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I am trying to plot Driven Double Pendulum with a control Parameter "Gamma". My understanding is that as this gamma approaches a critical value, the pendulum is pushed towards non-linear regime, and you eventually get chaotic behavior. So, for each value of gamma, the curve should look different. But for each value of gamma, I get the same curve with different amplitude. Even after I cross the critical value.

I made the equation linear (Sin[theta]=theta) and I observed the same effect. How is a linear and nonlinear equation behaving the same way? Where is the error?

ω = 5*π
m = 1
g = 9.81;
L = 0.5;
b = 1;
β = b/m;
Subscript[ω, o] = √(g/L);
solNonLinear = 
  ParametricNDSolve[{ϕ''[t] == -2 β ϕ'[t] - 
    Subscript[ω, 
    o]^2 Sin[ϕ[t]] + γ Subscript[ω, 
    o]^2 Cos[ω t], ϕ[0] == 0, ϕ'[0] == 
   0}, ϕ, {t, 0, 20}, {γ}];
Plot[Evaluate[
Table[ϕ[γ][t] /. solNonLinear, {γ, .9, 
1.4, .125}]], {t, 0, 15}, PlotRange -> All]
(*When I change Sin[ϕ] = ϕ*, I still get the same graphs)

enter image description here

enter image description here

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  • $\begingroup$ Parameter m is not defined. $\endgroup$
    – Alex Trounev
    Jan 15, 2019 at 12:33
  • $\begingroup$ @AlexTrounev I edited the code with the value of m. $\endgroup$
    – Rumman
    Jan 15, 2019 at 14:16
  • $\begingroup$ I noticed you have not received a welcome to Mathematica.SE! 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. And please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$
    – Michael E2
    Jul 28, 2019 at 22:50

1 Answer 1

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Your simulation to be chaotic needs a lower $\omega$ as for instance $\omega = \frac{\pi}{2}$. Follows the results in this case

enter image description here

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  • $\begingroup$ Thanks a lot! Would you happen to know why lower value of omega causes chaotic behavior and not higher value of omega? $\endgroup$
    – Rumman
    Jan 15, 2019 at 14:19
  • $\begingroup$ @RummanPlease. Have a look at this publication. math.colostate.edu//~shipman/47/volume12009/bevivino.pdf $\endgroup$
    – Cesareo
    Jan 15, 2019 at 16:22
  • $\begingroup$ That link was very helpful! thanks! $\endgroup$
    – Rumman
    Jan 16, 2019 at 1:04

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