Topological method for symmetric periodic orbits for maps with a reversing symmetry
Daniel Wilczak Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - A 2007, 17(3): 629-652 doi: 10.3934/dcds.2007.17.629
We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.
keywords: symmetric periodic orbits reversible systems computer assisted proofs.
Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system
Daniel Wilczak
Discrete & Continuous Dynamical Systems - B 2009, 11(4): 1039-1055 doi: 10.3934/dcdsb.2009.11.1039
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

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