I was simulating a chaotic system (driven damped anharmonic oscillator) on my pc:
$$ \begin{align} \dfrac {dx}{dt} &= v \\ \dfrac {dv}{dt} &= -x-x^3-0.3v+F \cos 2t \\ \end{align} $$
For various parameters of F I could see the period doubling and a very similar behaviour to the logistic map (even though not exactly). Now I tried to make a phase portrait and get a lorentz-section (like in Strogaz: ‚nonlinear dynamics‘ in chapter 10.6; red points in phase portrait). I should get a nearly unimodal map, wich represents the universality (https://vd.univie.ac.at/fileadmin/user_upload/p_vd/VDS_Mathematics/Minicourse_Anusic_neu.pdf). But my map is really not unimodal. Nevertheless I thought this could happen, because it is a higherdimensional differential equation. But now also my calculated delta feigenbaumkonstant converges to 4 instead of 4.66 like it should. Now I am wondering: Am I doing something wrong or is this really because it’s a more complex system or because I have wrong initial conditions or constants?
Edit: I don’t expect anyone to find the mistake (with the given informations it would not even be possible). I just thought someone with a bit more experience could tell me what this means and if this system should behave this way.
I hope its ODEs help you, first for $x(t)$:
$$ x'' = (x')' = v' = -x-x^3-0.3v+F\cos (2t) = -x-x^3-0.3x'+F\cos (2t) \Longrightarrow $$
$$ \Longrightarrow x'' = -x-x^3-0.3x'+F\cos (2t) \Longrightarrow x''+0.3x'+x+x^3-F\cos (2t)=0 $$
Alghouth $v(t)$'s ODE can be found with similar way, but its ODE will be of degree $4$ and hard to solve. So maybe finding $x(t)$ and substituting it will be better.