6
$\begingroup$

There are dynamical systems which have regions of phase space that are both chaotic and integrable, e.g. small perturbations of integrable systems as in KAM theory. Are there any tools for bounding the proportion of phase space which is chaotic or integrable (say, in terms of "local" properties like lyapunov exponents?)

Improve this question
$\endgroup$

1 Answer 1

Reset to default
6
$\begingroup$

You will want to study the Poincaré map, by numerically integrating the equations of motion on a grid of starting positions. The regular regions show up in the Poincaré section as regions of zero Lyapunov exponent.

As a worked out example, see Regular and chaotic phase space fraction in the double pendulum. The dark blue regions are the regular regions (MLE = maximal Lyapunov exponent $\lambda=0$).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .