(ChaosBook.org blog, chapter Fixed points, and how to get them) — Predrag Cvitanovic 2010-04-09

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2010-05-24 François Mauger (mauger snail cpt point univ-mrspint point fr) For understanding the atomic systems we are dealing with, with T. Uzer and C. Chandre, we have been looking for periodic orbits. We have been using probabilistic metaheuristic algorithm such as the Simulated Annealing to do that. Are you aware of methods based on stochastic algorithms to find periodic orbits for Hamiltonian systems or other deterministic flows?

2010-06-10 Predrag to François We have not used any stochastic algorithms to find periodic orbits. We use methods described in chapters 13 Fixed points, and how to get them and 29 Relaxation for cyclists, and sometimes we use nearest recurrences for initial guesses, as for example in http://chaosbook.org/tutorials/recurr.html.

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Read Morita, Y. and Fujiwara, N. and Kobayashi, M. U. and Mizuguchi, T., “Scytale decodes chaos: A method for estimating unstable symmetric solutions”, Chaos 20, 013126 (2010). They say:

“A method for estimating a period of unstable periodic solutions is suggested in continuous dissipative chaotic dynamical systems. The measurement of a minimum distance between a reference state and an image of transformation of it exhibits a characteristic structure of the system, and the local minima of the structure give candidates of period and state of corresponding symmetric solutions. Appropriate periods and initial states for the Newton method are chosen efficiently by setting a threshold to the range of the minimum distance and the period.”

2010-03-12 PC I have studied the paper - see repository siminos/blog/dailBlog.tex for my notes.

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