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GLf3ffMMM___!@3f3f"ʚ;18ʚ;g42d2dH7F0ppp@ <4!d!d@0.<4dddd@0. <4BdBd(A000___PPT10 ___PPT9An undamped Pendulum undergoes simple harmonic motion if the amplitude is very small.? %y 'rWhy?  _ Equilibria of a dynamical system!!$  d.Fixed points of First-order autonomous systems//  / L   h Vector fields    mPhase flow in 1-D phase space :  !$jFirst-order autonomous systems   n Exercise 1:   l Exercise 2:   !oSecond-order autonomous systems   <}Phase space portrait >~#Example 1: A damped simple pendulum$$  *uA damped simple pendulum 3y,v2Phase space portrait of the damped simple pendulum33@  BVan der Pol oscillator , F@Phase space portrait with  = 1.8!bb,HThe Limit Cycle  KHigh order autonomous systems  &qHomework      0` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> PH(    60[7 P 7 X Click to edit Master title style!!   0\1  7 RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S   0b7 `` 2 d*b    0h7 `  7 f*b    0|  5 %* H  Zp@ ??p P`v  It s either on top of a hill or at the bottom of the track. To find out which, push it (perturb it), and see if it comes back.    S 0e0e B CDEpF0A@ A>8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?\ 8p`H`8@@ `@8P`H0@PpP @ 0@         0  @ 2   `!?? x8 2   `!?? (@ H  0޽h ? ̙33___PPT10i.ћnN+D=' = @B +d 0 {s  (  x  c $0@  @  0 NA4  ? ?p  c   @  0 ZA5  ? ?S" pN   @(  6t@0 6  By a fixed point we mean that x doesn t change as time increases, i.e.:HH!   6`@  j6So, to find fixed points just solve above equation. 77!  0 ZA6 ? ?S"  `  @   f@ ??`p v ^autonomous systems    f@ ??`P  ononautonomous systems"   H  0޽h ? ̙33___PPT10i.Ɖ+D=' = @B + 0  (    6@C"?:_@ &xE.g.: dx/dt = Ax(1  x) with A=6. What are the fixed points?p=!%faefA!2  6@ 0:Set: dx/dt = 0 ie: 6x(1-x) = 0 so either x = 0 or x = 1" 2! %f! %f! ff!,(m  6 } [Use difference equation: x(t+h) = x(t) +h dx/dt from various different initial values of xJ\ 2f+f>-   6zL  DTherefore 2 fixed points, and how about the stability? Perturb the points and see what happens under the system dynamics& 2| 2, PPGH  0޽h ? ̙33___PPT10i.yc+D=' = @B + 0 P \(  r  S  p   X  0A.?e   ZT ??` _ (t, x(t))  B  3 rDԔ??  Z@ ??P`p z(t+h, x(t)+hdx/dt),    6ٹ P`  E}So x=0 is unstable and x=1 seems to be stable. Change the value of parameter A to see the influence of parameters on systems~!%f! %PP!%f ! %PP(a%AaH  0޽h ? ̙33___PPT10i.rPϘ+D=' = @B +: 0 QI0 $(  $x $ c $1   f $ s *A E? ?0  EB  $  fD!Ԕ?? pp`B  $  fD!Ԕ?? `  $ 67  g hdx/dt ( #!v $ NA+ ??` 2  $  `GL f??C"?0 @B  $  fD3f>??@00 $ 6tu3f pP X1-D phase space , 2 $  `??0P $0 TA H? ? H uH $ 0޽h ? ̙33___PPT10i.E+D=' = @B +i 0 xp (  r  S xAPp     0 TA8 '? ?   ' f  s *A9 6? ?m  6   6C0P0 P h1. Find fixed points: 4! %!!   6I Pv  2. Identify stability using phase flow as above using the gradient at the fixed points Ba!a*A!  6P@ P`  E &!a  ZT ?? p ` xk is stable xk is unstableJ! J !$   0 TA7 8? ?|  8 H  0޽h ? ̙33___PPT10i.u38+D=' = @B +E 0 PHP ,(  ,x , c $f      ,0 TA: B? ?@p0  B  , NA< ?? f  @c0c0v , NA= ??   , 6|ip  f1. Fixed points: x=0 2!!!  , 6o  I2. Stable or not? !   , 6t  If a<0, d(ax)/dx<0. stable L!!!!   ,0 TA? @? ?P & @  , 6y@ `  If a>0, d(ax)/dx>0. unstable 2!!!!H , 0޽h ? ̙33___PPT10u..+D=' = @B +d 0 {s (  r  S Ȅ`0   v  NA> ??Mv 0 TA; E? ?0P  E  0 TA@ H? ?  H   6l 0 n 1. Fixed points: x=0, +1, -1 2!!!!  6`P I2. Stable or not? !H  0޽h ? ̙33___PPT10i.!+D=' = @B +N 0 e]< (  <x < c $   l < 0AB M?PP? m M < <@ v&Suppose we have the following system: ''!&  < 6Ԝ  u%1. Find the fixed points by setting: &&!%  <0 TAD P?PP ?   P D  < 6L 0 ^2. Identify stability by Jacobian matrix (will not talk here) or 2-D phase space portrait6_F!%! !&= H < 0޽h ? ̙33___PPT10i.+D=' = @B +  0  1(  x  c $lP    _  Zp ??  sThe 2-D space of possible initial conditions in which each solution follows a trajectory given by the vector field"t6>6= 7  6   +(x(t+h),y(t+h)) =(x(t)+hdx/dt, y(t)+hdy/dt)(,aAZ          dB  <DPPov y X2  03fa r   B V S (x(t),y(t))  !  X2  03fcRB  s *DzRB  s *DU  B t  Ix!    BP `4p  Iy!   0 TAB I?PP ?c`hP I uH  0޽h ? ̙33___PPT10i.`o:+D=' u= @B + 0 (  r  S 7`P`0    H  0޽h ? ̙33___PPT10i.П+D=' u= @B + 0 \v(  \P \ C (AIpenillLp x \ c $`0   f  \ s *A %? ?G  %H \ 0޽h ? ̙33___PPT10i. d'+D=' u= @B + 0  x(  x x  f@ ??{  Second-order autonomous systems"  2 x Z3f?? x  f@ ??0  k/First-order autonomous system in two variables  0.f x s *AG ? ?    x0 TA$ ? ?   @< x 6@p  TFind the fixed points: (0, 0), (, 0), & & 4+`@`*!  x c \A 6Simple_Pendulum_Oscillatorz  c0 x0 TA &? ?P  & @H x 0޽h ? ̙33___PPT10i.A(+D=' u= @B + 0  $`<(  `V  ` C .AOpendulum_3`@ V !` C .APpendulum_2 W "` B     #`0 TAN '? ?  '  $`  f! ??0`* l(Created by AI Lehnen)fH ` 0޽h ? ̙33___PPT10i.Ɛ+D=' u= @B + 0 8(    6H  &Example 2: the Van der Pol oscillator $'% 0, H  0޽h ? ̙33___PPT10i.нɑ+D=' u= @B +E 0 \T  (  r  S 8p    0 TA ? ?P   B   `D!Ԕ??pjq   6t  p NThe second term is not a constant like that in the damped simple pendulum with(O/ f   s *AQ ? ? @     f ??p   Second-order autonomous systems"  2   Z3f?? `P    f| ??   y/first-order autonomous system in two variables .0.f  s *AR ? ?  f  s *AS ? ? P  H  0޽h ? ̙33___PPT10i.`+D=' u= @B + 0 @ (    6[ 0 -Find the fixed points: (0, 0). Stable or not?4.`@`-  0 TAR ? ? (  [v  NAT ?? 0P   Z??[   B [     [ H  0޽h ? ̙33___PPT10i.:+D=' u= @B + 0 `(  r  S |  p 0  [   c A(:VanDerPolOscillatorFile:VanDerPolOscillator.png0  c0H  0޽h ? ̙33___PPT10i. +D=' u= @B +u 0  (  r  S [0  [   6[  . f  s *AU ?PP?`  >  6 [ pz Suppose we have the following system (imagine a neural network & ):CC#B     f[ ??m .We basically need a n-dimensional phase space.:/!H  0޽h ? ̙33___PPT10i.,<+D=' u= @B + 0   H(  H H 6y[P>  h" Find fixed points of a simple system Classify fixed points as stable/unstable. Especially, use graphical methods (vector field plots in phase space) to analyze and elucidate the behaviour of simple systems):a! !  !  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