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I'm trying to plot the homoclinic tangle that can be observed following the evolution of the unstable and stable manifolds of the standard map. The map I am using is defined as:$$ \begin{cases}p_{n+1}=p_{n}+k\cdot \sin\big(\theta_{n}\big) \\ \theta_{n+1}=\big(\theta_{n}+p_{n+1}\big)\mod(2\pi) \end{cases} $$

Generating a given number of initial conditions in a neighbourhood of the point $(\pi,0)$ I am able to follow the evolution of the stable manifold (I suppose that it is the stable one, given the fact that I iterate the map forward in time),as it approximates the starting point, but eventually explodes as it became closer to it.

I am instead unable to plot the evolution backward in time of the set of initial conditions, which should be controlled by the inverse of the map. I tried to calculate the inverse of the map by insulating the term $p_n$ and $\theta_n$ in the previous definition, but I found the same behavior of the direct map. Where I am wrong? I was unable to find a definition for the inverse map online, and if it is possible to use the direct form of the map for the backward evolution.

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A possible error in the calculation of the inverse map is to miss that $\theta_{n+1}$ is being defined not in terms of $p_{n}$, but of $p_{n+1}$: the latter should therefore first be expanded in terms of $\theta_{n}$ and $p_n$, before an attempt to invert the map.

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