SharedTA Sandbox / Hao's Sandbox / Lab10_slides / Lab10_slides.ipynbOpen in CoCalc
Author: HAO LEE
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Chaos

We have learned Two Stable System Behaviors

  1. Converge to Equilibrium
  2. Stable Limit Cycle

Is there any other type?

Chaos System{Bounded OutputNot Periodic\begin{cases} \text{Bounded Output}\\ \text{Not Periodic} \end{cases}

Romeo Juliet System (Continuous Time Example)

R=R+0.1J R' = R + 0.1J

J=JT J' = -J-T

T=0.1cT+RT T' = 0.1-cT+RT

Let c=14,R(0)=5,J(0)=5,T(0)=1c = 14,R(0)=5, J(0)=5, T(0)=1

In [1]:
_=var('J,R,T') c = 14 jprime = R+0.1*J rprime = -J-T tprime = 0.1-c*T+R*T
In [2]:
time = srange(0,100,0.1) sol = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [5,5,1])
In [85]:
list_plot(zip(time,sol[:,0]),plotjoined=True,axes_labels=['time','J'])
In [86]:
list_plot(zip(time,sol[:,1]),plotjoined=True,axes_labels=['time','R'])
In [87]:
list_plot(zip(time,sol[:,2]),plotjoined=True,axes_labels=['time','T'])

It do not converge to any equilibrium point, nor stable limit cycle

However, it is still bounded!

In [62]:
list_plot(sol,plotjoined=True)

Initial Values Matters a Lot

R=R+0.1J R' = R + 0.1J

J=JT J' = -J-T

T=0.1cT+RT T' = 0.1-cT+RT

In [69]:
time = srange(0,500,0.1) sol1 = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [5,5,1]) sol2 = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [5.1,5.1,1.1])
In [76]:
fig1 = list_plot(zip(time[0:1000],sol1[0:1000,0]),plotjoined=True)+list_plot(zip(time[0:1000],sol2[0:1000,0]),plotjoined=True,color='red') fig2 = list_plot(zip(time[4000:-1],sol1[4000:-1,0]),plotjoined=True)+list_plot(zip(time[4000:-1],sol2[4000:-1,0]),plotjoined=True,color='red') show(fig1) show(fig2)
In [78]:
list_plot(sol1,plotjoined=True,color='blue')+list_plot(sol2,plotjoined=True,color='red')
In [115]:
time = srange(0,100,0.1) sol = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [5,8,2]) fig1 = list_plot(sol,plotjoined=True) sol = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [1,4,3]) fig2 = list_plot(sol,plotjoined=True,color='red') sol = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [1.3,-1.6,50]) fig3 = list_plot(sol,plotjoined=True,color='green') show(fig1+fig2+fig3)