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dxi = fi (x1 , ..., xN ), i = 1, ..., N dt
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+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
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I λ − Sλ + J = 0, 2
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FF't+367) n(('1q/$5B(:&'&'qon-# X )'+%n+ 0f&C5(onp5$5B67& r3(-H (/q/C3r5(,$'$5B67&E&'#3q/C .67& λ = A A & Fxr3(-H (/q/C3r5(,$'$5B(4&'&'qon-# &t! .'( 8 (,$&'8eC3+3CS0 (3A α + jω λ = α − jω α > 0 ω &C3+ &Sr:)'.(o! 5 ! "9H (,6 )"t$'C3(3A$+:)'+#' # #%C3("np4B-'1q,)'+5$(,$C#5 F )'+n+ 0 &C3("np3$5B8+~# F ,+5r 5B8 )+5.C5.(/Cm)'.(o!-q"C#3r%/n .(/Cmq/+ 5+58S. #3q('."=&'&'r ##V HA"=V@q . q,)'&'.'#np .'&'q# oAC3&') q/+5q/C3+ .$'& .:. #3r5$+5r5(,q/& .F B6;=SJGM9>.CK:EFA6012 KaYP.-ITSJGHe
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$+5r5(,q/&.a&'67((V>C r5&1! *
λ1 λ2
1
x˜ = ξ1 exp(λ1 t) + ξ2 exp(λ2 t) y˜ = ζ1 exp(λ1 t) + ζ2 exp(λ2 t)
K
2
K
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λ1 λ2
1
2
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λ1 , λ2 , λ3
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λ1
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3
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be
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KJI
dx = m − x2 , dt
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2
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K 3G
dx = mx − x2 dt
'& q"C5(,6#C %G &67(,(/Cx!-r5( C3+&''&.'#3r5$'+5r3(,q,&?. x = 0 & x = m 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/.VH (,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 . $ ##%nV !-#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(,$( x ↔ (−x)
5
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K 5
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3
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[m − (x2 + y 2 )]x − ωy = 0 [m − (x2 + y 2 )]y + ωx = 0
7K9
+567$+ 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# #
&t!$'+'A&C5+fq-"9H (/q/C3r#"5("C!r #V.(-8 (/$'&. * +,!-$'+V& $'&?F x = 0, y = 0 & n & z = 0 &ar3C5+3.'+5(3A ! n/. '1+3C3+5.'+54/+ x + y = |z| = m Q+g(,q"C5p ).'&I3&%O" .' # ; &'+5$'$+56 $'#&(,$'&'& m = 0 r + ''.(,q/C3$'+5q"C5& q/+5q/C3+.$'&.. #3r3$'+5r5(/q,&.q0'1+5+5.3!-&'$'#%C#367& .'+ 02!#3(/C3q .)'.'(o!("np%F x = 0, y = 0 √ (,6# 5&%O"t.'t# ; && $t!-.'+3$'+5r #:FU +5)%x#a! n. $ 5B8 ; & 'tnA . # !& "5q '1+3C3+5.'+54/+ |z| = m $'+5.6#np5$'+38 %x+3.'76 5 5 "A )'.'( !-q/C#3rn(,$'$+58 r~)' . .'67!+ " 4/+%np5$ 561'1+5+5.3!-&'$ #CM# A-& +5.'#%02("F $ #2$ #~.'&'q! ?
2
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.&'#q J*.& + 5# q,+ 5q"C5r5(/$'$ 56,$ # &'(,$& .a+3C5.'& ; #%C5(onp5$ 5^&aq/+5q/C3 + .'$'&'(*. #%r5$'+5r5(oF < −2 q,?& .:).'(o!-q"C#3r%/n .(/C q,+ 5+3C 8 "3q/C5+38 &'&'r 5B8 "t,("n .'&'!q # X
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Q P 4
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a(/C5+,! 5eC5(,+3.'&'&'1+%n (( #3$&'8 67+54 "1C )'.&'67(,$. C5p5q . . #3$'(/$'&. * x + y + s = xn/. B/C3+54,+~!-+5q/C#C5+&'$+q"n+ 0f&C5pf)'. #3r5B(F& #3q"C5& C3.'( "t. #3r5$'(/$'&'8m&C"?3(o!-&C3p5q .AD&C5+ const. Rqon& &$C5(/.')'.'("C5&'.'+3r #%C5p & ' !# 'S+3C3$'+5q/&C5(onp5$5B( '1+5$ ; (,$C3. # ; &&AC3+ x˙ + y˙ + s˙ = 0 x, y s ?& -q "t67L6 "_67+ 0f$'+)'.&'$ .C3p #( !-&'$'& ; "D* DGB/C5+36Uqo/n "=& #30 ( &'&q"n+ "t. #3r3$'(,$'&8 x+y+s = 1 67+ 0 $P+ "t67(/$'#p 8 & C5p~!+2!-!r "9D*
x˙ = x(β − βx − (α + β))y y˙ = y(−γ + (α + γ)x + γy)
EK9
&'q"C5(,6# "t. #3r5$'(/$'&'8 K K9+3)'&'q(5Br5#3(/C !-&'$'#367&'"2.( ##' ; &'8fqT"=&'("C5+56 "5q"n +5r5&.Vq,+> . #3$(,$'&. ^. #%67' # +3)'&'q #%$'$'+58m67+,!-(on&_) #3.'#367(/C3.5 q/&'q/C3(,675j!-+%n 0f$5 5C5+ 't#> # !#3("C5q .*!-& % %*(,.'(/$ ; & #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(,$C3+5r uq/& n&C3("np K "t)'.'#3r%/n .(/C5q . !-+3)'+%n$'&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)
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I = (S0 − S1 u2 )u,
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1
I = (S0 − S1 u2 − bV )u, ,
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dV dt
B/C5+36
= −γV + ϕ(u).
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d du M S0 − RC 1 ( − u+ dt dt LC LC
Z
udt +
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!"#$%& ')(+*,$!,-.0/12*3#4 Q+34 !# du M S0 − RC 1 − u+ dt LC LC
4 !(
A
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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
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udt.
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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
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4 !-( √ √ √ x = (u/u0 LC), y = W/u0 , z = V (M b/ LC), Φ(x) = ϕ(u)(M b/γ LC),
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00
0
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x = A cos(pt + Θ),
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0
K9
$ '2$ &4 x00 + 2α(x, x0 ) + ω02 (x, x0 ) = F (t)
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−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-% *% & ; &'(,$C#5O)'(,.(o!
cos pt
&
sin pt
A')'+%n"?&&'6 q,&q/C5(/6L"1" .'#3r5$'(/$'&'8+*
3 (ω02 − p2 )a + 2αpb + µ(a3 + ab2 ) = B 4 3 (ω02 − p2 )b − 2αpa + µ(b3 + a2 b) = 0 4
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a = ρ cos ϕ, b = ρ sin ϕ
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A-4 !-( ρ FB#367)tn&C#" !# A
ϕ
3 (ω02 − p2 )ρ cos ϕ + 2αpρ sin ϕ + µρ3 cos ϕ = B 4 3 2 2 (ω0 − p )ρ sin ϕ − 2αpρ cos ϕ + µρ3 sin ϕ = 0 4
K%I
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$ '2$ &4 u 67$+ 0f&'6 ? S $ #
u 67$+ 0f&'6 ? S $ #
cos ϕ
A # ? LI $ #
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3 (ω02 − p2 )ρ + µρ3 = B cos ϕ 4
sin ϕ,
#C = I $'#
cos ϕ
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2αρp = B sin ϕ
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9 2 3 3 µ A + µ(ω02 − p2 )A2 + ((ω02 − p2 )2 + 4α2 p2 )A − B 2 = 0 16 2
= #
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$ '2$ &4 C a tu,& #3q"C5+' #3. ##'C3(,.'&q/C5&''&A-+54,. #%$'&&'(/$'$5B8C5+&'t#367& abcd A?.'r%n/.(/C3q .+%n-#3q"C5pVO4/&'q/C3("F .'(" &'q,#A )'+5q-'t+np'"m)(,.'(/q('1+''&S. ##5+&(,8mC5+&''&S67("0~!/"_r5("C5r#.67& #3. ##'C3(,.'&q/C5&''& r5r5(/.g& r5$'& ).'+5&'q@5+ !/.C $'#>. # $56 & #%q/C5+%C#MDU #3. ##' C5(,.$5B( &,67(,$'(/$'&. r%x#%,+5r5+56)'.+5q/C3. #3$'q"C5r5( q,&q/C5(/76 5|A).'+5&'@q 5+ /! .H &( rV+ %n-#3q"C5&a4,&'q"C5(/.'(/,&'q #A)'.'+5&tn'n V@q"C5.'&.'+5r #3$5O$ #2.'&'q# ? Q## ''&'6:+ 5. # +56BA )'.'& "tr5("n& &'(,$'&0 & & #3q/C3+3C 5h)'.'& p = ω )'.'+5&@q +,!-&C 5&% "t. 't# ; & . .+ 02!-(oF $'& .~)'#3. 5 ; & ' n+5r *"5q/C3+58 &'&r5+54,+ |&V$-( "5q"C5+58 &'&'r5+54/+~? J|.'& p = ω F 5&%O"t. 't# ; & .Vqon& .'$& . "5q"C5+58 &'&'r5+54/+ ; & ' n-# q $'( "5q/C3+58 &'&r 5B6 ; & 'tn+56 A%q,+5)'.+5r5+ 0~!-## VH !# .'q . ?& &(q &'(" $'+5r3(,$'&'(/T 6 J|.'4 & "t67(,$'!p 8 (,$&'3 & &'#3q/C3+3C 5 +3C,$ # &'(,$'&3 8 5+n#p 8 =& ω )'.'& p = ω )'.'+3&'@q +,!-&Ca.'+ 02!-(,$&'( ) #3. 5 ; & ' n+5r *"3q/C5+38 &'&'r5+34,+ _& $'( "5q/C3+58 &'&r5+54,+e A #C5(,6 )'.'& p = ω F2qon& .'$&'(m& &'qoF &'(" $'+5r3(,$'&'0 ( "3q/C5+38 &'&'r5+34,+ ; & ' n-#&m$'( "5q"C5+58 &&'r5+54/+m: .&'#q ? #H & %O"t. 't# ; & ..'+ 02!-(,$'?& . & n&2qon& .$'& .~) #3. 5 ; & ' n+5r *!"5q"C5+58 &'&'r5+54/+ & $'-( "3q/C5+38 &'&'r5+34,+ $'#%> 5Br #%(/C5q .~q/(o! n +%F "1/n +5r5+5P8 5& F & p = ω )'.'+5&'@q 5+,! .C %O"t. 't# ; &'(/+8 Q## ''&'6Q+ 5. #%,+56BA )'.'&S $ # &'(/$'& .?S) #%. #367(/C3.'+5r p=ω q,( ! n+%F "1/n+5#r 5B$ # (,67'1+5q/C3&2)'.'( !-q/C#3rn/.'("C|q,+3+58 6# n5B( '1+%n (( #3$&. +%C5$'+5q/&C5(onp5$'+@+5)'.'(oF !-(on(,$'$+54,+V $ # &'(/$'& . q = q(E ) *
= # q = q + q˜ $
o
s
0
s
0
!-(on-#3rVq,+5+3C3r5(/C3q/C3r#"=VHA"=V )'+,!-q"C#3$'+5r#'"r4 ? K9 )'+n/"=&'&'6T* L
J|+5q('1+%n p'"
d2 q˜ d˜ q + R + U (q0 + q˜) = Es + V0 cos pt 2 dt dt
6 #n+A 67+ 0 $+. #%/n + 0f&C3p U (q) r.9.1! Q(,8tn+5. #)'+q"C5(/)'(,$.6 q˜A +54/. #3$'&&'&'r%F 8 &q,p n&'$'(/8'$5B67&Uq"n-#%4o#3(,675B67&+* TJ|+,!-q/C#%r5&'r B/C3+hr#5B. #%02(/$'&'(mr U (q) ' U (q ) + q˜ "t. #3r5$(,$'&'(x&ar5q,)'+367$'&'r'A & C5+ A')'+%n/"=&'&6 "t. #3r5$'(/$'&'( ! n/.)'(/.'(,67(/$'$'+58:q,+5q"C#3r%n/.tF U (q ) = E VH (,8a #3 . .t!-# * q˜
? I d˜ q 1 d q˜ q˜ = V cos pt +R + L q˜
0
o
dU (q0 ) dq
s
2
!-(/q,pgr5r5(
!-(,$'+g+3+3 $ # &'(/$'&'(!*
dt2
dt
Cd = dq/dU
Cd
0
F !-&% *% (,.'(/$ ; & #n p5$ #!.h(,67'1+5q"C5p F )'(/.'( 5+,!#
$ '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!## VH (,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$'(/$'&' +,!-&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 #5st(,$'(/. #%C5+3. 4o#3.'67+5$'&&'(,q-'t+34,+g$ #3)'. . 02(,$'&?.A.'##5+3C##V H &'8 r!-& #3) # +5$'(& #3q/C3+3Cg+3C G !-+f ' s ; A q & #%q/C5+%C5+567(/.'+56 &a$'&('t+3+567$5B6 r565+,!-+56 >|q,qon(o!/"5(,675B8 '1+5$C!" .mq r56 +,!-$5B6 #"=%*(,.$5B6 ' #%q('t# !-+56c&_q/&'q/C3(,67+58 1' +5676L"1C# ; &&A '1+ B-% %*& ; &'(,$C )',( .(o!# &'&:r56 +,!-$'+34,+ #"=%x/( .'$'+54/+'t#3q-' # !#C 3G *v # )'(/.'(o!$'(,8) #%$'("n& "3q/C#3$+5r''&. #%q,)'+%n + 0 (,$5cq,("C5(/r5+58C#"t67%n (,.q@&$t!-&'t#%C3+5.'+56 r' n/V&'(oF $'&. A .'(,4 " n/.C3+5.&V&'$t!-&'t#%C3+5.$ #%)'. .02(/$'&.fq,67(-H (,$'&?. E A C5.'( )+3 & ; &'+5$'$5B8)(,.'(-' nV&'#F C5(onp -$ (/0f&'6 %A 4/$'(/o!#2!'n/.a)'+,! ' nV&'(,$'&?._r5+%n p C367(/C3. # &a+5q ; &tn'n+54/. ##%x#A )'(/.'((' n/V& #%C3("np ( +np C567(/C3. %A . "=&''2 & "t)'. #3rn(,$'?& . #367) n& #C " !-+58 3 & & #3q/C3+3C5+38hq/&'4,$'#n-# 4/(,$'(/. #%C5+3. #A 8A't#n-# & #3q"C5+3C3+567(,.'# HEn#+ '_)'&C#3$& .DA ''.'+567( $'(,+ >5+ !&'76 56 ! /n . . ## 5+%C 5G"5q"C#3$'+5#r ''&g$ #3)' . .0 (,$'&'8 A r 5B. !# #%C 5Br5#3(/C|$'#3)' . .02(/$'&'(FEG !H> I 'D >5E >: .EA JLKM '63B 5'6>NDOP I!34 F'67!Q "R 5%9 S!?K!N7#>T@ A !*UV'6W+ !V> X TJ@%+ (?'6WYJ+ ZE *E E 0F@ >?'6 ( F'6[ '\ Y E AH ]NG^JJ_ >_=* `a '6 W+b =?>F dcfe) s
I
$ '2$ &4 ?= > &cfe) + 0 E & .* >@ >Q@ >T f0 ,EG ' F[ /J%9 ?'6WYJ N A & ¯ ) = C /C ¯ FE)'.'&'r3(o!-(/$'$ #!.a(,67'1+5q"C5p9 J|+5q"C5.'+5& C5p 4/. ## %*& ''& f (E ) C (E ) C(E o
s
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p ') )'.'+'1q/&'67&'.'+5r5#%C5pc #3r3&'q,&'67+3q/C5p C(E %O"t$' ; &(,8 p A ¯ ) (U − E ) / (U − E ) +5).'(o!-(on&'rV $ # &'(/$'&.a$'(,& r5(/q/C5$+54,+ '1+B-% %x& ; &'(,$ C# U ) J T ' d ? / ?E '6 Y.5 FEG +.? ? 1 /J%+J,?'6WYJQ T [ T C(E ?E ,.Y H[4.? ? 0 2T ' >FE >T, .? ?E JS.*'6 J ¯ )
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K
s
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I5G
$ '2$ &4
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"
'$ ("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 ##' nV& ##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
)'.&%*&'1q,&'.+5r #3$'$+56 $'#F β−γ &'(/$'&'& |C3+:+3)'.'(o!("/n .(/C_4/. #3$'& ; " #3q,q/(,8'$+5r:).'& C .0 (,$'& .Sq,+3-q "9H (/q/C5!r "?V H ?& α>0 #%C%C5. !# 'C5+5.+5r 3
I
'
! # #
J &/q(' (5Br #V:q'q75 "! 5'$##%56&.V5q .r%#r% J #5q'%(?8)*!#%+! '%&,q-',#-#.#3q ; nn/./#% .'r%n.3 q .01 %1 % &/, q('/(21 43 ./'q5/6.V H)(q .7 n8/#%(+ J,"5q'5p97 n8 :1 !.;% ', : ./3 q .6 &. l !#5-? #&/q('$#%L"@6 #'$#%"qO&% q%#%#%( ω r2q"n/"=& A! '% &/, q('/? #5q ; n'n/.;%#%/#5rF& 3q##"B/.B! % C ( 5Br# V EX*QGM9>.C9>.-K B:*/IT*\EXITA@ n8lEF*\G=9>.CV 9>.CKi0/. , W S%D&@1;=B:AXR+Emq> ! n8/! "F78#%( ν/l N* 3r5Jk6A:9UST@-* B:*\IT*\EXITA n8 *C3D5Jk/ 7 n.:/6.V )H (q .07 n8/5O* K
3 , ,3! ,#$ * $! x¨ +
g x=0 l0 (1 + ν cos ωp t)
%# & ', #' ω 1!/?7 !#n -? !.*1 n8#5q'5p< ! n8/!"67 5 #', &' ! #q 3r%V q# q/?7 /+ &/ ! '% * ν 1 ; /n# H %O",' ; V r) .,7 Q ( n8#% C!# r%n8/ ? " n. #%- &/'r#8)q,p~n8(5J &tn8 4/n8#FH>.+* ν Y
g l0
2 0
1 1+ν cos ωp t
#7-q4 3rn/.. J
?
1 = 1 − ν cos ωp t + ... 1 + ν cos ωp t
r ? ?!#%n"?&
#
; 3r% "
x¨ + ω02 (1 − ν cos ωp t)x = 0.
5# q ; n'n/.;%#% ? ! '%A./3q . r%#BrC!/#-? #%&/,q('$#%L" 6 #'$#%"#-> 5J( r%,7 ", 3r%/.0 4>5Br '5q .y ", 3r/ `2 45p%o} R-#B#FH.#>6 !'q 5pVrfr,7
x¨ + ω 2 (t)x = 0,
MK9
- 7 ω(t) = ω p1 − ν cos ω t 3 6 3r%'q .?H> !..# r%q#5q3r%/ !. & 3q#4 )#5q ; n'n/.;%#% ? R7q"n8 r ",3 r/W MK9 F 7 #% F7 ; ##%( %A##* Y
2
dx dt
,#,7 q6 #3 !/#( '
x¨ + 2αx˙ + ω 2 (t)x = 0
8 7'$#5q3p .'5q . !# F !',#" C(t) = C (1 + νc(t)) - 7 C 3fq=7 6 &/ 3 '$#B-% % ; / - n/"=%/ 9#67/" n. ; / ! / c(t) 3 %O",' ; ?. JF 7 #VH) !. F #'$#% ν /.)! '% / !/& max|c(t)| = 1?6%#G!/ 1 n8# '$#B(% % ; // #7" n/. ; / / q# 5q'5 r%#%( &3 q# 5 ',#/#", B# H #5!=7-q4 %r%/5p rVr,7 #* R7q"n
0
ω(t) = ω0
p 1 − νc(t).
KK
0
3 , ,3! ,#$ * $!
'5p@! ω(t) .'5q . !#7 &/,q('/mq !#7 #% T * c(t + T ) = c(t) z #'0! B#%2 #"67 'xr%,q'%q( .Vq'q'%1 6 3r%'q#5q'%.# ! n8/!"F75 F! ',# " y ,7 }
q ω(t) = ω0 1 − νSign(cos ωp t)
3 %O";' ; .A6 !'; ;F!/1 #VH) !.C &/ +1 ! !/#%n8#FH.;%"n p%#%2 --",/% x −1 31!9#% ; %onp%#%TB'%#% qon/"=& *q#3q5r M. & 3q'%#4 P'$#%/#", >!#7 &/,q('/ !/1 ' -7 r &6 & .+* ω = ω √1 + ν ω = ω √1 − ν F !/#7 ! '%&'3 q('$#%-2# r%#?7 '( q'5 r%? . T = ?*%- &/x r93 r%/#%-#)!/%# n85# r%/E!#7 2 r%#?7 ( q5 r . &3 q#4 A3# q ; nn/./#% 5# q4 '5 q . !5# q'% # ./#%+( B#% q"n/"=& '7 n/.P', HB3 [0; T /2] [T /2; T ] 7 #%:( &3 q !#7 9#FH./#@ (/ %# &/#%5 n/% &/, q('$#%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'$#%/! "; Sign(x)
1
p
p
p
0
2
2π ωp
p
J
2 x¨ + 2αx˙ + ω1,2 x=0
(,7 ':q'q ," 3r%( !'r#%-#B!#% .,7'; ?* K
0
3 , ,3! ,#$ * $!
x˙ = y
?
2 y˙ = −2αy − ω1,2 x
J k1,2
#/ r%#7.+!#7-q'4 #5r#'" x ∼ exp kt #%!/?7 on8'r &/. !/#', F "n/. B('1q!/#%;5 !%# n"?&/ # #H (ar%,7 -8 .97 n/.q'q75 * ' −α ± jω 1,2
x = exp(−αt)[A exp(jω1,2 t) + B exp(−jω1,2 t)] y = exp(−αt)[A(jω1,2 − α) exp(jω1,2 t) − B(jω1,2 + α) exp(−jω1 t)
) #0/-8)0F 3r%q/ # 7-r#" !#/ r%#np%56*!#5q'%# ./56 A B 7'$##%5J #%--"$ 5 3 p_r5J H) 5 &/'9 & np5J 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...
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