CoCalc -- Collaborative Calculation in the Cloud
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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

_=var('J,R,T')
c = 14
jprime = R+0.1*J
rprime = -J-T
tprime = 0.1-c*T+R*T

time = srange(0,100,0.1)
sol = desolve_odeint([jprime,rprime,tprime],dvars = [J,R,T],times = time,ics = [5,5,1])
list_plot(zip(time,sol[:,0]),plotjoined=True,axes_labels=['time','J'])
list_plot(zip(time,sol[:,1]),plotjoined=True,axes_labels=['time','R'])
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!

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

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])
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)
list_plot(sol1,plotjoined=True,color='blue')+list_plot(sol2,plotjoined=True,color='red')
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)