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+5)'.'( !-("n(/$'&.mr3&t!-$'+'A &C3+q,+5q"C5+ .'$'&(2. #3r5$'+3r5(,q/&.:F6B/C3+q,+5q"C5+ .'$'&(2r '1+3C5+5.+56 +3C5q "1C%F q/C3r#"5("C r5q . '1+5(!-r5&0 (,$'&'(!7xn/.hC5+34,+ & C5+5 $'#38C5& 't+3+5.3!-&'$ #%C#5 q,+5q"C5+ .'$'&?. . #3r5$+5r5(,q/&.A $'(/+ > +,!-&'67+ )'.'&. #3r5$. C5p ' $" n V )'. #3r5B( & #3q/C3& !-& % %*(,.'(/$ ; & #n p5$ 56 "t. #3r3$'(,$'&8 K"A +5)'&(q 5Br ## V H ?& !&'$ #367 & &(,(q '?"?V q,&'q"C5(/L6 " v*#34 n .t!- $ 5B6 ).'&'67(,.+56 q,+5q"C5 + .'$'&8. #3r5$'+3r5(,q/& ..r%/n .'("C5q .$&0 $(,( &r3(,. $'(,( )+%n+ 02(,$&'( 6#%C3(,6#%C3 & &'(,-q '1+54,+~6!# . C5$' & 't# * ϕ = 0 & ϕ = π ϕ F "t4/+%n+3#C ' n+5$'(/$'& . 6!# .C3$' & 't# +3C r5(,. C5 & 't#F n & MRqon& $'# &'#np5 $ 5B( "5qon+5r5& .x # !#%C5p r q/+5q/C3 + .'$'&'& . #3r5$'+3r5(,q/& .A C5+>)'.'& +3C5q "1C5q/C3r5&'&~r5+3 ,L6 " F H (/$'&'8fq,&'q"C5(,6# #" !-(/C $ M# 5+,!-&C5p3q . r*$'(/6 't## 'P" 4/+,!-$'+ !-+%n4/ + !-$ ## '1+'A r*.'(,#np5$'+3q/C5&fr5q,(/4 !# Y )'.'&-q "1C5q"C5#r "=V Cgr3+3 L6 "9H2!# V H &<( %x## 'C3+5 . 5|A '1+3C3+5 . 5B(V+ . #C5("n p5$'+ar 5Br5(o/! "tC!-&'$'#367 & &'(/(q '"=V q,&q/C5(/L6 " & @q/+5q/C3+ .$'& . . #3r5$'+3r5(,q/& .: $ #3)'.&'67(,.=A % /n "='#C "5# ; &'/& B/C3+56 q"/n "=& #%(@(,(
!" #$%& ')(+*,$!,-.0/12*3#4 C5+#& 't# . #3r5$'+3r5(,q/&.6#!. C5$'&' #A.r%n/.(/C5q .$'(-"3q/C5+38 &'&'r5+38+)` #n(/88 &(,% n/"='C#"5# ; &'&a)'.'&'r3(o! "1C ' C3+5L6 "5A &C5+ 6#!.C3$'& ' r5B8t!(/Cm& B/C3+54,+mq/+5q/C3+.$'&. & $ # &'$"1C5q . #%C#" ##V H &'( '1+%n((5#3$'&. A )'+5qon( '1+3C5+5.56 q/&'q/C3(,6#V)'(,.'(/8t!-("C 'a$'&0f$'(,6L" "3q/C3+58 &'&'r3+56L"aq,+5q"C5+ .'$'&V . #%r5$'+5r5(/q,&. `a(/C5+,!:+5).'(o!-(on(,$ & . "5q"C5+5 8 &&'r5+5q"C5&q,+5q"C5+ .'$&.. #3r3$'+5r5(/q,&.: C##'0 (,'t##'&fn V 5+54/+ &'$'+54/+ .'( 8 (,$& . +5q,$+5r #3$*$ #B&'q/q"n( !-+5r #3$&'& )'+3r5(o!-(/$'& .|n&'$( #3.'& +5r #%$'$'+58xq,&'q"C5(/76 5 r>#+ ''.'(,q"C5$'+3q/C5& C5#+ & ''& .'#3r5$'+5r3(,q,?& .< & n&x&'$'+38 +5q/+ 5+58|C3. #3(( ' C5+5.'& & n4/+5.'&C36 !# ' n V &'#3(/C3q . r q"n( ! "=V H (,T6 J,"3q/C5p2F r %x#% +5r3+56 )'.'+3q/C5.'#3$'q/C3r5(*q,&'q"C5(/76 5 7C3+ & 't# P q '1+5+5.3!-&'$'#%C#367& x = x¯ , x = .'r% n .'(/C3q . q/+5q/C3 + .'$'&'(/6 . #3r3$'+5r5(/q,& . !#%$'$'+58aq,&'q"C5(/76 5|A-C3+f(/q/C5p * x¯ , ..., x = x¯ 1
2
N
1
N
K
fi (¯ x1 , x ¯2 , ..., x¯N ) = 0, (i = 1, ..., N )
65T5(/.'(,6
$ # & #np5$5B( 3" q"n+3r5&.:rV+#''.'(,q"C5$'+3q/C5& q,+5q"C5+ .'$'&?. . #%r5$'+5r5(/q,&. *
K#
x1 = x¯1 + x˜1 , x2 = x¯2 + x˜2 , ..., xN = x¯N + x˜N
4 !-( x¯ , x¯ , ..., x¯ F '1+5+5.3!-&'$ #C5jq/+5q/C3+.$'&.:. #3r5$+5r5(,q/&.A $'& . J|+ !q/C#3r5&6 K# rP"t. #3r5$(,$'&.aq/&'q/C3(,675 K# * 1
2
2
N
x˜1 , ˜x2 , ..., ˜xN
F6#n 5B(*)'.'&. #!H (oF
d¯ xi d˜ xi + = fi (¯ x1 + x˜1 , ..., x¯N + x˜N ), dt dt
K#LK9
i = 1, ..., N
J|.'&h6#n 56 C5+#& '' &
+3C#' n+5$(,$'&.= x˜ %O"t$ ' ; &'& f (x ) 67+ 0f$'+g. #% "n+ 0 & C5prS. .t!Q(/8 n+5.'#g+#'t+n+ A'+54/. #3$'&&'&'r #!.q,p2n&'$'(/8'$ 5B67& & n(,$'#367& . # /n+ 02(/$'&. n&$'( #3.& +5r5#%C5p9 * (¯ x , ..., x¯ ) 1
i
i
i
N
N X ∂fi (x1 , ..., xN ) fi (x1 , ..., xN ) = fi (¯ x1 , ..., x¯N ) + x˜j ∂xj x ¯1 ,...,¯ xN j=1
4 !-(| $'#&(,$'&.a)'.+5& r3+ !$ 56ar5T&'&'qon/. V>C5q .:r2C5+& '1(*. #3r3$'+5r5(/q,&. J*+,!-q"C#3r%n/.?.3 K r &a)'.'&'$'&6!# .ar5+~r5$'&'6#3$&'<( K "A #3)'?& 8 (,I 6 " .'#3r5$'(/$'& .! /n .CB,r3+%n V ; &'&:r3+3 6L"9H (/$'&'8+* I
K#
K#LK9
!" #$%& ')(+*,$!,-.0/12*3#4 N d˜ xi X ∂fi (x1 , ..., xN ) x˜j = dt ∂x j x ¯ ,...,¯ x 1 N j=1
K# I
+5q"C5+ .'$'&(f. #%r5$'+5r5(/q,&. #" !-("C1"5q/C3+58 &'&r5B6BA (/q"n&r5q/(Vr3+3 6L"9H (/$'&. x˜ #" ! "1C: #%C#" #%C5p:r5+ r5.'(/67(,$'& J|.'&:'t##''&? "5q"n +5r5&.=:B/C3+1#" !-(/C)'.+5&'q@ +,!-& C5pfu@. #3r5$'(/$'&. I N.r%n/. V>C5q ._n&tF $'(/8' $ 5B67& & ?& .-( 8 (/$'& . &'67-( V C %*+5.'L6 " x˜ ∼ exp(λt) 6'1q,)'+3$'(,$ ; & #np5$'#!. % "t$ ' ; &. #" F !-("CV #%#C " ## V H (,8m)'.'& "tr5(on & &'(/$'&'&_r5.(,67(,$& t A'(/q"n&:)'+ 't#% #%C5(on
r3-( H (/q/C3r5(,$'$ 5B 8 a& n &Q&'67(/(/Ce+3C5.& ; #%C3("np5?$ "?V r5-( H (,q"C5r5(/$'$ "=V & #%q/C5p '1+567)tn(( '1q,$ 5B 8 C5(/+5.'&'&n&$'(,8'$ 561" .'#3r5$'(/$'&'8& r3(,q/C3$'+'/A &C3+2)'+ 't#% #%C5(on& λ +5).'(o!-(o/n . V>C5q . 't## ' '1+5.'$'& 1# F . ## ' C5(,.&'q/C3 & &'(,-q '1+54,+ "t. #3r5$(,$'& . * i
i
"
# ∂fi (x1 , ..., xN ) det − δij λ = 0 ∂xj x ¯1 ,...,¯ xN
#
F q/&'67r5+%ncz*.+5$'(('1(/. # f& n &A & C5+ C3+ 0 (_q #367+3(3A' #!' q,+3q/C5r3(,$'$ 5B(_ ,$ # &'(,$&. 6#%C3.'& ; 5 δij _O'1+5& ?Q ##''&6 +5. # +56BA5! n/.fC5+54/+0&C5+#5U&q,q"n (o!-+5r5#%C5p0"5q"C5+58&'&'r5+5q"C5pVq,+5q"C5+ .'$&. [∂fi /∂xj ] . #3r3$'+5r5(/q,&. A $'(,+>5+ !&'67+ )'.+ !("n-#%C3pfqon(o!/"?V H (/(#* #3)&'q #%C3pf6#C5.'& ; "5A B"n(/67(,$C#%67&4'1+3C5+5.+58 .r%n/. V>C5q .C& #3q"C5$ 5B( )'.'+5& r5+,!-$ 5B(x+3CV)'. # F r56 & #3q"C5(,8 "t. #3r3$'(,$'&8:q/&'q/C3(,675^)+ !-&'$ #367&&'(,q-''&'6 )(,.'(/67(,$'$ 5B6 ∂fi/∂xj 6#%C5.& ; # _O1' +5&
?
65T&&'q"n&
C5pf ,$ # &'(,$&. 1' +B-% x% & ; &'(,$ C5+5rf6#%C5.'& ; j 5 r~C5+& '1(x. #3r5$+5r5(,q/&.+
Y ') .'( !-("n &C5pq/+5q/C3r5(,$$ 5B( &'&'qon-# )'+n/"=&'(,$'$+584&'&'qon+5r5+38_6#%C5.& q !-(on-#%C3p'A.'(-8 &'r~! n .$'(,(O1#3.'##'C5(/.'&'q"C5& &'(/q('1+5( "t. #3r3$'(,$'&(#* det[aij − λδij ] = 0
4 !-(
δij
FE(o!-&$'& &'$ #M.:6#C5.'& ; #
G
; 5
[aij ]
C3+67+ 0 $+
!" #$%& ')(+*,$!,-.0/12*3#4
P
P
P
O
"
|J .'+3r5(o!-(/6 #%$ #n& >q/+5q/C3+.$'&'8f. #3r5$'+3r5(,q/&. ! n/.~)'.+5q/C3(,88 &?V!&'$ #367& &(,q('&?Vq,&'q"C5(/6 r 't+F C5+3. 56 r5+3 67+ 0f$ 5 '1+%n (( #3$&.|F q/&'q/C3(,6mq. #% ,67(,.'$+5q/C3pV %x#% ,+5r5+54/+ )'.+5q/C3. #3$'q"C5r # N = 2 C5+ (,q"C5pa$ # x% #% +5r3+58m) n+5q-'1+5q/C3& U|#%. ##'C3(,.+5q,+3+58mC5+& '& q,+5q"C5+ .'$'&?.S.'#3r5$'+5r3(,q,&?. +5).'(o!-(oF n/. (/C5q . #3. ##' C5(,.+56S)+5r5(o!(,$'&.A%x#% +5r#56 C3. #3((' C5+5.'&8Vr@(,(<6#n +58 +''.(,q/C3$'+5q"C5&+9J|+5r5( !-(,$&'( x% #% +3r56* C5. #%((' C3+5.'&'82rn& &xC5+&'' & . #3r3$'+5r5(/q,&. +5)'&'q-B5 r #3("C5q .At" . #%r5$'(,$&'. 67& r r #3.'&'# ; &.= A C5+(/q/C5pa)'.'&6#n 56_ +3Ct' n+5$'(/$'&.? +%Cq,+5q"C5+ '. $'&?. . #%r5$'+5r5(/q,&g . 675 )'.'(/$'((3.'(,4o#3(/6j$'(on&'$'(/8 F $ B5 67& qon-#34o#3(/67B5 67& )'.'& . #% /n + 0 (,$'&'& r . t. ! Q (,8tn+5. # K Q ##'' &'6 +5. #% ,+56BA)'+5r5( !-(,$&'( q,&q/C5(/675 )'.'&%n&02(/$'$'+ +5)'&q(B5 r #3("C5q c . n&'$'(/8'$ B 5 67& !-& % * % (,.'(/$ ; & # np5$ B 5 67&E" .'#3r5$'(/$'&'. 67& A P .'( 8 (,$&'( 1' +3C3+5. 56h &'67(/(/C3x % +3.'6L" A C3+S(/q/C5pq,+#5q/C3r5(,$'$B5 (: $'#& (,$'&. λi x˜ ∼ Ci exp(λi t) )'+%n $'+5q/C3pV +5)'.'( !-("n/.V CP1 #3.'##' C5(/.:C3+& '' &:. #3r5$'+3r5(,q/& . !-&'$ #367&'& (,q-t' +38 q,&q/C5(/675O $
#3q,q/67+3C5.&'6h!-&'$'#367& &'(/q('"=V q/&'q/C3(,6L"ar5&t!-#*
dx = f (x, y), dt dy = g(x, y) dt
LI
&V&'q,qon(o! "3(,6 r5+3 ,67+ 0 $ 5B(BC3&') 5 C3+&'(('f. #3r5$+5r5(,q/&.V$ # % # +5r5+38V)tn+5q('1+5q"C5&+?J,"5q"C5p C3+& 't# P q '1+5+5.3!-&$ #%C#367& x = x¯, y = y¯ .r%n/.(/C5q .~q/+5q/C3+ .'$'&'(/6S.'#3r5$'+5r3(,q,&?.+5z*##'F"t0 ( 5 n+|)'+#' # #3$'+ r5 8 1 ( K Br + '.'(,q"C5$'+5q"C5& B/C3+58 C5+#& ''&m675O67+ 02(,6 ).'(o!-q"C#3r5&C3p !-&'$ #367&&'(,q-''&'(2)'(/.'(,67(/$ F $ 5B( q,&q/C5(/76 5 r|r5&t!( x = x¯ + x,e y = y¯ + ye 4 !-( xe, eyF6#n 5B( r5(on& &'&'$5 9J|+,!-q/C#3rn/..V$'+5r#5B(
!" #$%& ')(+*,$!,-.0/12*3#4 )'(/.'(,67(/$'$ 5B( r
I
&:&'q/)'+%np% ("#.:. #% "n+ 02(,$&'( r~. .t!4Q(/8 n+5.'#A')'+%n "?&&'6 ∂f (x, y) de x ∂f (x, y) x e+ ye = f (¯ x, y¯) + dt ∂x x¯,¯y ∂y x¯,¯y ∂g(x, y) de y ∂g(x, y) x e+ ye = g(¯ x, y¯) + dt ∂x ∂y x ¯,¯ y
J|+5q('1+%n p'"
f (¯ x, y¯) = 0, g(¯ x, y¯) = 0
q,&q/C5(/6#f)'.&'67(/Cfq"n( ! "=V H &'8mr5&t!
K# G
x ¯,¯ y
Atn&'$'(,#3.'& ,+5r #3$'$'#!.arf+'.'(,q"C5$'+5q"C5& C3+& ''&:. #3r5$+5r5(,q/&.
*
de x = Ae x + Be y, dt de y = Cx e + De y. dt
#
#3$'$"=V q,&q/C5(/6L"a67+ 0f$'+~)'.'( !-q/C#%r5&C5p rV6#C5.'& &$'+58 *% +5.67(
de x dt de y dt
e A B x = , C D ye
K
J|+%n/"=&'(/$ #_q,&q/C5(oF 4 !-( A = , B = , C = , D = 6#*!-r!"9Vn&'$(,8'$ 56V!-& % %*(,.'(/$ ; & # np5$ 56" .'#3r5$'(/$'&'8q@)+5q/C3+.$'$ 5B67&1'1+B(% %*& ; &(,$C#367& A q )'+567!+ H p V)'t+%C5+5.'+3867+ 0f$'+ ).'+ #3$ # n& &.'+5r #%C3pA1#3.'##'C5(/.q,+5q"C5+ .'$'&8a.'#3r5$'+5r3(,q,&?.+ x#%q/C5$+5( .'( 8 (,$&'( q/&'q/C3(,76 5G"t. #3r5$(,$'&'8 K# & H ("C5q ._r~r5&1!-(
8
∂f (x,y) ∂x
Y
x ¯,¯ y
∂f (x,y) ∂y
∂g(x,y) ∂x
x ¯,¯ y
x ¯,¯ y
x e = γ1 exp(λt), ye = γ2 exp(λt).
∂g(x,y) ∂y
x ¯,¯ y
/( (@.'( 8 (/$'&'( q/&'q/C3(,675 n&'$'(/8'$ 56<" .'#3r5$'(/$'&'8(,q"C5p n &'$'(,8$ #!. 1' +5673&'$ # ; &?. & #3q"C5$ 56 .'(oF (/$'&'8+
#H
!" #$%& ')(+*,$!,-.0/12*3#4 J|+,!-q"C#3r%n/.?.2
r
KoA')'+%n "?&&'6 (A − λ)γ1 + Bγ2 = 0
K# K
Cγ1 + (D − λ)γ2 = 0.
xn/.hC5+34,+ &C3+5 q,&q/C5(/6# n&'$'(/8'$ 56 +,!-$+5.'+,!-$ 56 #n4,(-5. #3&&'(,q-''&?"t. #%r5$'(,$&'8 K K9 +3C3$'+5q,& C5("n p5$'+@$(,& ,r5(,q"C5$ 56 γ & γ &'67("n-#B$'(/$" n(,r#5B( .'( 8 (/$'&.A $'(/+> +,!-&'67+ &@!-+5q/C#C5+&'$+'A &C3+ 5^+3)'.'(o!("n&C3("np (,( 5 n:. #3r5(/$:$" n V 1
2
K# #
B A−λ det = 0. C D−λ
@u . #3r5$(,$'&'( K $ #% >5Br5#3(/C3q .41#3. #!'C5(/.'&'q/C3& &'(/q(''&'6G"t. #%r5$'(,$&'(,6Uq,&'q"C5(/675 # Y $'+ )'.'( !-q/C#%r%n/.'("Cq/+5+58 & #%q/C5$5B8 q"n "?&'#38C"t. #%r5$'(,$&.3 K J|(,.'(/)'&8 (,6I #3. ##'C3(,.'&q/C5&&'(,q-'t+3( "t. #3r5$(,$'&' ( K# # rVr3&t!-(
I λ − Sλ + J = 0, 2
4 !-( S = A + D, J = AD − BC Y C367(/C3&'6BA & C5+ S & J F>qon(o! &_!-(/C3(,.'67&'$'#3$C_6#C5.'& ; |5 A !r 5+ !/.H &( r0"t. #3r5$(,$'&'( K# # z|+5.'$&1 #3. ##'C3(,.'&q/C5&&'(,q-'t+34,+ "t. #3r5$'(/$'&. K# I λ1,2
S = ± 2
r
K# #
S2 −J 4
+5)'.(o!-(on/. V>Cf.(-8 (/$'&'(*q,&q/C5(/675 #"A q"n( !-+5r #%C3("np3$'+~&)'+5r5( !-(,$'&(O% # +5r56 C5. #%(('C3+5.'&'8 r + '.'(,q"C5$'+5q"C5&cC5+&''& . #3r5$+5r5(,q/&.+ r5+3)'.'+5q/( + "3q/C3+58 &'&'r3+5q/C3&Qq/+5q/C3+.$'&. .'#3r5$'+5r3(,q,&?. +5)'.(o!-(o/n . V HA"=Vj.'+%n p &'4/. #3("C $'##'~!-(/8'q/C3r5&C3("np5$+581& #3q/C3&q,+#5q/C3r5(,$'$+54,+ ,$ # &'(,$&.: ; λ R7q"n& +3 C . 5"g+,!-$'+34,C + 't+3.'$ . 1#3. !# 'C5(/.'&'q/C3 & &'(/(q '1+54,+ "t. #3r5$(,$'& .m!-(/8'q/C3r5&C5(onp5$ #M. & #3q"C5p )'+%n + 0f&C3("np5$'!# . Re λ > 0 &tn& Re λ > 0oA
2
1
2
!" #$%& ')(+*,$!,-.0/12*3#4 Q &')5^q,+3q/C5+ .$'&.:. #3r5$'+3r5(,q/&.a+5)'.'( !-("n/.V C3q . C5(/6BA ' #!'t+3r5G'1+5.'$'& A 1#3.'##'C5(/.'&'q"C5& λ 1 λ2 &'(/q('1+54,+ "t. #3r5$'(/$'&. vx#0%x#% ,+5r5+58) n+3q('1+5q/C3&_67+54 "tCf$ ##n V !#%C3p5q .aq"n( ! "=V H &( q"n/"=& #%&+*
A
F
F 1' +567) n(-'1q,$ 5B(xq|+3C5.& ; #%C3("np5$5B67& r5( H (,q"C5r5(/$'$ 5B67& & #3q/C .67&+* λ = −α + jω A A & Fr5(-H (,q"C5r5(/$'$ 5B( &'&'q"n# J|+,!-q/C#3$+5r't# !#3$'$ 56 q,+3q/C5r3(,$'$ 56 λ = −α−jω α > 0 ω ,$ # &'(,$'&8 rS#+ #H (/(:.'( 8 (/$'&'( ).'&'r5+,!-&C ' qon(o! "=V H (,8 #3r5&'q/&'67+5q"C5&h!-&$ #367& &'(/q(''&= )'(/.'(,67(/$' $ 56:+3CVr5.'(/67(,$'& * λ1 λ2
1
2
x˜ = X exp(−αt) cos(ωt + Θ1 ) y˜ = Y exp(−αt) cos(ωt + Θ2 )
4 !-(fr5(-H (,q"C5r5(/$'$ 5B( '1+5$'q"C#3$C5 X, Y, Θ , Θ +5)'.(o!-(on/. V>C5q . & f$ # & #n p5$ 56 "5qon+5r5&'8 &1!aC5. #3(-'C5+3.'&'8r +''.(,q/C3$'+5q"C5&aq,+5q"C5+ .'$'&?.. #3r5$'+3r5(,q/&.)'.(o!-q"C#3r%n/.(/CVq,+3+58 ##'." F &'&r ## V HF"=V@q .~q,)&'. #np< .'&'q# #Q &') q/+5q/C3+.$'&.~. #%r5$'+5r5(/q,&. FPSJG=9>.CK:EXA60/2 K YP.CITSJG- 1
?
2
A
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λ1 λ2
1
2
A F !-(,8q/C5r3&C5(onp5$ 5B(*+3C3.'& ; #%C3("np3$ 5B( &'&'q"n#* λ < 0 A λ < 0 B"C5+56 qon/"=& #3( .'(oF 8 ,( $'&'(|n&'$( #3.'& +5r #%$'$'+54/+ "t. #3r3$'(,$'&?. r~+''.(,q/C3$'+5q"C5&_q/+5q/C3+.$'&.:. #3r5$+5r5(,q/&.a&'67((V>C r5&1! *
λ1 λ2
1
x˜ = ξ1 exp(λ1 t) + ξ2 exp(λ2 t) y˜ = ζ1 exp(λ1 t) + ζ2 exp(λ2 t)
K
2
K
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λ1 λ2
1
2
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λ1 λ2
1
1
2
1
A B ~a = λ2~a C D
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λ1 , λ2 , λ3
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λ1
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2
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& #%4,. #3676# q/(o! n +%F "1 /n +5r5+58 5&% "t. 't# ; &'&+NJ|.'& m < 0 q,+5q"C5+ .'$'&?.c. #3r5$'+3r5(,q/&. +%C5q-"1C3q/C5r!"?V>C= 5& %O"t. 't# ; &'+3$'$'+58 C3+& '1( m = 0 r5+% $'& 't#3("C*q,( ! n+%F "t ,("n #J|.'& m > &'67(/(/C3q .Q!-r5(r5(/C3r5&^q/+5q/C3+ .'$'&'8^. #3r5$'+3r5(,q/&.+*T"5q"C5+58&'&'r #!. q/) n+#8 $ #!.cn&'$'&. A 0 $'# 't+%C5+5.'+38mn (/0~#%CC3+& ''& x &g$'( "5q"C5+58 &&'r #!. 8 C5.'&= +5r #M.gn &'$'&. A$ # '1+3C5+3.'+58 n (/0~#%C~C3+ & ''& x 1
2
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K 3G
dx = mx − x2 dt
'& q"C5(,6#C %G &67(,(/Cx!-r5( C3+& ''&.'#3r5$'+5r3(,q,&?. x = 0 & x = m <"3q/C3+58 &'&'r3+5q/C3pV+3)'.'(o!("F n/.(/C5q .q/+5q/C3r5(,$$ 5B6Q ,$ # &'(,$'&(,6 ρ = m − 2x /xn/.mC5+&''&g. #3r5$'+3r5(,q/&. x = 0 )'+n/"=& #3(,6 &g! /n .SC5+ &''& x = m )+%n/"=& #3(/6 ρ = −m &t!$'+'A &C5+:C5+& 't# x = 0 .'rn/.'("C5q . ρ =m "5q"C5+58 &'&'r5+58 )'.'& &$'( "5q/C3+5 8 &'&r5+58 )'.'& m > 0 Dxn/.gC3+& ''& x = m r5q,(~$ #%+5+5.'+%C m<0 J|.'& 5&% "t . 't# ; &'+5$'$'+36 ,$ # &'(,$'&&V)'#3. #367("C5. # C3+ & ''&~. #%r5$'+5r5(/q,& . q"n&r ##V C3q .~&2+567(oF m=0 $'&'r5## V C3q .3"3q/C5+3 8 &'&'r5+3q/C5#p VA+ &'#34,. #%676#C3. #3$'-q ''.'&C3 & &'(,-q '1+58 5& %O"t . 't# ; &'&S& + 3. #%02(/$ #$ # .'&'q! # ? 1
2
i
1
2
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x
x2 x1
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& #%4,. #3676#2C5.'#3$'q('.'&C5&&'(,q-'t+38:5& %O"t. 't# ; &'&
"!#%$&')( " %
v|+5.6#np5$ #M. *% +5.'6#A +5)&'q(5Br #!V H2#M.!-#3$'$"=VW5& %O"t. 't# ; & V A&'67(,("C r5&1! dx = mx − x3 dt
*
K ?
B"C5+582q/&'q/C3(,67(B)'.'& &'67(,("C5q .~+,!-$ # C3+& 't#@. #%r5$'+5r5(/q,&. x1 = 0 #J|.'& m > 0 q-"9H (/q/C3r#" F m<0 (/CVC5.&_q,+5q"C5+ '. $&. .'#3r5$'+5r3(,q,&?.+* x1 = 0 A x2 = √m & x3 = −√m 'uq"C5+58 &&'r5+5q"C5pq/+5q/C3+.$'&'8 . #3r3$'+5r5(/q,&. #3. ##' C5(,.& (3" (/C5q j . q,+5q"C5r5(/$'$ 5B6 $ # &'(/$'&'(,6 6 &t!-$'+A &C5+ ! n . ρ = m − 3x2i C5+#& '' & x1 = 0 q,+5q"C5r5(/$'$'+5( $ # '& (/$'&'( ρ = m A3! n/. x2 = √m & x3 = −√m q,+3q/C5r3(,$'$'+3(@ $ # F '& (/$'&'(>F ρ = −2m ?h .(/ (" np C#%C3(|)'+%n/"= & #%(,625&% t" . t' # ; &'+5$'$"=U V !& #34,.'#3676L"5A1& ,+5. #%0 (,$'$"=V $ #.'&'q! # I+*J .&)(,.'( +,!-(P" ). #3r%n/.V H (,4,+a) #3.'#367(/C3. #C& (,.'(" f$+%np 5" q"C5+58'& &'r5+5(Vq,+3q/C5+ . $'&'( . #3r3$'+5r5(/q,&. q"C#3$'+5r3&C5q h . $'( 5" q/C3+58 '& &rB 5 6 & r_(/4,+m+'' .(,q/C3$'+5q"C5& .+ 02!#3("C5q . ) #%. # 5" q"C5+58'& & F r56: q,&6767(/C5.& '& $ 56a !-.t" 4x!-.t" 4 : " q/+5q/C3+ '. $'&'8m.'#3r5$'+5r3(,q,&?.+ n(o! 3" (/C+3C367(/C3&C5p'AD& C5+~!#3$'$ #M. 5& %Ot" . t' # ; &?g . $ ##%n V !-#3(/C3q g . rq,&'q"C5(,6#g q q,&6767(/C5.&'(,8+ &'q/C3(,6#3 K ? q,&'6767("C5.'&'& $ #)'+ +3C3$'+#8 (/$'& V a ' #367(,$( x ↔ (−x)
5
!" #$%& ')(+*,$!,-.0/12*3#4
x
x2
x1
x1 m x3
$7&'q## I*?
& #%4,. #3676#:5&% "t. 't# ; &'& r5&tn ) n+#8 $ 5B(an &'$'&'&hq,+5+%C5r5("C5q/C3r#"=V C 3" q/C5+38 &'&'r5B6 q/+5q/C3+ .'$'&.6BA/8 C5.&? +5r5#!.aFE$(-"5q"C5+58&'&'r5B6 q/+5q/C3+.$'&.6 "!#%$&')( "
$ !
|J .'& B/C3+58 5& %O"t. 't# ; &'& q/+5q/C3+ .'$'&. . #3r3$'+5r5(/q,&.x)'.'+3&'q@ +,!-&C@.'+ 0~!(,$'&'(E)'.'( !-("n p5$'+54/+ ; & 'tF n-#v|+5.6#np5$ #M. %*+5.'6#A'+3)'&'q(5Br5##V H #!.!-#3$'$"=VW5& %O"t. 't# ; & V A&'67(,("C r5&1! * dz = (m + iω)z − z|z|2 , dt
K 5
4 !-( z F 't+367) n(('1q/$ #!.:)'(,.'(/67(,$'$'#!.A i = √−1 A ω F )'.+5& r3+%np5$ #M. '1+5$q/C#3$C# AD'1+3C3+5. #!.:+5)'.'(oF !-(on/.'("Cq,+5q"C5r5(/$'$"=V & #3q"C5+3C!"mq,&q/C5(/675 &_$( &'4,.'#3(/C.'+n&:5& %O"t. 't# ; &'+3$'$'+54/+) #3. #367("C5. #? v|+5.'6#n p5$ !# .3%*+5.'6# 3O.r%n/.(/C5q .2't+367) n(('1q/$ 5B6 B-''r5&'r5#n(,$ C5+56 $'+5.6#np5$'+382%*+5.'675 5& %O"t . 't# ; && r5& n J|(/.'(,)'?& 8 (,6 K 5 *rmr5-( H (,q"C5r5(/$' $ 56h)'(/.'(,67(/$' $ 56A)'+%n-#34o#M. L .'(" (" np C#%C3( z = x + iy )'+% n "?&&'6 q/&'q/C3(,L6 "!-#r " 1"t. #3r5$(,$'&'8 * dx = [m − (x2 + y 2 )]x − ωy, dt dy = [m − (x2 + y 2 )]y + ωx, dt
3
K# 3
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x
m
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7K9
+567$+ 02#!.V)'(,.'r3+5( t" . #%r5$'(,$&'( $ # x A r3C3+5.'+5(BF$ # y & q(' n-# ! B5 r #!.f)'. #3r#5B(@&Vn (,r5B(,&'#3q/C3&A )'+%n "?&&'6 [m − (x2 + y 2 )](x2 + y 2 ) = 0.
K# #
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K# I x¨ + αx˙ + βx + γx = 0. 3
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1
dϕ dt
1
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t$ ' ; & V
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sin ϕ
ϕ3 6
ϕ5 120
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x
Fd
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F = k 0 x + k 1 x3 ,
#2q,& n#~C3.'(,$'&?. )'.+5)'+5. ; &'+5$ #n p5$ #~q('1+5.'+5q"C5& Fd = δ
Q+34 !#2)'+%n/"=&'&'6I"
dx . dt
.'#3r5$'(/$'&'( !-r5&02(/$'&. m
d2 x dx +δ + k 0 x + k 1 x3 = 0 2 dt dt
& n& x¨ + αx˙ + βx + γx3 = 0,
4 !-(
α = (δ/m), β = (k0 /m)
&
γ = (k1 /m).
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m
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1 2 1 4 ax + bx , 2 4
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m¨ x+
∂V (x) = 0 ∂x
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α
X + Y → 2Y,
K# 5G
4 !-( α Fq-'1+5.'+5q"C5p .( ##' ; &'&+ R7q"n &VC##'1+58Vr5q"C5.'(-&'& $'(B).'+5&'q@5+ !&C A C5+x+5q-"9H (/q/C3r%n/. V>C5q . .'( ##'tF ; &'&!-r#" C3&')'+5r c67(/q/C3( $ # + 0~!(,$'&.a67+%n (('" n:)'(,.r5+54,+~C5&) #V).'+5&'q@5+ !&C&?4,(,$(,. # ; &. & x$'(/.'( #34/&'. "=V H ?& 't+367)'+5$'(/$C5+5rfq,.'( ! 5O* β
X + S → 2X,
K#
#frf67(,q"C5( $ # + 0~!(,$'&.:67+%n(-'" ngr3C3+5.'+54/+r5(-H (,q"C5r # ).'+5&'q@5+ !&C&?:. #% "n+ 02(,$'&( $ #V$'(,.'(oF #34/&'. "=V H &'(A'1+567)'+3$'(,$C#5O* γ
Y + S → 2S,
K# #
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$'#n+54/& &'$'+V67+ 0 $'+2 #3)'&'q,#%C5p " '. #3r5$'(/$'&'( ! n/.& ,67(,$'(/$'&. t' +3$ ; (,$ C5. # ; &'&ay ! 5B.'+' } *
Y
ds = −βxs + γys dt
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5
!" #$%& ')(+*,$!,-.0/12*3#4
&'67& &'(/q(''&= .( ##' ; &'8+*
x˙ = x(−αy + βs) y˙ = y(αx − γs)
K 5
s˙ = s(−βx + γy)
x(,4-'t+<"
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x˙ = x(β − βx − (α + β))y y˙ = y(−γ + (α + γ)x + γy)
EK9
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+3C n&&'&'(~+3C+#5T&'$'+54/+4,(,$(,. #%C3+5. #$ # C3.'&'+,!-(2q '1+%n(- #%C3("np5$5B6 '1+5$C!" .+56Qr ; (,)'&mq/(/C'&A 4 !-( .'+5q/C#367) n& #C " ! 5 #3r3C3+ '1+%n(( 5#3$'&'8_+34,. #3$& &'&'r #%(/C5q ._$'("n&$'(,8'$+5q/C3pV r5+%n p CVF #367)'(,.$'+58 1#%. ## 'C3(,.'&'q"C5 & '& C5.'&+ !-#A3 "!(,q,, p #3. ## 'C3(,.'&q/C5& ' # "5q,&tn&C5(o/n . .'r%/n .(/C3q . n&'$'(/8'$'+58A5#, #n-#3$q 67(/02 ! "g)'+3q/#C "t) !# V H (/8 & . #3q/q,(/&'r #3(/67+58 r:q/&'q/C3(,67( B/$'(,.'4/&'(,8 #+ 5(,q/)'(( &'&r #3(/C3q . $'(on&'$'(/8'$'+58 #%r5&'q,&67+5q/C3p V q,+3)'.'+3C3&'r%n(/$'& .m+3C)'.'+3C3(( 't## V H (,4/C + &(,.'(" 2$'(,4/+ C5#+ ' ?# 67(/$'(,$'&( $ # &'(/$'&'8 q,+3)'.'+3C3&'r%n(/$'& . ).'+5&'@q 5+ !&C $'(a674,$'+5r3(,$'$'+A #g&'$'(/. ; &'+3$' $ 5B6 + 5.'#% +5T6 #%r5&'q,&67+5q/C3p q,+F )'.'+%C5&'r%n (,$'& . +3C>C5+ 't#> # !#3("C5q .*!- & % %*(,.'(/$ ; & #n p5 $ 5B4 6 "t. #3r5$(,$'&'(/T6 #$ #% 67(/.'$'+5q"C5Op %x#% ,+5r5+54/+ )'.'+3q/C5.'#3$'q/C3r #~!#%$'$'+58a#3r3C3+ '1+%n-( #%C5(onp5$'+38 q,&q/C5(/76 5O. #3r5$'#2C3.'(,T6 $ #3q,q/67+3C5.&'6 67+,!-& %x& ; &'.'+5r5#3$'$ "=V @q (/L6 " 4,(/$'(,. #C5+5. #mq&$'(,. ; &'+5$'$'+38h$("n&'$(,8'$'+3q/C5#p V A & ,+ 5. #%0 (,$'$ "=V $ #g.'&'#q G !-(/q,p '1+%n (( #%C3("np3 $ 5B 8 '1+5$!C " . rS+3C n & &'&'(_+%3 C 'tn-#3q,q/ & &'(/(q '1+%F 4,+mqo/n "=& !# . $'(q/+,!-(,. 0f&Cg$'("n &'$'(,8 $ 56 B"n(/67(,$C3+5 r uq/& n&C3("np K "t)'.'#3r%/n .(/C5q . !-+3)'+%n$'&tF C5(onp5$'+38 ; (/)'p V + 5. #C5$'+58aq,!r . & A q,+,!-(,. 0~M# H (,8an&'$'(/8' $ 5B8 "3q,& n &C5(onp K &:&'$(,. ; &+5$' $ 5B8 )'.'(/+ 5. # +5r #C5("n 4 p K #%)'& 8 (/6 " .'#3r5$'(/$'& . ! /n . 't+n(( #C5("n p5$'+54/< + '1+5$!C "t. #2+3C5$'+3q,&C3("np5$+f$ #%)' . .02(/$'& . u(t) $ # '1+5$t!-(/$'q #C5+5.'( D *
LC
J|.'(o!)'+%n+
dI du d2 u +u = M . + RC 2 dt dt dt
0f&'6BA6&C3+ #3r3&'q,&'67+3q/C5p C3+'t#
I(t)
K
$ #Sr56 +,!-(4"3q,& n& C5("n .^+3C r! +,!-$'+34,+ $ #3).9.tF
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u(t)
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I = (S0 − S1 u2 )u,
K 5I
4 !-( S & S F)+5q/C3+.$'$ 5B(B)'+n+ 0 &C3("np3$ 5B(T'1+B(% %*& ; &'(/$C5O9J|.'( !-)'+%n + 0f&'6BAM&C5+@67( 1#%$'& 6 r5+3 o!-(,8q/C5r3&. ; (,)'&a&'$'(/. ; &'+5$$'+58 +#5. #%C3$'+58aq,r#. &a)'+,! &'&'$.(/C3q . #!'t+3$'+567(,.$'+5q/C3& 0
1
I = (S0 − S1 u2 − bV )u, ,
K# #
4 !-( V = V (u) F $ #3)'. . 20 (,$'&( $ # r565+,!-(|&'$'(/. ; &'+5$$'+54,+~)'.(,+5.'#% +5r5#%C5(on/.A b F )'+5q"C5+ .'$'$5B8 '1+ B(% %*& ; &(,$C6JO"3q/C3p '& $'(,. ; &'5+ $'$+5( )'.(,+5.'#% +5r5#3$'&'(:+5q-"9H (/q/C3r%n/.'("C5q .Ur q/+5+3C3r5(/C3q/C5r3&'&cq "t. #3r5$(,$'&'(/6
dV dt
B/C5+36
= −γV + ϕ(u).
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*I
t& !$'+'A+&C3+ B"C#aq/&'q/C3(,6#)'.'( !-q/C#3rn/.'("C_q,+#5+58g+5q ; & n'n/. C5+5.A!&'q,q/&') # ; &?.S& & #%q/C5+%C# 1' +%n(oF
#3$&'8:rP'1+3C3+5.'+56e&'$'(,. ; &'+5$'$+56 +5.'#% +56 #%r5&'q .Cf+3CV&?a#%67) n&C!" ! 5O
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d du M S0 − RC 1 ( − u+ dt dt LC LC
Z
udt +
M S1 3 Mb uV + u ) = 0. LC LC
!" #$%& ')(+*,$!,-.0/12*3#4 Q+34 !# du M S0 − RC 1 − u+ dt LC LC
4 !(
A
F 't+3$'q/C#3$ C# >& n &
Z
udt +
Mb M S1 3 uV + u = A LC LC
du M S0 − RC Mb M S1 3 1 = u− uV − u + (A − dt LC LC LC LC
!-(on-#3(,6e #367(/$")'(/.'(,67(/$'$ 56 1 W = A− LC
S.(/ (" np C#%C3(E! n/. q@ (/675 4,(,$(,. #%C3+5. #@qE&'$'(/. ; q"n (o! "=V HF"=V q,&q/C5(/6L"1" .'#3r5$'(/$'&'8
Z
Z
udt).
udt.
&'+3$'$'+58 $("n&'$(,8'$'+3q/C5p#V $ # .'&'q# G )+%n/"=&'&'6
du M S0 − RC Mb M S1 3 = u+W − uV − u, dt LC LC LC dW 1 = − u, dt LC dV = −γV + ϕ(u). dt
MK G
5(/ ,. #% 67(/.'$ 56a)'(/.'(,67(/$'$ 56 t" . #3r5$'(/$'&.a4,(/$'(,.'#%C5+5.'#f67+ 0 $+V).'(o!-q"C#3r5&C3pr~r5&t!-( dx = mx + y − xz − dx3 , dτ dy = −x, dτ dz = −gz + gΦ(x), dτ
4 !-( √ √ √ x = (u/u0 LC), y = W/u0 , z = V (M b/ LC), Φ(x) = ϕ(u)(M b/γ LC),
*I
KMK
!" #$%& ')(+*,$!,-.0/12*3#4 √ √ √ m = (M S0 − RC)/ LC, g = γ LC, d = M S1 LCu30 .
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Φ(x) = x2 , Φ(x) = ex , Φ(x) = x, Φ(x) = ax − bx3 ,
G K
)'.'+ #3$'#n& ,&'.'+5r #C5p & )'.'& d = 0 q m g
!" #$%& ')(+*,$!,-.0/12*3#4 0,
Φ(x) =
x2 , x ≥ 0,
0, Φ(x) = x,
Φ(x) =
x<0
ax, bx,
x<0 x ≥ 0,
x<0 x ≥ 0,
ax2 , Φ(x) = bx2 ,
x<0 x ≥ 0,
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x¨ + (ε −
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'
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= (/q/C5p -q "t676# + !H (,4/+ .'( 8 (/$'& .#%r3C5+5$+567$'+54/+F"t. #3r5$(,$'&. A q,+5+%C5r5("C%F x + 2αx + ω x = 0 q/C3#r "=V H (,4,+gq/r5+ 5+,!- $ 5B6 '1+%n (( #3$& .'6j+3q ; & n/n .C3+5. #A & & #3q"C5$'+34,+m.'( 8 (,$&. $'(,#3r3C3+5$'+567$'+34,+ "t. #3r5$(,$'& . ? "A-q,+3+3C5r5("C5q"C5#r "=V H (,4/+_r 5B$ "102!-(,$'$ 5BE 6 '1+%n (( #3$& .'6c+5q ; & n'/n . C5+5.'# J|.'&m$ #F n & &&'& !-&'q,q/&') # ; &'&eq/r5+ 5+,!-$ 5B( 't+n(( #%$'& . ##C " 1## V>C & +5q"C## V C3q . C5+%n#p 't+gr 5B?$ "t02!-(/$' $ 5B( '1+%n-( #3$'& . #A '1+3C3+5 . 5B( "102(<$'(E #3r3&'q .C*+3C $ # & #np5$ 56A"3q"n+5r3&'+8 Q## ''&'( 't+n(( #%$'& .2$ #> 5Br ## V>C "5q"C#3$'+5r3&'#r 8 &67&'q .+ R7q"n& F4o#3.'67+3$' & &'(/(q 't!# .1%O"t $ ' ; & .r3.'(,67(/$'&r5&t!# * F (t) = B cos pt A F (t) C5+V.'( 8 (,$&'F( "t. #3r3$'(,$'?& .2 ? )'+%/n "=& #%(/C5q . q"n( ! "=V H &'T6 * 0
00
0
2 0
?
x = A cos(pt + Θ),
4 !-( A =q Fx#367)tn&C#" !#:'1+%n(- #3$'&8A Θ = arctan F q !-r5&'4 %x#% (5 67("0VF ! "ar5B$"102!##V H (,8mq/& n+58m& '1+%n((5#3$'&.67&m+5q ; &tn'n/.C3+5. # A%*+5.'6L" n 5 ? # qon(o! "3(/C A+& C5+ #367) n &#C " !0# '1+%n(- #3$'&'8r5+% . #3q"C#3(/CV)'.&q/C5.(,6n(/$'&'& &'#3q/C3+3C5Qr5+3 "!(,8'q"C5r5&.1'q/+5q/C3r5(,$$'+58 & #3q"C5+3C3(~+5q ; &tn'/n .C3+5. # 2#%$'$'+5A ( .r%n(/$'&'(2$ #%( 5Br #3("C5q ..'(/ ,+5$ #3$q,+56BA # #3r5&'q/&'67+5q"C5pa#367) n&tF #C " ! 5 #r 5B$ "10~!-(/$' $ 56<'1+%n-( #3$'&'8f+3F C & #3q/C3+3C 5cr5+% "!-(/8'q/C3r5& .fF .'(" +5$'#3$'q,$+58 ''.&'r5+5+8 = &1!.'(oF +3$ #3$'q/ $ 56 '.'&'r 56~! /n .~n&'$'(/8'$'+54/+2+5q ; & n'/n . C5+5. # ).'&. #% "n & &'$ 56 α )'.'&'r3(o!-(/$$ # .'&#q ? # n( ! "5("C+3C567("C5&C3p'A &C5+f6## '1q,&'6#n p5$'+5(| $'# &(,$'&'( #367) n& #C " ! 5jrV+5q ; & n'/n . C5+5.'(xq*)'+3C3(, . .'67& !-+5q"C5&'4o#%(/C5q . $'( )'.'& p = ω A #|$ # %n&( 't+3<8 ' ω & #3q/C3+3C5( ω A5)'.'& &'(,6S. #% $+5q/C3p |ω − ω | C3(,6 5+%n #p 8 (%A &'(/6 5+%n!p 8 ( 't#+ B(% %*& ; &'(/$C !-&'q/q,&') # ; &'+& Q+ &'$+5(~q,+5r5)'# !-(,$&'( ω = ω !-+3q/C5&4o#F (/C3q .C5+%n#p 't+~r~q" n "?&'#3( 't+3$'q,(/.'r #%C3&'r5$'+54/+f+3q ; & n/n .C3+5. #A'C3+V(/q/C3pfr2qo/n "=& #3(3/A '1+54 !-#~r~q/&'q/C3(,67( +3C3-q "1C5q"C5#r "=V C)+3C5(/.'& & n &:)'+,! 't # & 't<# B/$'(,.4,&'2 & α = 0 @.'-( 8 (,$'& .3 ? 7n(/(4 '1+2r 5Br5(/q/C3&a+3q,$'+5r3 $ 5B( q,r3+58'q/C3r # '1+%n-( #3$'&'8rxn&'$'(/8'$'+56h+5q ; & n'/n .tF C5+3.'(#* B2 (ω02 −p2 )2 +4α2 p2
2ωo p2 −ω02
0
0
p
0
p
>z|+%n(- #3$'&?.fq/+5r5(/.82##V>C5q .$'#F& #3q"C5+3C3( r 5B$"102!##V H K*K
p
0
(,8 q,& n 5O? 5+%n(/( +!H (,6 qon/"=& #3(%A
$ '2$ &4 1
=0
A
2
3
4
p 0 1
$7&'q! = * $
<
2
<
3
<
4
(/ ,+5$ #3$q,$ 5B(F''.'&'r5B(|n&'$(,8'$'+34,+V$'( #3r%C5+5$'+367$'+54,+V+5q ; & n'n/. C5+5.'#4 =
'1+54 !#:r_q,)'(-'C5.(r5$'( 8 $'(/8 q,& n 5 )'.&'q-"1C5q"C5r#"=V>C $'(/q('1+%np#'t+:&'#3q/C3+3C #" ! 1" Cf)'.'&'q 1" C5q/C3r5+5r #C5pr~q,)(('C3.'( r5B$?"t02!-(/$'$ 56 1' +%n((5#3$'&'8+
A C3( 02( & #3q/C3+3C5
? 76 ) n&C!" !-#&4%x#% #P"5q/C#%$'+5r5&'r!8 &?5q .3'1+%n (( #3$&'8m+5)'.'( !-("n . V C3q .#367) n& C#" !-+58m&4%x#F ,+58 r5B$?"t02!##V H (/8_q,& n/5j& $'(* #%r5&'q .C +3Cf$ # & #np5$56 "3q"n+3r5&'8_r2n&'$'(/8'$'+56 +3q ; & nF /n . C5+5.#( 67) n &C#" !#P'1+%n((5#3$'&'8:)'. .'67+V)'.'+5)'+3. ; & #3$'#np5$ #~#367)tn&C#" !-( r5+3 o!-(,8'q"C5r5&?.+ >v*#P& #3q/C3+3C#MA n& >'&? ' q,+#5q/C3r5(,$'$+584& #3q"C5+3C3( +5q ; & n'n .C5+3. # )'.'+3&'q@ +,!-&C.(/ +3$ #3$'q%A C3+f(/q/C5pf.'(/( '1+5( r3+3 . #%q/C#3$'&( #367)tn&C#" ! 5^)'.'& p → ω p
J|.'(o!)'+%n+
f0 &'6UC3(,)'(/.'p'A&C3+) #3. #367("C5. 5 n&'$'(/8'$'+54/++5q ; &tn'n/.C3+5. #3 q/6TD"t.+ ?@ '1+B-% %x&tF ; &'(/$C !-&q,q,&) # ; &'& α &q/+5q/C3r5(,$$ #!. &'#3q/C3+3C# ω .'rn/. V>C5q .$'( )'+5q"C5+ .'$'$5B67&r5("n&&'&'$ #367& A #| #3r5&q .C +%C*!-&$ #367 & &'(/(q ''=& )(,.'(/67(,$' $ 56 q,&q/C5(/675 x & x 5u . #3r3$'(,$'&(0 = r B/C3+56 q"n "?&'#3( )'(/.'(,)'?& 8 (/C3q .mq"n (o ! "=V H &'6 + 5.'#% +5T6 * 0
0
K9
$ '2$ &4 x00 + 2α(x, x0 ) + ω02 (x, x0 ) = F (t)
?
|r5$ 5B8 r3&t!E'1+B(% %*& ; &(,$C3+5r +5)'.'( !-("n .'(/C3q .1#3.'##'C5(/.'&'q"C5& 't#367&Q$'("n&$'(,8'$56 B"n (,67(,$ C5+5r '1+5$ ''.(/C5$+58 q,&q/C5(/675O u@. #3r5$'(/$'&'( ? ra+3C n& &'&( +3C2 ? .'rn/.'("C5q .S$'("n&$'(,8'$5B6 &+5qoF ; & n/n .C3+5. A '1+3C5+5.5B8_&'6 +3)'&'q(5Br5#3(/C3q ._$ #% (5Br #3("C5q ._$("n&'$(,8'$ 5B6c+5q ; & n'n .C5+3.'+56T $#3q/q,67+3CF .'&'6BA 't## '& 67(/$ .C3q . ra$'("n&$'(,8'$+56Oq"n/"=& #%(fq/r5+58'q"C5r # ~F@A )'+n/"=&'(,$'$5B(~! n/.gn&'$(,8'$'+34,+ +5q ; & n'/n . C5+5.'# Y !-$'&'6h& *q,r3+58'q/C3r~$'("n&$'(,8'$ 56 q/&'q/C3(,6 .'r%/n .(/C3q .:#+ 5+54o!# H (,$'&'(*q/)'(( 'C3. P# 't+F n-( #3$'&'8m$+5r 5B674 & & #%q/C5+%C#367&A$'( q,+,!-(,. 0~M# H &'67&'q .rfq,)'-( 'C5.'(~r5+% "!-(/8'q/C3r5& .+ Q+ (/q/C3pq,)'-( ' F C5.'#np5 $ 5B8q,+3q/C#31 r '1+%n-( #3$'&'8g67+ 02("C:+%C n & & #%C3p5q .+3Cq,)(( 'C3. #np5$+54,+aq,+5q"C#3r #r 5B$ "102!## V F H (/8 q,& n 5OL & #3q/C3$'+5q"C5&A7(/q"n& r5$'( 8 $'(/(r5+3 o!-(,8'q"C5r5&( # !#3$+m4o#3.'67+5$'& &'(,-q 't+38 %O"t $ ' ; &'(,8 r5.'(/67(,$'& F (t) = B cos pt A C5+ r_q,)'-( 'C5.C ( '1+%n (( #3$&'8 +5q ; & n'/n . C5+5. : # #" !-("Cm)'.'&-q "1C5q"C5r5+5r5#%C5p 't## ':& #3q"C5+3C# AC## 'gr+ #H (,6cq" n "?&'#3(2&:!-. "t4,&0 ( & #3q/C3+3C 5Oz|+%n (( #3$& .mq,&q/C5(/76 5 )'(/.'(,q"C## V C p 5+ .'+5$$'+5q/C3: p '1+%n(( 5#3$'&'8A r 5B)'+%n $ . V HA"=V@q . rn&'$'(/8'$'+56O+3q ; & n/n .C3+%F .'!( + 5q"C5r5(/$'$ !# .P&'#3q/C3+3C#x+3q ; & n/n .C3+5. # q/C#%$'+5r5&C3q . % "t $ ' ; &'(,8f#367) n &#C " ! 5hr 5B$ "102!-(,$'$ 56 '1+%n-( #3$'&'8 B(% %*(( 'C$'(,& ,>+ .+5$'$'+5q"C5 & v*(/& >+ .'+5$$'+5q/C31 p 't+n(( #%$'&'8m$'(on&'$'(/8'$'+54/+a+3q ; & nF n/. C5+5. #f+ 't#%> 5Br #%(/Crn& .'$&'( $ #Vr5&t!m(,4,+ .'(/ ,+5$ #3$'q/ $ 56 ''.'&'r 56DA %n-#34/+ !-#3 . ./$("n&'$(,8'$'+3L6 " 67( #3$'& ,L6 "U+54,. #%$' & &'(/$'& .G'1+%n (( #3$&'8 ! ## ' /n V & #3("C5q .^+5$Qrhq"n (o ! "=V H (,T6 OJ,"3q/C5I p & #3q"C5+3C# r5$'( 8 $'(/4,+r5+3 o!-(,8q/C5r3& . %n&( 't# '_q,+ 3q/C5r3(,$'$'+38 &'#3q/C3+3C5(2+5q ; & n' n .C5+3. # B"C5+56cqo/n "=& #3(2q,&tF q/C3(,6C # B(% %*(( 'C3&'r5$'+:+3C 5&. #3(/C B,$(,.'4/ & V +%C:r3$'-( 8 $'(,4/+ &'q"C5+ &'$ & 't#:& 't+n(( #%$'& .r$'(,8Sr5+3 /F . #3q"C## V C 9$ +5q/Cx#%67) n&!C " ! 5er 5B?$ "t02!-(/$' $ 560'1+%n (( #3$&'8~)'.'&'r5+,!-& C ' +3C5q"C5.'+3 8 '1( q/+ 5q/C3r5(,$$'+58 & #3q"C5+3#C 5O+5q ; & n'/n . C5+5. # DA '1+3C3+5. !# .C5(/6 5+%n!p 8 (3/A &(, 6 5+n#p 8 (x#367)tn&#C " !?# c.'(/- " np C#%C5( B-% F _
I K
$ '2$ &4 %*(('C3&'r5$'+3q/C5p r3 #3&67+ !(,8'q"C5r5&.S+5q ; & n'n/. C5+5.'#:q~r5$'( 8 $'&'6^&'q/C3+&'$'& '1+56)" 67(/$'p#8 #3(/C3q .+ &tF q/C3(,6# r#56 +,!-&C& .'(/ ,+5$ #3$q # &:+5q ; &tn'n/.C3+5.m)'(/.'(,q"C#3(/C #367((& #C5pr5$'( 8 $ V V q/& n/"DQ ##'&'6 +5.'#% +56BAr qon/"= & #3(_$("n&'$(,8'$'+34,+ +5q ; & n'n/. C5+5.'# 67+ 0f$'+S+ 0f&t!#C5p'A&C5+ r5&t!U.'(" +5$'#3$'q,$56 ''.'&r56 !" !(/C #3r5&'q/(/C5p~$'(>C5+%n p'1+2+3C ) #%. #367(/C3.'+5r2q/&'q/C3(,675|A $'+ & +3C2#367)tn&C#" ! 5Ur5$'( 8 $'(oF 4,+r3+3 "!-(/8'q/C3r5&. |J .'&m6#n 56_ #367) n& C#" !#MA'1+54 !#+3C3q/C3.'+58 't#6#n-#A-.'(/ ,+5$ #3$q,$ 5B(P''.'&r5B( q@ + 0f&eq.'(" +5$'#3$'q,$B5 67& '' .'&rB5 67& n &'$'(,8$'+54,++5q ; & n'n/. C5+5.'# |J .'&I3+%np#8 &? #367)tn&C#" !# .'(" +5$ #%$'q,$ B5 ( '' .'&rB5 ( 67+34-1" C ,$ # '& &C3("np5$+ +3C n& & #%C5p3q . +3Cx$'&?D0 ` #!t' q/&'6#np3$ #!f . #367) n& C#" !# rB5 $1" 02!-(,$$ 56
x00 + 2αx0 + ω02 x + µx2 = B cos pt
r|q,)(('C3.'( '1+%n (( #3$&'8P#" ! "1C*)'.'&'q "tC3q/C3r5+5r #%C3pA&'("C5$ 5B(>4o#3.'67+3$'& ''&~qT& #3q"C5+3C#%67& 2p, 4p & C ! v* # B"C5?& & #3q/C3+3CM# #" !-(/C C!# '02(<)'.+5&'@q +,!-& C5p B(% %*(( 'C3&'r5$'+3(Br3 #3&67+ !(,8'q"C5r5&'(B+3q ; & nn/.C3+5. # q@r3$'-( 8 $'&'6h&'q/C3+ &'$' & '1+56BA'# $ # &'& C2.'(/ ,+5$ #3$q,$'+5,( "tr5(on & &'(/$'&'(*#367)tn&#C " ! 5E'1+%n-( #3$'&'8 A(,qon& &C ! ω ≈ 2p, 4p 0
mn&'$'(/8'$'+56m+3q ;
& nn/.C3+5.'(6"3q/C#3$'+3r5&'r#8 &'(,q .P'1+%n(- #3$'&?. $'(E #3r5&q .C +%C $ # & #n p5$ 56x $ # &'(oF $'&'8 x(0) & x (0) A #|+5)'.'( !-("n/.V C3q . C5+np'1+ r#5B$"10~!#!V H (/8 q/& n+58+1z*##'~)'+#' # >5Br ##V>C &q,q"n ("F !-+5r5#3$'&. A r$'(on&'$'(/8'$'+56Q+5q ; & n'/n . C5+5.'(~)'.&m+ !$'+56c&_C5+56c02( r3+3 "!-(/8'q/C3r5&'&g67+54 "tCaq/+5q-"9H (oF q/C3r5+5r #C5pa$'(,-q 't+np '1+ "3q/C#3$'+3r5&'#r 8 ?& q . A "5q"C5+58 &'&'r 563'1+%n-( #%C5(onp5$ 56.(/0f&'67+5r v*# 't##'1+58 &O B"C5?& r5+3 67+ 0f $ 56f.'(/0f&'67+5r2)(,.'(/8t!-(/CVq/&'q/C3(,6#2+5)'.'( !-(" n .'(/C3q .a$ # & #np5$ 5B671 & "3q"n+3r5& .'67& #3$'$'+3( .'r%n(/$'&'(%A1 ## ' n V & ## V H (,(/q .r*q,+3-q "9H (/q/C5r3+5r #3$'&&r % # +5r5+36S)'.+5q/C3. #3$'q"C5r5( q,&'q"C5(,76 5 0
K9
$ '2$ &4 $'(/q('1+%np'&? "5q"C#3$'+5r5&r#8 &?5q .A"5q/C3+58 &'&r56e.'(/0f&'67+5r:'1+%n(- #3$'&8A ' '1+3C3+5. 5B6 +5$ #g#3q,&6F )C3+3C5 & &(,(q '&Qq"C5.'(/67&C5q .U)'.'& r3+3 . #%q/C#3$'&& r5.'(,67(/$'&A<$'#% >5Br #%(/C5q .c6L" np C3&'q/C##3& np5$'+3q/C5p#VA J|.'+5r5( !-(,$$'+5( q/. #3r5$(,$'&'( $'("n&$'(,8'$+54,+ +3q ; & nn/.C3+5. # q7n&'$'(/8'$ 5B6g)'+'t#% #n+ $ # n& &'&'( )'.'&'$tF 6 ; &')& #np5$+54,+'/A 't # &'(/q/C5r3(,$'$'+34,++3C n & &'& .:r~q,r5+58q/C5r5M# CB/C3?& aq,&'q"C5(/T
C " O
" Q
< / <
" O
3# q,q/67+3C5.&'6 C3(,)'(/.'pg)'+,!-.'+3$'(,(r3&t!h.'(" +5$ #%$'q,$ 56 ''.'&'r56 $'(on&'$'(/8'$'+54/++3q ; & nn/.C3+5. # )'.'&a4o#3.'67+3$'& &'(/q('1+56 r5+3 o!-(,8q/C5r3&'&_$ #~)'.'&'67(/.'(x+5q ; &tn'n/.C3+5. # A"=% %*&'$'4o#2FE+ !$'+58a& A5#% +%F r5667+,!-(on(,8aC5(/+5.'&'& 't+n(( #%$'&'8+ R74,+P"t. #3r5$(,$'&'(* #%)'&'q(5Br5#3(/C3q ._q"n (o! "=V H &'6 +5.'#% +56T* $
?MK9
x00 + 2αx0 + ω02 x + µx3 = B cos pt
& n&
x˙ = y,
?
y˙ = −2αy − ω02 x − µx3 + B cos z, z˙ = p.
J|.'(o!q/C#3r%n (,$'&'(r %*+5.'67(Vq,&'q"C5(,675 "t. #3r3$'(,$'&8E ? )'+'t#% (5Br #3(/C A &C5+ ; & nn/. C3+5. &'67(/(/CVC5.'( 67(,.'$'+3(A%x#% +3r5+5(|)'.'+3q/C5.'#3$'q/C3r5+q '1+5+5.3!-&'$'#%C#367&
$'( #3r%C5+5$'+367$ 5B8 +5qoF x, y, z
J|.'( !-)'+%n + 0f&'6BA=&C5+2$("n&'$(,8'$'+3(xqon-#34o#3(/67+5( 3 6# n+ )'+2q/. #3r5$'(/$'& Vjq>n&'$'(/8'$ 5B67&q"n-# F µx 4o#3(/67B5 67&F"t. #3r5$(,$'&. ?LK Y q ; & n'n/. C5+5. q76#n+58x$'(on&'$'(/8'$'+5q"C5pVh$ #% (5Br #3(/C3q .A'r #% &tn&'$'(/8 F $ B5 6e+5q ; &tn'n/. C3+5.'+56T- z|+%n(- #3$'&?.:r~C##'1+56e+5q ; &tn'n/.C3+5.'(F%n& (''& 'C'1+%n((5#3$'&.6hn&'$(,8'$'+34,+ K
$ '2$ &4 +5q ; & n'n/. C5+5.'#A)+B/C3+56L":#367) n&C!" !# +5q/$'+5r5$'+38 4o#3.'67+3$'& ''&AC5+f(,q"C5p4o#3.67+5$'& ''& $ #<& #%q/C5+%C5( A !-+%n 0f$ # 5
= I
x(t) ≈ a cos pt + b sin pt
xn/.+5).'(o!-(on(,$&. '1+B-% %x& ; &'(,$ C5+5r a & b )'+,!-q/C#%r5&'6 ? I r ?MK9 &).'(,$'(-5.'("02(,6eq"n-#34o#%("F 76 5B67&Vq '. #%C5$5B67&& #3q"C5+3C#%67&+#2#%$'$ 5B8V)'+,!/ +,!1'~+5).'(o!-(on(,$& V^)'.& %n&0 (,$'$ 56V.(-8 (/$'&'8 ! /n ._$'("n&$'(,8'$+58gq,&'q"C5(,76 5 $ #%( 5Br #3(/C3q .s3g;M9>.C@/.83"]^*^7J3r.CB:A:E ;=G=IJ.-]^. W * k6*\B6GN*Tej.'(/ -" np C#%C5( )'+% n "?&'#3(/C3q .
−ap2 cos pt − bp2 sin pt − 2αpa sin pt + 2αpb cos pt + ω02 a cos pt + ω02 b sin pt +
3 + µ(a3 cos pt + b3 sin pt + a2 b sin pt + ab2 cos pt) = B cos pt 4
4,.?" ))'&'.'+5r5#3r'1+B-% *% & ; &'(,$C#5O)'(,.(o!
cos pt
&
sin pt
A')'+%n "?&&'6 q,&q/C5(/6L"1" .'#3r5$'(/$'&'8+*
3 (ω02 − p2 )a + 2αpb + µ(a3 + ab2 ) = B 4 3 (ω02 − p2 )b − 2αpa + µ(b3 + a2 b) = 0 4
J|(,.'(/8t!-(/6Ur = T' )+%n/.'.$ 5B6 '1+5+5.3!-&'$ #C#36 # Fx% #% #0't+n(( #%$'&'8+*
a = ρ cos ϕ, b = ρ sin ϕ
?
A-4 !-( ρ FB#367)tn&C#" !# A
ϕ
3 (ω02 − p2 )ρ cos ϕ + 2αpρ sin ϕ + µρ3 cos ϕ = B 4 3 2 2 (ω0 − p )ρ sin ϕ − 2αpρ cos ϕ + µρ3 sin ϕ = 0 4
K%I
?S
?LI
$ '2$ &4 u 67$+ 0f&'6 ? S $ #
u 67$+ 0f&'6 ? S $ #
cos ϕ
A # ? LI $ #
sin ϕ
&:qon+ 0 &6T*
? G
3 (ω02 − p2 )ρ + µρ3 = B cos ϕ 4
sin ϕ,
#C = I $'#
cos ϕ
&:r#5T&C5(/6 r3C5+3.'+5(*r5B. #%0 (,$'&'(*& x)'(/.'r5+54/+ *
?
2αρp = B sin ϕ
Y
C5+#5 & >5("0~#%C3p@ #3r3&'q,&'67+3q/C5&2+%C ϕ ?$ "t0f$'+ r3+3 r5(/q/C3&1 ? G &1 ? #r,''r # !. #%C & q"n+ 0f&C5p 5+3 ,$ # & #!.1&'(/.'(/ A ''r # !-.'#%Cf#367) n&!C " ! 5G'1+%n(- #3$'&'8 * ρ = A )'+%n "?&&'6T* 2
9 2 3 3 µ A + µ(ω02 − p2 )A2 + ((ω02 − p2 )2 + 4α2 p2 )A − B 2 = 0 16 2
= #
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$ '2$ &4 !-&'+,!#?'u . #3r3$'(,$'&( ? I 67+ 0 $+V)(,.'(/)'&'q #C5pr~r5&t!(#* d2 q˜ R d˜ q + + ω02 (Es )˜ q = V0 cos pt dt2 L dt
?
$'+|q,+3r5) # !#%(/C*q6"t. #3r3$'(,$'&(,6_! n/. n&'$(,8'$'+34,+*+5q ; & n'n/. C5+5. # A5q,+3q/C5r3(,$'$ #M.P& #3q/C3+3C# 't+%C5+5.'+F 4,+ +5)'.'( !-("/n .(/C3q .! & % %x(/.'(,$ ; & #np3$'+58(,67'1+5q"C5pVQ!-&+ !-#A $'# 't+%C5+5. 5B8)'+,!#%$'+ $ #3).9. 02(,$&'( q,67( H (/$'& . E * ω (E ) = 1/pLC (E ) 6Q ## ''&6 + 3. #% +36BA )'.'&c6#n +58U#367) n&C!" !-(Sr#5B$" F 02!## V H (,8Uq,& /n 5 $'("n &'$'(,8 $ 5BG 8 '1+5$!C " .Qr5(o!(/Ceq,(( .E't## ' n&'$'(/8'$ 5B8^+3q ; & nn/.C3+5.AN& #3q"C5+3C# q,#+ 5q/C3r5(,$'$ 561'1+%n(( 5#3$'&'1 8 '1+3C5+3.'+54,+~+5).'(o!-(o/n .'("C5q .a$ #3)9. . 02(,$&'(,6hq,67( H (,$& .A )'+,!#%$' $ 5B6h$ # F )(,.'( +,! $#3q/q,67+3C3.'&'6jC5(/)'(,.p 5+%n (,(~+ #H &'8Sqo/n "=& #38S)'.'+5& ,r5+%np3$'+58S#367) n&!C " ! 5O n . C5+34,P+ &C5#+ 5Q.-( 8 & C5Pp " .'#3r5$'(/$'&'<( ? K9oA $(,+ > +,!-&67+V # !#%C3pVr .r5$'+56 r3&t!-O( %O"t $ ' ; & V U (q) A +5)'.(o!-(o/n .'(/L6 "?VUq,r3+58'q/C3r #367& F2)(,.'( +,!# =J|+5(q '1+%n p '"2& ,67(,.'& C5p $'#3)' . .'L6 "=V #3 . .1!$'# '1+5$tF !-(/$'q #%C3+5.'(>$'(,).'+5q/C3+' A B/!C "2 #3r3&'q,&'67+3q/C5px+ 5T&$'+*)'+%/n "=& ## V>C 't+3q,r5(/$' $ 5B6 67("C5+,!-+56BA &'@q 5+, ! .V& #%r5&'q,&67+5q/C3& C (E ) A 't+%C5+5. M# . n(/(4 '1+ q,67+ 02(" C #5
s
0
d
s
d
s
s
0
Θ Cd = p 3 (Upn − U )2
?
4 !-( Θ F<$'(-'1+3C5+5.'#!.4'1+5$'q"C#3$C#A# U F6't+3$C##'C3$ #!.m. # $'+5q"C5p)'+%C5(,$ ; & #n+3rV! n/.: #3)'(/.C5+34,+ )'(/.'( 5+ !-# Y #PB/C3&?_)'#3. #367("C5. #V$'(,+>5+ !&'6#+5)'.'( !-("n &C5p B-'1q,)'(/.'&'67(,$ C#np5$'+& 1#3.'##'C5(oF .'&'q"C5 & ' !-&+ !-#' |$C3(,4,.&'.'+5r #%$'&'(,6 "t. #3r5$'(/$'&. ? E67+ 0f$'+ )+%n/"=&'&C3p~ #3r5&'q/&'67+5q"C5p2 #3. .tF !#2$ # !-&'+,!-(*+3CV)'.'&tn+ 02(,$'$+54,+f$ #3)' . .0 (,$'& . * pn
4 !-(
p q = −3Θ 3 Upn − U + D
D
F)'+5q"C5+ .'$$ #!.&'$C3(,4/.'&'.'+5r5#3$'&. A'+5)'.'( !-("n .'(,6#M.& |$ # & # np5$ 56"5q"n +5r5&'8+'t#&(,q/C3r5( 3I
$ '2$ &4 B/C3&?1"5qon+5r5&8:
# !# !-&'6T* C0 = Cd (Es )
J|.'&:B/C3+56c)'+%n/"=&'&6 q"n (o! 3" (/C A & C5+
A +%C'" !# D = q + 3ΘpU 1+5$ #%C3("np3$'+f)+%n/"=& #3(/6T*
p q0 = D − 3Θ 3 Upn − Es p ' & Θ = C0 3 (Upn − Es )2 Y
0
3
pn
− Es
C ?
D = q0 + 3C0 (Upn − Es )
& n& q p q˜ = q − q0 = 3C0 (Upn − Es ) − 3C0 3 (Upn − Es )2 3 Upn − U
= JI
!-(/6h.'#3q,q/6#%C5.'&r #%C5pP'1+%n(- #3$'&?.$ #3).9. 02(,$&.$ # (/67'1+5q/C3& v +3C3$'+5q/&C5(onp5$'+~$ #%)'. .02(/$'&. q,67( H (/$'& . E :U = E + v 65B. #% ,&'6 & = JI #3r5&q,&'67+5q"C5p v(˜q) * "
H
s
s
? 3G
q˜2 q˜3 q˜ − + Co 3C02 (Upn − Es ) 27C03 (Upn − Es )2
v=
J|+,!-q/C#3rn/..e #3r5&q,&'67+5q"C5pE ? %G ~r3" .'#3r5$'(/$'&'(a! n .'1+5$C!" .'# ? K92)'+%n/"=&'&6 "t. #%r5$'(,$&'( ! n/a . )'(/.'(,67(/$'$'+58:q,+5q"C#3r%n/.V H (,8 #3. .t!#~$ # F)'(/.'( 5+,!-(*!-&+ !-#Vr2 #%67''$"1C5+56 r5&t!-(!* L
d2 q˜ d˜ q q˜ q˜2 q˜3 + R + − + = V0 cos pt dt2 dt Co 3C02 (Upn − Es ) 27C03 (Upn − Es )2
r5( !-(,6e$'+5r#"=V )'(/.'(,67(/$'$"=V LC0
!-(/q,p
β =
)'+%n "?&&'6
1 3(Upn −Es )
A
x = q˜/C0
&a)'+,!-q"C#3r5&'6 (/(*rP"t. #%r5$'(,$&'(#*
d2 x dx + x − βx2 + µx3 = V0 cos pt + RC0 2 dt dt
µ=
1 27(Upn −Es )2
J|.'&S)'(,.'( +,!-( 'myH67(o!'n(,$'$+56L"
x¨ + 2αx˙ + x − βx2 + µx3 = V0 cos Ωτ,
r3.'(,67(/$'& }
τ = ωo t
? =
A
$ '2$ &4 "
" " " #"$ "%
! '
$7&'q!-? =*HEn+' F q@ (/6#2n-#!5+5. #%C3+5.'$'+38C"3q/C#3$'+3r''&
4 !-( x˙ +3 $ # & #%(/C A α = √ A Ω = p/ω F )'.&'r5(o!(,$'$ #M. & #3q"C5+3C# # !##V H (,4/+ 4,(/$'(,.'#%C5+5.'# J|+%n/"=&'(/$'$'+5(0"t. #3r5$(,$'&'(C ? ? F 67+,!-(onp+5q ; & n'n/. C5+5. #q2$'(on&'$'(/8'$'+58(,67'1+5q"C5pV A'1+3C5+5.'#!. 67+ 02(/C 5
2
R L/C
0
"$ $ &
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*
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K
s
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I5G
$ '2$ &4
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6xn/. 6L" np
Q
O
"
'$ ("n&$'(,8'$+58 q,&'q"C5(,675 1#3.'##'C5(/.'$ 5B67& .'rn/. V>C5q . B(% %*((' C5 $'(,& ,+>.+5$'$'+5q"C5&h& C5&q/C##5&tnp5$'+5q"C5&+ E&(,6e ##' n V & ##V>C5q . B/5C & B(% %*(('C#5
?> |q,)+%np3 -"#. 67(/C3+,! 67( ! n(,$$'+S67(/$. V H &?5q . #%67) n&C!" ! q,6T . # "!-(on/K ? ? x+5).'(o!-(oF n& C5p)'+5)'.'#3r'" ':& #3q/C3+3C5(P'1+%n(- #3$'&8S#%r3C5+5$+567$'+54/+ 't+3$'q,(/.'r #%C3&'r5$'+54/+:+3q ; & nn/.C3+5. # q '"=5& &(,(q '1+58:$'("n &'$'(,8$'+5q/C3pV x¨ + ω x + µx = 0 2 0
3
J|+'t#%0f&C5(%A>&C3+ )'.&x+3)'.'(o!("n(/$'$ 56x $ # &'(/$'&.=*) #3. #%67(/C5.+5r rT%x#% ,+5r5+56)'.+5q/C3. #3$'q"C5r5( +5q ; & n'n .C5+3. #* x¨ + αx˙ + βx + γx = 0 q,+5q "9H (,q"C5r#"=V>Cm!r #:#C3C5.'##'C5+3. #LJ|+5q"C5.'+38C5( B"#C "V+ %n-#%q/C5p* $'# &(,$'&'8 $ # ) n +5(q '1+5q"C5&) #%. #367(/C3.'+5r
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<
0
p
0
0
p
g x¨ + x = 0 l
n8I ,q"n2!# 7-q'4 3r%/3p r5J 4H> / 7 n .:/6. V )H (q .07 n8/ 5O* K
3 , ,3! ,#$ * $! x¨ +
g x=0 l0 (1 + ν cos ωp t)
%# & ', #' ω 1!/?7 !#n -? !.*1 n8#5q'5p< ! n8/!"67 5 #', & ' ! #q 3r% V q# q /?7 /+ & / ! '% * ν 1 ; /n# H %O", ' ; V r) .,7 Q ( n8#% C!# r%n8/ ? " n . #%- &/'r#8)q,p~n8( 5J &tn8 4/n8#FH>.+* ν Y
g l0
2 0
1 1+ν cos ωp t
# 7-q4 3rn/.. J
?
1 = 1 − ν cos ωp t + ... 1 + ν cos ωp t
r ? ? !#%n "?&
#
; 3r% "
x¨ + ω02 (1 − ν cos ωp t)x = 0.
5# q ; n'n/.;%#% ? ! '%A ./3q . r%#Br C!/# -? #% &/,q('$#%L" 6 #'$#%"#-> 5J( r%,7 ", 3r%/ .0 4>5Br '5q .y ", 3r / `2 45p%o} R- #B#FH.#>6 !'q 5pVrfr,7
x¨ + ω 2 (t)x = 0,
MK9
- 7 ω(t) = ω p1 − ν cos ω t 3 6 3r%'q .?H> !..# r% q#5q3r% / !. & 3q#4 )#5q ; n'n/.;%#% ? R7q"n8 r ",3 r /W MK9 F 7 # ",% r% / V ! r%# 7 /@6 7 & @# '$%# n( " np#%E ',#/# ",0 %')!#% p I7 ',3# q5 p '$##%#%-#9 ./3 q .&!# 0
p
2
K
3 , ,3! ,#$ * $! i
R
C(t)
UC
L
UL
1 2n8(#%-#<'$#%;#", q / .V H ('q . #fr 7'$#5q3pV !# 7 &/q('$#%L":F !',#" $ 3 q,q#%/ q@ L "Vn8(#%-#0'$#%n8(% %onp%#%-#0'$#%/#", ; / 7'$#5q'5p0'$##%#%-#)F 3r%q / # r% 2 q# *q@%# 7 .2;A6# '$#%5# rVz ;- # %A / 8#FH #)F !/' q6 3 p<",3 r /# * $ 'q!? *
L
#
di + Ri + Uc = 0 dt
v ! ./H) J ,',#,7 q6 #% U #FH #|r5J 46/3p &/ 'J6 &.B 7'$#5q'% 5 6 .,7 B B- # #' n F7 ', MD* U = q/C J / !8 ", 3r / ? 7 n . c
q
c
*
L
K !#%n/"=& ", %r% /n8/ (/#%- # = x˙
d2 q dq q +R + = 0. 2 dt dt C(t)
%# & !. R/2L = α, 1/LC = ω , q = x, #5q ; n'n/.;%#% fr>% F7 ; ##%( %A##* Y
2
dx dt
,#,7 q6 #3 !/ #( '
x¨ + 2αx˙ + ω 2 (t)x = 0
8 7'$#5q3p .'5q . !# F !',#" C(t) = C (1 + νc(t)) - 7 C 3fq =7 6 &/ 3 '$#B-% % ; / - n/"=%/ 9#67/" n . ; / ! / c(t) 3 %O", ' ; ?. JF 7 #V H) !. F #'$#% ν / .)! '% / !/ & max|c(t)| = 1? 6%#G!/ 1 n8# '$#B(% % ; / / # 7 " n/. ; / / q# 5q'5 r%#%( &3 q# 5 ',#/#", B# H #5!=7-q4 %r%/5p rVr,7 #* R7q"n
0
ω(t) = ω0
p 1 − νc(t).
KK
0
3 , ,3! ,#$ * $!
'5p@! ω(t) .'5q . !# 7 &/,q('/mq !# 7 #% T * c(t + T ) = c(t) z #'0! B#%2 #"67 'xr%,q'%q( .Vq 'q'% 1 6 3r%'q#5q'%.# ! n8/!"F7 5 <& %q%#5 ! 43 &/, q('$#%(<# ', & ' r0 S( r%# # H #0 ,r%/%/.! &/q('$#%( -"3q#%( &'r#5q L'$#%#43 ! . ! r%# 7 /: ' / #%- &/ #L " 3 q4 / V ! n/# "67 5 '$%# n8- ( Q !',# .r%n8 / r2q q%1M aq !# 7 &/ q(' .V H ' q .0! % > 5Br 5 q .0! &/ q('/ '6#% ' q #%T J,"5q
p
p
P
P
< " xn/.+ 8) . 6 7 & #r%# '#3r% / ! %/ &,q-'$#%( -"3q%#( &'r%#3q%Urmn8(;3 #%<7 q,q /! % r%#%*! '%&,q-',# #5q ; n'n/.;%#%< # /q('$#%np'$#P",!#5q'%eq 'q'% L" !=7 !%# n# H ' r/ &;%#B! '%2 q ## 5q3 r% ! .C&3 q#4 J ./3 q . !#>F! ',# " y ,7 }
q ω(t) = ω0 1 − νSign(cos ωp t)
3 %O";' ; .A6 !'; ; F!/1 #V H) !.C & / +1 ! !/#%n8#FH.;%"n p%#%2 --",/% x −1 31!9#% ; %onp%#%T B'%#% qon/"=& *q #3q5r M. & 3q'%#4 P'$#%/#", >!# 7 &/,q('/ !/1 ' -7 r &6 & .+* ω = ω √1 + ν ω = ω √1 − ν F ! /# 7 ! '%&'3 q('$#%-2# r%#?7 '( q'5 r%? . T = ?*%- & /x r 93 r%/#%- #)!/%# n85# r%/E!# 7 2 r%#?7 ( q5 r . &3 q#4 A3# q ; nn/./#% 5# q4 '5 q . !5# q'% # ./#%+( B#% q"n/"=& '7 n/.P', HB3 [0; T /2] [T /2; T ] 7 #%:( &3 q !# 7 9#FH./#@ (/ %# &/#%5 n/% &/, q('$#%B 8) /5 !# n8/ 5# r 45 p !5# r?7 F2 r%, q/p !/ # 7 -v (,78 *# &#A/- 8) 7 n/.1",3 r% .C'$#%/! "; Sign(x)
1
p
p
p
0
2
2π ωp
p
J
2 x¨ + 2αx˙ + ω1,2 x=0
(,7 ':q'q ," 3r%( ! 'r#%- #B!#% .,7 '; ?* K
0
3 , ,3! ,#$ * $!
x˙ = y
?
2 y˙ = −2αy − ω1,2 x
J k1,2
#/ r%# 7 .+!# 7-q'4 #5r#'" x ∼ exp kt #%!/?7 on8'r & /. !/#', F "n/. B('1q !/#% ;5 !%# n "?&/ # #H (ar%,7 -8 .97 n/.q'q 75 * ' −α ± jω 1,2
x = exp(−αt)[A exp(jω1,2 t) + B exp(−jω1,2 t)] y = exp(−αt)[A(jω1,2 − α) exp(jω1,2 t) − B(jω1,2 + α) exp(−jω1 t)
) #0/-8)0F 3r%q / # 7-r#" !#/ r%#np% 56*!#5q'%# ./ 56 A B 7'$##% 5J #%--"$ 5 3 p_r5J H) 5 &/ '9 & np 5J<"3q"n8#5r. J 3B%# -8 x = x(0) y = y(0).
!/ 6 ! 8 5 q . rfr,7 # * ω=ω 0
0
1
x(t) = exp(−αt)[x0 cos ω1 t + y0
sin ω1 t ] ω1
? I
y(t) = exp(−αt)[−x0 ω1 sin ω1 t + y0 cos ω1 t]
|q !#np("!. r5J H> / ? I ? n8 --',#
F !q6 3pmq #%###8 .H>7 " & np%5J<6 &?3 ./ x y & /. & /B!#%n3 ! /# 7 @! %/ &,q-'$#%- #:r%#'78 ('q'5r%. t = J ?7q4 3r%/B# q ##%#!8) / rVr%,78A1 4% ; 5O* 0
n8
π ωp
0
πω π π cos ωp1 x( ωp ) = exp(−α ) ωp 1 −ω1 sin πω y( ωπp ) ωp
x( ωπp ) = [M1 ]x(0)
1 x sin πω ωp 0 1 y cos πω 0 ωp
1 ω1
n8#- &/#+#FH #<6 !'q 5p<7 n/. / 'r n %r # K%K
π ωp
3 , ,3! ,#$ * $!
7 # * x( ) = [M ]x( ) - 7821 ; " [M ] # H #!#%n/"=&;5p / 1 ; 5 [M ] F 3 /#%(3& 3q#5 ω ω r:q n/" #%- # &/%#0!#5r%=7 / :!/ 'r%# r%#%/#% //%'r n r% +#%! q(5Br #V 5q .*# 7 #'$#5r#5J", 3r%J #Sq9 6 5J+6 &./*! 43 ω(t) = ω `0# H # n8<#!?78" n8/3 pmq #%#%## 8 /@H> 7 "& & n p% 5J< & ? . 7 &/ q('? ! 56 x , y C? A6 & ./ & / !/ # 7B! '% &/, q('$#%-*# r%#3 7 '( q3 r% . Y &/ r%,7 #/ &/%#C, q"n. /, r%, q'% 5 1 ; 5 [M ] [M ] '$##% 5J 6 7 # V C4 3 'B q#%### 8) .) 7 n/.)! ' r#%(52 r%#%/#%(5!%# n85# r 5+! /#67 /5 q ##5 r%'5 q'5 r%# %# 1 ; ; .'rn/. V H>! .q .9? .!#%/, r%?78 J/ #"67 q,r#.;> 5Br 3 p>6 & ..7 &/ q('? @! / / 56 &C! /# 7&! &/ q('$#%- # r%#'78 '( q'5 r% . * 2π ωp
2π ωp
π ωp
2
1
2
1
2
1,2
0
0
1
n8 *
x( ω2πp ) y( ω2πp )
= [M2 ]
-7
x( ω2πp )
x( ωπp )
2
x0 = [M2 ][M1 ] y0 y( ωπp )
x0 = [M ] y( ω2πp ) y0
[M ] = exp(−2α
π )× ωp
πω2 ω1 πω2 πω1 πω1 1 2 1 2 −ω1 cos πω sin πω − ω2 sin πω cos πω cos ωp cos ωp − ω2 sin ωp sin ωp ωp ωp ωp ωp × πω2 πω2 πω1 πω2 ω2 πω2 πω1 πω1 πω1 1 1 cos + sin cos cos − sin sin cos sin ω1 ωp ωp ω2 ωp ωp ωp ωp ω1 ωp ωp
) #-8 #"67 ' 3q'4 #V H J ,q"nO%# .F5 # 7 /#>q#5q'5r%#%6 &/ 11 ; 5 !#B#67/" n/V !"F78<%#%np#8 ?7 / ; 5O [M ] !/?7 o n8 q #3q5r 5JG & /.91 ; 5 [M ] Y @ M%# 7 ./3q .9/ #'/ q%&'3 Y q('$#%-# ",% r% / . '$#%##%* r>7 r#", # q"n "?& C/ fr%,7 * K
3 , ,3! ,#$ * $!
- 7 S = M + M 3Eqon8?7 1 %/ ; 5 r #%hq"n/"=& !* 11
µ2 − Sµ + J = 0 [M ]
22
8
J = det[M ]
31 #%!/?7 on8/"np 3q/q 1 ;3
ω1 ω2 πω1 πω2 πω2 πω1 π cos − + sin S = exp(−2α ) 2 cos sin ωp ωp ωp ω2 ω1 ωp ωp π J = exp(−4α ) ωp
|q !#np("!. ' ,F !/8) q #3q5r 5JA6 &. µ1,2
- 7 s = 2 cos r%# ', ' r>%#
πω1 ωp
q"n "?& 8,qon8+* 2 cos πω − ωp
*
" # r π s2 s = exp(−2α ) ± −1 ωp 2 4
ω1 ω2
+
ω2 ω1
2 1 sin πω sin πω ωp ωp
%/ &,q-', !. -"3q%#( &'r%#3q5p
J
s r s2 π − 1 > exp(2α ) ± 2 4 ωp
3 q,q#%/ "3q"n8#3r%3 %#%n8@!# 7 ##%# JO"3q3p |s| < 2 LB'%#% q"n "?& 7 'q-';3 /:''r F7 #%- #4", %r% /. # ; "n8 q ##5q3r% 5J@6 &. 3O'$#%!tn8('1q /#3 q #! ./H> 5J# :# 7 " n80# 7 #'$#5r#5O* $
s r r q 2 s s2 π s s π |µ1,2 | = µ1,2 µ∗1,2 = exp(−2α ) ( + j 1 − )( − j 1 − ) = exp(−2α ) ≤ 1 ωp 2 4 2 4 ωp
#% #%J ! "5qon8#5r% |s| < 2 '$#%n8- ?.er! &/q('$#% #5q ; n'n/.;%#%0 3q4 #V JO"3q3p |s| ≥ 2 +$ 3q,q#%/jq & n q"n/"=& ( '$#%'q 'r% %'r#%- # #5q ; n'n/.;%#% Q !'
K
3 , ,3! ,#$ * $!
α=0
'%#%eq"n/"=& "5qon8#5r%/ ! &/q('$#%( /-"5q'%#%(&'r%#5q'%:#"67 ' *
EB
|s| ≥ 2
n8 * '5p Q #- 7 J,"5q
ν →0 ω1 ω2
+
ω2 ω1
πω ω ω πω πω πω 2 1 2 1 2 1 ≥2 2 cos cos − ( + ) sin sin ωp ωp ω2 ω1 ωp ωp
? K
( $#%/ #21 n8#%( ! n8;#"67 5 ! &/q('$#%- #gr%#'7 ('q3r%. 8# % !,r %;5q ._r* →2
qon/"=& 2 ,q(' (& 1"5qon 5r ? K
πω πω πω πω 2 1 2 1 cos ≥1 cos − sin sin ωp ωp ωp ωp
/# #FH #5! !'q 5pr~r%$7 #* &
cos π(ω1 + ω2 ) ≥ 1 ωp
) #A/ 3r 'q'5r%#Vr5J!#%n.'5q .07'n/.
π(ω1 +ω2 ) ωp
= πk, k = 1, 2...
tn8+*
2ω0 = k, k = 1, 2, ... ωp
%# & & ' ζ = 31#%##8 / q#5q3r% /#%(aq =7 (C& 3q'%##5 '$#%/!", '1& %q%#% #', & ' s" ; 0##%n 3q'% (:! &/q('$#%( /-"5q'%#%(&'r%#5q'%07'n/. '$#%'q 'r 4%'r%/#%- # #5q ; n'n/.;%#%< #"6 7 "$ ''r#5J F 7 3r 75J ", %r% /.+* Y
z #'r%$7
ω0 ωp
cos √
πζ πζ πζ πζ 1 cos √ sin √ sin √ −√ = ±1 1−ν 1−ν 1+ν 1+ν 1 − ν2
K
#)/ "5q"n#5r%. ' 7 5J - ; 5 ;!#5q'%#% 5J !tn8#5q('$#5q'%:! #5r ,#%!/ #V 3q .: C#& ' (0.5; 0), (1; 0), (1.5; 0), (2; 0), .... $70#%n %q%(9-"3q#%( &'r#3 ζ −ν KI
3 , ,3! ,#$ * $!
"
" !!"# $ % ""$ % ')
,7:##%n 3q'% (.! '%&,q-',#(9-"3q%#( &'r%#3q%.7 n .'$#%'q 'r% %'r#%- #>#3q n/3 /%# n8/y }K @7 n/.@7 'q/q ! %'r#%- #.!:6 &.?97 q,q /! ; 2 n8/; y }K α/ω = 0.001 + n8/ y X } α/ω = 0.002 n y t}K α/ω = 0.003 n8/
$ 'q#? ?* n .
y "}
0
0
0
α/ω0 = 0.004
q27 n/. '$#%'q 'r 4%'r%/#%- #0! %/ &,q-'$#%- #0#5q ; n'n/.;%#% :!r%?7 & 9'q# : n8/ y K} 9hq"n/"=& 7 'q/q ! %'r#%- # #5q ; tn'n/./#% / #> # 7 /#2r% ?",3p5q . '<%#%n8E#!H) L"~r#5 3 H) V TJ E%/q('$#%-&# 1 n8#%( ! n8/!"67 Sr#'7 ('q3r%. ν → 0 3q'4 '$#%8n - (2!#%'@q %# 7 /59 p /F#"67 ' !#5(q '$#%n p '" exp(2α ) = exp(2πζ ) > 1 I#%5 ! 3 &/(q '$#%( "5'q %#%( &/'%r #5'q %27 /n . #5q ; n'/n .;%#% X !/#%9./ #', (5B r #V 3q . !/!# 7 .;5 3 F7 #5q,p V #5q ; 'q,!q + o8n &/ @!#%#-? @#%!=7 "/n ./3q . & / #%/##8). ! 43 @7 'q,q ! ; ' q #5q3%r /#%(2& %q %#% '$#%/!", α/ω ,7 #n 3q ( ! '% &/,(q '$#%( "5q #%( &%r #5q
α ω0
0
0
min
0
(C #% ! # H #J##" %$ //#&' %#( 6!$)" +* 6 -, - .+ /# '#& !/ I 0 $## (&:7 ! %1#%- #2 ,#+*8 " 4%3*4&%#%-#5 $#%/(/, 0 !$/( 6$ !#:F ( $#%7/28 '3
9
3 , ,3 ,#$ * $ 0,25
b
0,2 νmin 0,15 0,1
a
0,05 0
0
0,001 α/ω0
0,0005
0,0015
0,002
% /## (& /1 +* &%#- # 6 !$@- */7" # 7 / *4$ /@! C#C!,3 1?7 #%(@7 ! :7*4$0##" * ( '%( ζ ≈ 1 . ζ ≈ 0.5 ,78 %#%#%( 8 1%#%( /67 1%#%#%( (# "# ( 1% /#%2( %#% $#%;(/, .!.3. =7 !# *# H J /#@! %/ $#% 1%#?7 !( # 1)5 $# *8 " 7 $21 ,#/( /,) # 7 $;# 6$ 1A ## 1 ##% # ) /! $ I!/## , F / 3/, $( . ( / 1% # , /- ! $ , *8 ;% /'3 $#%-# !# *$ $#%,7 6 4%#% ( 1% 3 F $,7" 3G0 $## ( & 3."! *8 12# / 1 ;3 q /2C q C /1+ * &%#- #&6 ! $ F $$7 /, &# );# & 0 ,(# # & 2∆C %# , - 7 $ F !( $ 1 $#%,7 6 4%#% / 1% *8 ;# 6$ (.7. 1
2
2∆C q02 1 1 q02 ∆W = ( − )= 2 C0 − ∆C C0 + ∆C 2 C02 − (∆C)2
1 +*#%- #B/6 /!$20 $## ∆C C
$ 3*4/
0
∆W '
q02 2∆C = 2W0 ν 2C0 C0
- 7 ν = ∆C/C 3 - *4/ "% 2# 7 / *4$ ! '% !.%$ #% /21% $#%- 7 6 $,7 $#%,7 6 4%#%E (1% !/ *&C $ 0 $## (& , $#%##" 6/#/ 1%3* 1 # 6$@ . ' 7 ( $# ,'%# 2∆C 0
9 (
3 , ,3 ,#$ * $ q
t
U
t
C
C
0
t
" 9!# 7 $#%-#B;6 / !$.F $,7 / ; ! $/H>!$9. 0 $## $#%,7 ;3 %#
(.7. #
C!1%=7 2/, &# & , - / 1 $#%,7 6 #%1( 3*8=7 (1%>#( /$# '#1%!$ 6 $,7 . (# ,%# /15 C#678#%- #5!# 7 ##"# '#1% $# *8 " ( $#%,7 6 4%#%97 1 H>7
!# */7 !#% / & , - I *8?7 (# 1 % * &%# ! )), - 0F 5! /# 7 #( 4 (1%; ∆WT = 4W0 ν
!. ( 9
#%21 $#%/ /,C6 >7 ! # 26 >! /#67 $ # *8 "% ( ## '4 (1 *4$& ∆Wd =
W0 Q
- 7 Q = 3/7 ##"#%/## (& $#%/ /, !.E!7*/(/67 ' $# *8 "% ()"(/67 ( 4 4#& 1 #%2 *4/ *8&!/## , - /!/ 1 ) 0!#% / .% ##,#% /% /# 3*##1% 1%#6 /##1%!$ ! 43 / $#%-#.'6#% 6 "(/67 ' ω0 2α
9
4W0 ν ≥
7*8
ν≥
#%$
& ( '
5*)5,+
8-:9' <;>=?+
C0 *8<- *4/
!. ( !
W0 Q
!. ( "
α 1 = 4Q 2ω0
- / .) 102-3+
54 76
@ ' B A 5)
%/ 2# 7 / *4$ / ! ν "%# * & ) / $#%- # 6 / !$I #2 , H>7 J( !# 7 $# * " /(1 $#%;(/, " /F78)7 ##" 1 *4$/(&# 6$, / -!$ . E! *8/(/67 $# *8 " (: / #%- # 1%# ( 4 4# &.ED #- / /#% 1%# ( '4 & ! *8/(/67 35' / * &F4 + ?3 !# *4&6## 1% ! $*8(#%(9# 7 *8 74*4$.#%! !$.! %/ $#%- #B #% ! 6 !. $ +* &%#43 #%- #%-# 1%# ( 4 ! $ 7! *8/ /F 7 /C!/#% # 7 ; I!## $# * &# !/0!/2 "%# *4&# ) 7! */( /67 $# *8 " /( ! %1 &7 #% ! / *8( J 1%#%!( # 1% / 1! 1%# 7 $ ) #%- / & 7! */( /6 7 ,+# *8 " ( . ' 7 /:4( +!# $1 * (+ *8(## '% #3 H>' 3*4/$H /# & 6( 1% ## ( & 1%3 * ( / 1% !# & 1 ## 7 **$/%#9#& ! *8/( /6 7
$# *8 " ( 1 .7 /( # &/ / !/ I7 $## , / /A7 !# /9## /7 **4$;%#% AF( 1%! / # $# *8 " ( 1. 3*?7 / & (# "% #% . $ ,#% 3*4/ ?7 /5!#% α = α +α x 1 $#%/ /, "(/67 /$9( '% )/## #% 7! *8/ /6 7 ,+# *8 " ( 1@ ! / $#%##% 6 ?3 . 7! *8;( /6 7 1%#6/ 'A7 $#%G 1%## 1% H>7 /, /- ( /!(# ( /,! & )( 1 $#%/ /; , - ( ( 1 #% ( .F *8=7 ## 1 3 * &# *& "#% $# * &/,- # 7 #B1+ *8#G6 $## , / /4 27 ! 4% 1%/#%( 3 *8/ (/## ! 1?7 ' <#%- & $# *8 " ( . α 'G! J(A / #%- ! $C 7! */( /6 7 ,+# *8 " (A 1% ( 6$*GHJI IKH L M N7HJOQP RTS. U ! /,- !/ # 3 *8/ (/## #%- ! 1 # & ( $# *8 " 7 $51.! '% / $#% ## ?3 "
0
1
1
9
2
A Amax
ζ
1
ζ
0.5
2
ζ
% /## (& !7*8/ /67 ! ,#+*8 " ( A # #%## 7$ ( '%# !+7 ! /! %! 1%#% / #%- !$ !7*8/ /67 . ζ ζ 3)- ,3 ζ (# " * ( '% ! $#%( /( %#( 1%#( % ! 1 " # 6 &- *4/ 3 " #64 7 / *$ &! ν . %## 1 J *8&/ ##"% 4H5 #& 0 !7*8/ /F78#3 ( #%! / & $ /% !/94 7 *4$*8/ (#- #.(# 7 *!*4$/#% !. 7*!*4$/#% 8#FH>' 3*4/$H /# &5F( 1 (# # & (# "# ( 1% /#%( ( # (# 7 *!*4$/#% 5#B 7! *8,3 (/67 ,+# *8 " ( 1 (.7. 1
1
2
ω 2 (t) = ω02 (1 − νSign(cos ωp t) + µx2 )
'%#% *4/ #( !7*8/ /F7 $# * " /( 1 ## **4$/#%@!/1%=7 2 ) / & -# #("# (1% /#%( ( '%## +I *8?7 #(1 % * &%#/ ( 1# # 7 /C/ #% ! $#%- # '6#% 6 7. U) J(0 / #%- / ! $2 7! */( /6 7 1 # 6$M I I N P RTS. ( #%/ ,( / , /4 +! 1%# 7 ; ( '%#% J( $ /6+#%- / ! $ 3 7! *8;( /6 7 $# * " / ( . ! ) /,( 1%/ C7 *4$&3 * (/#%- # $#% 1 4% 1%/#%- #0## /7 **4$,3 %# B!/ -? #% / $#% ! '%! ,# 1%#'78 ( '# 1% $ ,
x¨ + ω02 (1 + ν cos 2ωp t)(x + µx3 ) = 0
7.
# 7 ##"%#/, (1%/ # H) #%!/ 1 4#& $#%/ /, " !/#%& (1 , #%#%#0 $## (& # * &# $# 9
!# 7 !$/( 6$ #1% J #> @F (1%! /B#>!/7*8#FH>#%-#B ! $/H>!$ . U)*4$ *8/F /, (1%/ !$ 7. !/# * & /% '%# 7 -? #% / $#%-#2" * / !# 7 4 1%1 ( #% J (1%#( !# *8#(1% ( #
1 /,( 1 / / x = a cos ω t + b sin ω t ( , %#C '# &) 1 4% 1% *4/ (@## ##1%/#%(.##"+* ( .! $#%-#B/6#43 ! 6 . " -? $ 3*8 -? J % J ( #4 2ω t, 3ω t !# */7 / '% / /,( 1%/ ( p
p
p
p
ν 3µ 3 3µ 2 a + ab )(1 + ) − aωp2 = 0 4 4 2 3µ 3µ ν ω02 (b + b3 + a2 b)(1 − ) − bωp2 = 0 4 4 2
ω02 (a +
7. (
1% +* &%# / ) !. ( a = 0, b = 0 #%##1# '#1(/) #( %# $/ & (1%##1 7$1 +*8 $#%#%7 # #& ## 3/,( (1% & $# *8 " /( .A- $# / 1%,78#&/ /# !/ # 7 ##1 #% 1 J!+# *8 /)( 1% ( 1 1 '4 ##1%;# 6$)/ #(1% '%#%( . 0##"% #%J a 6= 0, b 6= 0 ! $# *8 " 7 $ #FH> ### 1%'# '# 1%## 1% # & # 7 C/- *4/ a 6= 0, b = 0 7*8 a = 0, b 6= 0 .'G!=7 3 * 7! */( /67 / $# *8 " ( ,( A = √a + b . #% 3*/7 ( 3.E( 1%/ E47 *$2 7! *8/ /F7 $# *8 "% ( !' 1%$7 2
2
!.
ωp2 3 A = −1 4µ ω02 (1 + ν/2) 2
F (1%! ## '%C# ( , ! '% 3* (/## µ ! '%! - $# *8 " 7$ 1!/ 3 p *8 1%#%( 7! *8/ /F78#%( 3 1% (1% /# 3/ '#1(/ & *8!"%#2!/ 3 A ω > ω = ω 1 + ν/2 74*4$5!/# *8#FH.;%3 *4&% > /( µ *8 "%# !/ ω < ω = ω p1 + ν/2 3I74*$5#% % * &% 6 / ( µ . U)*4$ 3*/7 $ !# */7 / /,( 1 / 47 *$2 7! *8/ /F7 $
p
p
1
1
0
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# % 7
3* #1 !$ 4*4$ 3/
) '#1%#(1 !$
) (1% /#%- # 7$ 3*8=7 / & # 8* !"%# 1
9
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/# '%#%!( 1 J(+ 3 /# #%( ! 1 J(*!/?7 * &% J / 7*
ω = ω 1# '% # & ( 6$1 A ## 1%#%!(# % ( 1% B 'F# & . ' 4 # 6$@?7 ! # 1 J(0 % /%#%B3 (# %# $/ ( 1%(# 1% ! $ . $ , ! / / 1 ## " * ( '%) ( '%# ω < ω < ω "(/67 ( " * & 7 # &43 ω0
p 1 − ν/2
p
2
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c
9
1
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p
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1 2
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ωc
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