Periodic windows distribution resulting from homoclinic bifurcations in the two-parameter space
Abstract
Periodic solution parameters, in chaotic dynamical systems, form periodic windows with characteristic distribution in two-parameter spaces. Recently, general properties of this organization have been reported, but a theoretical explanation for that remains unknown. Here, for the first time we associate the distribution of these periodic windows with scaling laws based in fundamental dynamic properties. For the R\"ossler system, we present a new scenery of periodic windows composed by multiple spirals, continuously connected, converging to different points along of a homoclinic bifurcation set. We show that the bi-dimensional distribution of these periodic windows unexpectedly follows scales given by the one-parameter homoclinic theory. Our result is a strong evidence that, close to homoclinic bifurcations, periodic windows are aligned in the two-parameter space.
Cite
@article{arxiv.1012.2241,
title = {Periodic windows distribution resulting from homoclinic bifurcations in the two-parameter space},
author = {Rene Orlando Medrano-T. and Iberê Luis Caldas},
journal= {arXiv preprint arXiv:1012.2241},
year = {2010}
}
Comments
The results of this work were presented in the conferences Complex Systems 2008, as an invited speaker, (http://www.lac.inpe.br/WSACS/programme.jsp) and Dynamics Days South America 2010 (http://urlib.net/sid.inpe.br/mtc-m19@80/2010/08.11.01.46)