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June  2009, 11(4): 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system

Daniel Wilczak 1,

1. 

Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, Norway

Received  July 2008 Revised  December 2008 Published  April 2009

The four dimensional Rössler system is investigated. For this system the Poincaré map exhibits chaotic dynamics with two expanding directions and one strongly contracting direction. It is shown that the 16th iterate of this Poincaré map has a nontrivial invariant set on which it is semiconjugated to the full shift on two symbols. Moreover, it is proven that there exist infinitely many homoclinic and heteroclinic solutions connecting periodic orbits of period two and four, respectively. The proof utilizes the method of covering relations with smooth tools (cone conditions).
   The proof is computer assisted - interval arithmetic is used to obtain bounds of the Poincaré map and its derivative.
Keywords: rigorous integration of variations for ODEs., homoclinic and heteroclinic solutions, symbolic dynamics.
Mathematics Subject Classification: Primary: 34C37, 37B10; Secondary: 37N3.
Citation: Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039
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