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I am used to seeing bi-dimensional Poincaré maps, as the ones shown in this post:

Poincaré maps and interpretation

In that example, one manages to draw a bi-dimensional map because the number of degrees of freedom is quite limited.

My question is: does it make sense to draw bi-dimensional Poincaré maps also for higher-dimensional dynamical systems (e.g. a 5-body problems)? If yes, which is a good recipe to decide the section plane?

Or, instead, the Poincaré map is of higher dimensionality?

stafusa
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AndreaPaco
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1 Answers1

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A Poincaré map can be used to obtain a representation of the original, $N$-dimensional dynamics in any number of dimensions $n<N$.

That said, a 2-D map of a system with 4 or more dimensions is seldom used, because of the difficulty in interpreting the resulting figures -- unless, and that's an important exception, the original system trajectories are actually contained in a lower dimensional manifold (i.e., the system effectively has fewer dimensions). Also, if $n>3$, the Poincaré map of course won't be of much help for visualization purposes.

As for how to define the section, the more you know about the system (symmetries, bounds, invariant sets, etc.), the better the guess you can make. Apart from that, it's usually a matter of trial and error, where experience and luck will be helpful.

stafusa
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  • Thank you very much for your very good answer. Just one thing that I would like you to clarify: when you say "a 2-D map of a system with 4 or more dimensions is seldom used, because of the difficulty in interpreting the resulting figures", what do you mean with "because of the difficulty in interpreting"? Does it mean that it's not easy to distinguish between chaotic behavior and regular trajectories? – AndreaPaco Apr 10 '18 at 23:47
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    @AndreaPaco, Also, but I meant it more generally - after all there are better ways than plots to distinguish between chaotic and regular orbits (e.g., the Lyapunov exponent). The problem is that, even for regular orbits, say a quasiperiodic one, you typically don't get a simple curve in the Poincaré plot, but something that is either too sparse to make sense, or somewhat confusing, like a projection of points on a 3D surface (like some of the plots you find in, e.g., here). – stafusa Apr 11 '18 at 00:06