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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 at 12:33
  • $\begingroup$ @AlexTrounev I edited the code with the value of m. $\endgroup$ – Rumman Jan 15 at 14:16
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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 at 14:19
  • $\begingroup$ @RummanPlease. Have a look at this publication. math.colostate.edu//~shipman/47/volume12009/bevivino.pdf $\endgroup$ – Cesareo Jan 15 at 16:22
  • $\begingroup$ That link was very helpful! thanks! $\endgroup$ – Rumman Jan 16 at 1:04

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