Conley Index of Poincaré Maps in Isolating Segments
نویسنده
چکیده
We calculate the discrete-time Conley index of the Poincaré map of a time-periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment. As an application, we present results on the existence of bounded solutions of some planar equations.
منابع مشابه
A cup product pairing and time-duality for discrete dynamical systems
With each isolating block for a homeomorphism of an orientable manifold we associate a cup product pairing. Up to an isomorphism, this is a pairing of the discrete Conley indices of the homeomorphism and its inverse on the isolating block. We show that this pairing is nondegenerate. As a corollary we prove a time duality result for the discrete Conley index. 0. Introduction The observation that...
متن کاملConley Index for Discrete Multivalued Dynamical Systems
The deenitions of isolating block, index pair, and the Conley index, together with the proof of homotopy and additivity properties of the index are generalized for discrete multivalued dynamical systems. That generalization provides a theoretical background of numerical computation used by Mischaikow and Mrozek in their computer assisted proof of chaos in the Lorenz equations, where nitely repr...
متن کاملA Combinatorial Procedure for Finding Isolating Neighbourhoods and Index Pairs
Synopsis. A purely combinatorial procedure for nding an isolating neighbourhood and an index pair contained in a given set being a nite union of cubes in R s is presented. It is applied for a computer-assisted computation of the Conley index of an isolated invariant subset of the Henon attractor. As a corollary, it is shown that the Henon attractor contains periodic orbits of all principal peri...
متن کاملComputing specific isolating neighborhoods
A major challenge in applying the Conley index theory to a given dynamical system is to find isolating neighborhoods of the invariant sets of interest. In this paper we describe a computational method which allows for the efficient construction of isolating neighborhoods. As an example we compute an isolating neighborhood of a heteroclinic orbit connecting a fixed point to a period two point in...
متن کاملA Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems
We present a numerical method to prove certain statements about the global dynamics of infinite dimensional maps. The method combines set-oriented numerical tools for the computation of invariant sets and isolating neighborhoods, the Conley index theory and analytic considerations. It not only allows for the detection of a certain dynamical behaviour, but also for a precise computation of the c...
متن کامل