# ParametricNDSolve[] for Double Damped Pendulum

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, ϕ' ==
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)  • Parameter m is not defined. – Alex Trounev Jan 15 at 12:33
• @AlexTrounev I edited the code with the value of m. – Rumman Jan 15 at 14:16

## 1 Answer

Your simulation to be chaotic needs a lower $$\omega$$ as for instance $$\omega = \frac{\pi}{2}$$. Follows the results in this case • Thanks a lot! Would you happen to know why lower value of omega causes chaotic behavior and not higher value of omega? – Rumman Jan 15 at 14:19
• @RummanPlease. Have a look at this publication. math.colostate.edu//~shipman/47/volume12009/bevivino.pdf – Cesareo Jan 15 at 16:22
• That link was very helpful! thanks! – Rumman Jan 16 at 1:04