g(x,y)
/s x,~ X

g(x,y)>O

FIGURE 2.49 Relay as a singularly perturbed ODE. The solutions of the original system are then approximated as e ~ 0 by the solutions of the differential-operator equation

x' =fit, x, y(t)),

y(t) = V[to, yo]x(t).

(2.167)

The following observation is straightforward: LEMMA2.3.8. The limit operator V is rate-independent. Thus, the global analysis of singularly perturbed equations becomes, in the first approximation, an investigation of equations with hysteresis nonlinearities. This is the first point of contact between equations with hysteresis and singularly perturbed equations. We will demonstrate how this observation works below. Let the function g be described by the picture in Fig. 2.49. Suppose x(t) does not have local minima equal to a or local maxima equal to ft. Then the operator V is well defined on x(t) and

V[to, yo]x(t) = R~,fl[to, yo]x(t) for all t > to such that

x(t) # a

and

x(t) # fl.

(See Section 2.1.2 for the definition of Ra,fl.) This example shows that singularly perturbed equations can approximate equations with hysteresis. This is the second point of contact between equations with hysteresis and singularly perturbed equations. This approach is especially important if there are some physical reasons to believe that the relay (or any other discontinuous hysteresis nonlinearity) is an idealization of a limit operator similar to V. Sometimes, however,

2.3 EXAMPLESOF RIGOROUS CONCLUSIONS

231

discontinuous hysteresis operators are idealizations of quite different continuous operators. The application of methods used in systems with singular perturbations to these new types of singular approximations is the third point of contact between equations with hysteresis and singularly perturbed equations. We note in conclusion that numerous mathematical models of physical, mechanical and other systems can be described by the equations of the form

x' =fit, x, y, z)~ y = Fx(t),

ez' = g(x, y, z) where F is a continuous hysteresis nonlinearity acting on function x(t) (for, instance, Preisach nonlinearity P a with an absolutely continuous measure #). The analysis of such systems will require the combination of ideas both in the area of systems with hysteresis and in the area of singularly perturbed systems. The main technical problem lies in the fact that hysteresis operators are not smooth in a standard sense: they only satisfy a Lipschitz condition as operators in function spaces. Thus, even the notions of stable and unstable manifolds must be adjusted accordingly.

Generalized Play Ferromagnetic monocrystals can be considered as an idealization of an interesting hysteresis nonlinearity: the generalized play [15]. This will be denoted by L~, and is illustrated in Fig. 2.50. Formally, this nonlinearity can be described as follows. We denote by f2 the set of pairs (x, y) satisfying

?_(x)~y~7+(x). For a continuous input x(t) and for initial state r/0 satisfying (x(to), rio) ~ f2, the output y(t) = L~[to, r/0]x(t) is the unique continuous function such that, firstly, y(to) = r/0, secondly,

(x(t), y(t)) ~ f2,

t~to,

and, thirdly, the inclusion

(x(t), y(~)) ~ f2,

~t~r~

implies y(t) =_ y(r), z~tKr~. Thus, the output behaves 'lazily': it prefers to be unchanged when the phase pair (x(t), y(t)) belongs to f2. Fig. 2.51 shows a typical phase trajectory.

232

CHAPTER 2 Differential Equations with Hysteresis

V_(x)

,

1

='? / 13+~

x

(x) FIGURE 2.50 Generalized play Le.

Generalized play L~ can be also represented by the Preisach m o d e l with an appropriate measure. LEMMA2.3.9. Suppose x(t) has no local minima equal to ~ or local maxima equal

to fl, and let Yo equal either 0 or 1. Then the limit y,(t) = lim L~[to, yo]x(t) e-+ O

is well defined, and y,(t) - 2R_Kl~[t0, yo]x(t) - 1 for all t > to such that x(t) # ~

and

x(t) # fl.

In other words, for generic input functions x(t) the discontinuous nonlinearity 2R_K~x(t) - I is a limit of continuous nonlinearities LE[to, Y0] as E ----~ 0.

233

2.3 EXAMPLES OF RIGOROUS CONCLUSIONS y

ar

ill

v

X

x~

FIGURE 2.51 Phase portrait of a typical trajectory for the generalized play Le.

Hysteretic Canards We consider the equation x" + x - b sin(t) + y(t),

y(t) - 2 R _ K ~ x ( t ) - 1.

(2.168)

By L e m m a 2.3.9 this is to be considered as an idealization of the equation x" 4- x = b sin(t) 4- y(t)~

y(t) = L~x(t)

(2.169)

where Le is a generalized play from the previous subsection, and e > 0 is a small parameter. We will say that a pair (x,(t)~ y~(t)) is a limit periodic solution of Eqn. (2.169), if it can be approximated by some 2~-periodic solutions (x~ (t), ye (t))

234

CHAPTER 2 Differential Equations with Hysteresis

of the system (2.169) with arbitrary small e. That is, if for any a > 0 there exists e0 > 0 such that for e < e0 Eqn. (2.169) has a periodic solution xe(t), ye (t) satisfying max [xe(t)- x,(t)[ < a. THEOREM 2.3.8. Let 0 < b < 4/n, then Eqn. (2.169) has exactly three limit periodic solutions:

(xo(t), yo(t)),

(xl(t), yl(t)),

(x2(t), y2(t)).

(2.170)

We n o w give explicit formulas for the solutions (2.170). Introduce the functions b x0(t, b) = - f l cos t + ~ (sin t - t cos t)

nb +-v-(cos t - 1) sin t, 4

(

Xl t + n-~:.b 2 '

)

=-(l+fl)

+I+

cost+~

- n K t < n,

sin~sint-tcos

t+

2

nb (cos t - 1)(sin t + 1), ~ - 2nKt < ~, 4 sin(z/2)

x2(t) = - x l ( t - n). Here ~ = v(b), 0 < b < 4 / n is defined by the relationships 1-fl arccosl_Ffl
b=2

f l - 1 + (fl+ 1)cosz sin(r/2) (sin ~ + "c- 2n)

Introduce also 2n-periodic functions y0 (t), yl(t), y2(t) by the equalities nb ~ sint,

y0(t;b)=

Yl t +

n-r.b)= 2 '

1

-nKt
nb r --~-sin~(l+sint),

y2(t) = - y l ( t - n).

r-2n~
235

2.4 EXAMPLES OF TECHNICAL TOOLBOXES PROPOSITION2.3.29. The functions given above satisfy Theorem 2.3.8.

Theorem 2.3.8 and Proposition 2.3.29 are similar to those proved in [179]. Figures 2.52, 2.53 show x0(t), yo(t) w h e n / / = 1. Similarly, Figs. 2.54, 2.55 show the functions xl(t), Yl (t). We claim that these limit solutions are analogs of canard solutions, see [180-182]. In particular, (xo(t), yo(t)) is an analog of the symmetric canard and (xl (t), Yl (t)) is an analog of the asymmetric canard. 9 The first reason is that classical canards and these solutions are mathematical explanations of the same physical phenomena: the limit behavior of periodic solutions of a family of differential equations approximating a given equation with a discontinuous, relaytype nonlinearity. 9 Secondly, in both cases the limit solutions demonstrate strange behavior: in the classical canard situation for a time they follow unstable manifolds; in the 'hysteretic canard' situation, the solutions can stop in the course of a jump, and then hover for a while at an intermediate level. 9 Thirdly, similar methods could probably be applied to analyze both situations. For example, the topological degree approach used to prove the last theorem can be useful for the rough location of classical canards. On the other hand, the typical 'canard community' methods of investigation will be useful in a more subtle asymptotic analysis of hysteresis canards.

2.4 E X A M P L E S

OF TECHNICAL

TOOLBOXES

2.4.1 DESCRIPTIONS OF PREISACH NONLINEARITY Standard Representation States and Outputs Consider the set f2 on which the internal states ~/ : f2 ~ {0, 1} of the Preisach nonlinearity P are defined. Under natural technical restrictions this set can be chosen to be the two-dimensional half-plane FI = {(~, fl) : fl > ~} (see [15]). Moreover, the formula (2.3) is replaced by:

y(t) - y[to, r/0](t) - f~<~ Ra,~[to, r/0(co)]x(t) d#,

t~to.

(2.171)

Here # is a probability Borel measure in FI. Below, unless the opposite is stated explicitly, we suppose that this measure is absolutely continuous

236

CHAPTER 2

Differential Equations with Hysteresis

1

0.5

.

.

-3

FIGURE 2.52

.

.

.

-2

.

.

2

-1

3

Functions xo(t) (sine-like function) and yo(t) (step-like function).

l-

v

-1-

FIGURE 2.53

Parametric plot of (x0(t), yo(t)).

X

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

237

2

1.5 1

0.5

9

/-5

FIGURE 2.54

,

-4

-3

,

.

|

.

.

.

.

.

.

-2

.

.

.

1

Functions x l(t) (sine-like function) and yo(t) (step-like function).

1 I I I I I

-1

J

1

-1-

FIGURE 2.55

Parametric plot of (x l(t), yl(t)).

X

238

CHAPTER 2 Differential Equations with Hysteresis

with respect to the standard Lebesgue measure. That is, for S c II #(S) = L h(~x, fl)d~dfl, where h is nonnegative, and satisfies

/ii

h(~, fl) dadfl = 1.

It is sufficient to consider the case when the initial and varying states r/0, 1/(t) are characteristic functions of sets S, which look like the shaded set in Fig. 2.56. That is, S is located to the left of the line a - fl and below a continuous curve L = Ls, with a nonpositive slope. Additionally, in this standard case the coordinates of the intersection of L with the diagonal coincide with the current value of the input x. In other words such sets S admit the representation S = {(a,/~) :/~ + a ~ < ~ s ( / ~ - a)}

where 6s(U), 0~
- vl.

The curve Ls is connected with the function ~'s by

LS = {(a, t) : fl + ~ = ~/s(fl- a)}.

(2.172)

We denote by (~, the set of states r/of the Preisach nonlinearity the characteristic functions of which are described above (see Fig. 2.42).

Dynamics

The set ~, is convenient because the evolution of the every state r/e ~, admits a simple geometrical interpretation in terms of the area described by i/, see Figs. 2.57, 2.58. Here the input x(t), moves along the horizontal scroll-bar, and controls the point on the diagonal a = fl above itself. When moving toward the upper right corner, this point on the diagonal drags the horizontal line, and hatches the domain below this line and above the diagonal. (For instance, if, in Fig. 2.57, x increases from the value u to the value v, the filled area is increased by the hatched triangle.) When moving towards the bottom-left comer, the diagonal point drags the vertical line, and hatches everything to the right of this line and above the diagonal as in Fig. 2.58. The output y(t) is the area of both shaded and hatched domains with respect to the measure #.

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

239

FIGURE 2.56 The set S given by typical characteristic function r/E (~,. In particular the set @, is invariant with respect to piecewise continuous inputs. Clearly, after some time the 'tail' of the curve Ls(t) becomes a continuous piecewise linear curve L, whose links are parallel to one of the coordinate axes.

The Krej~i Theorem The 'picture definition' of the dynamics given above works formally only for piecewise monotone functions. There exists also a simple and very convenient general analytic definition, first discovered, probably, by Pavel Krej~i, which we will also use below. We recall that the play nonlinearity describes a mechanical device illustrated by Fig. 2.59. Here a bean B can be moved along the string S. We control a frame F which can be moved along a string S forward and backwards. The position x(t) of the handle is a varying input. The bean moves only if the left end of the frame pushes it forward, or if the right end of the frame pushes it backward. The position y(t) of the center of the bean is the output. This 'device' is the play operator with the width w = s - r/2 where s is the width of the frame, and r is the radius of the bean. We will illustrate the relationship between the play operator and the Preisach model first by Fig. 2.60 below.

240

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.57 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; increasing input.

Here we have many (ideally infinitely many) strings all parallel to the diagonal fl = a. There are also (infinitely) small beans, one on each string. There is no gravity, so beans do not move without external influence. We control the position of the angle-shape frame F, whose origin O is always at the diagonal a - ft. The position of the origin with respect to the scale x is the input. Initially, let the beans be located along a certain line L = L0, which bounds the set described by the initial state 1/= r/(0) of the Preisach system. We suppose that the initial position of O coincides with the intersection of L with the diagonal. Since r/~ ~, the whole line L is within the initial position of the frame. Then, when time passes, the beans will be moved (to the north-east when the input increases, and to the south-west, when the input decreases). It is easy to see that at any timemoment the beans will be positioned along the line L~l(t). This sketches the picture proof of the following Krej~i theorem. THEOREM2.4.1. Let P[to, x(t0); u] denote a one parameter family of the play nonlinearities with the initial state x(to) E ~ at to and the parameter u~O. Let rl(t) ~ if,,, t~to be the varying state of a Preisach nonlinearity for an input x(t), t~to and let the set S(t), corresponding to r/(t), be defined by a continuous

241

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

FIGURE 2.58 Evolution of the set S corresponding to the state r/of the Preisach nonlinearity; decreasing input.

FIGURE 2.59 The play nonlinearity.

function

~S(t), see

~l(t)(U)

(2.172). Then

-- ~S(t)(U)

-- P[t0, ~S(to)~ u]y(t),

where x(t) is the input of the Preisach nonlinearity.

y(t) - 2x(t),

t~to.

242

CHAPTER 2 Differential Equations with Hysteresis

FIGURE 2.60 The Preisach nonlinearity as a family of play nonlinearities. Interestingly, with some minor restrictions a play nonlinearity m a y be considered also as a Preisach nonlinearity.

Algorithmic Description Restricted Set of States In this text we are mainly interested in an even more restricted set of states of the Preisach model. These are the states ~/ corresponding to the sets S ~ FI b o u n d e d by the lines L with an infinite horizontal link, and with a finite n u m b e r of vertical and horizontal links, see Fig. 2.61. We denote the height of the infinite horizontal link by w (1). Let w (2k), 1 < 2kKK, be the ascending sequence of the horizontal coordinates of the vertical links. That is, the vertical links belong to the straight lines L2k = (W(2k), fl), 1 < 2 k ~ K . Let also w(2k+l), 1 < 2k + I ~ K , be the descending

2.4 EXAMPLES OF TECHNICAL TOOLBOXES

243

0

f

I

I

161

14

I21

i

w2

w4

w6

w5

i

w3

w1

FIGURE 2.61 State r/ ~ ~, that defines a set S bounded by the piecewise-linear curve L. sequence of the vertical coordinates of the finite horizontal links. By the definition numbers w (k), l <.k <<.K, satisfy the inequalities W (1) > . . .

> w(2k0 - 1 ) > w(2k0) > . . .

> W (2),

W (1) >

> w(2k0 +1) > w(2k0) > . . .

> W (2),

...

if K - 2k0; if K = 2k0 + 1.

(2.173)

The totality W of all states ~/, described by the finite n u m b e r of links as above, m a y be naturally identified with the family of all finite sequences w = {w k }lK

satisfying the inequalities (2.173). We emphasize that the lengths K - #w of different elements from W m a y be different. We introduce also the semiclosed intervals I0 = (cx~, w(1)], Ii=(-cx~, W(2)] and !k--[W (k-l), w(k+l)), 2Kk
CHAPTER 2 Differential Equations with Hysteresis

244

(TR) Let s be defined by x(a) ~ Ir Then assign K = K(a) = #w(a) := s + 1; afterwards assign w (K(~)) (a) := x(a). Note that the function K(t) = #w(t), zKtKa, is piecewise constant with the jump-points ti given by ti -- x - 1 ( w (ki) ('~)),

where

w (ki) ('~) E [x('~), x(o)].

Moreover, all components of w(t) except the last one are unchanged between the successive jumps. This rule (TR), together with the semigroup property, can be conveniently used as a base for an algorithmic description of the evolution of the state of the Preisach model, and in numerical algorithms for solving ODEs with the Preisach model.

Transformation Rule for Outputs

Let us also discuss a similar transformation rule to compute the varying outputs of the P