VALIDATED STUDY OF THE EXISTENCE OF SHORT CYCLES FOR CHAOTIC SYSTEMS USING SYMBOLIC DYNAMICS AND INTERVAL TOOLS
2011 (English)In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 21, no 2, 551-563 p.Article in journal (Refereed) Published
We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p <= 16 of the Poincare map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincare map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences.
Place, publisher, year, edition, pages
2011. Vol. 21, no 2, 551-563 p.
Periodic orbit, symbolic dynamics, interval arithmetic, Lorenz system
Engineering and Technology
IdentifiersURN: urn:nbn:se:uu:diva-152902DOI: 10.1142/S021812741102857XISI: 000289468300013OAI: oai:DiVA.org:uu-152902DiVA: diva2:414480