期刊
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
卷 21, 期 2, 页码 551-563出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S021812741102857X
关键词
Periodic orbit; symbolic dynamics; interval arithmetic; Lorenz system
资金
- AGH University of Science and Technology [11.11.120.611]
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.
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