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Abstract We present a new topological method for the study of the dynamics of dissipative PDEs. The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous… Continue Reading
We present a topological technique for analyzing dynamical systems with complex behavior, based on the general notion of covering relations. Our method can be used to study multidimensional dynamical… Continue Reading
Abstract We show how to effectively link covering relations with cone conditions. We give a new, ‘geometric,’ proof of the stable manifold theorem for hyperbolic fixed point of a map.
We describe a Lohner-type algorithm for rigorous integration of dissipative PDEs. Using it for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we give a computer… Continue Reading
There are many examples of complicated or chaotic dynamics, but the set of examples for which chaos has been rigorously demonstrated is quite small. In most cases where chaotic dynamics has been… Continue Reading
Abstract We present a method of self-consistent a priori bounds, which allows us to study rigorously the dynamics of dissipative PDEs. As an application we present a computer-assisted proof of the… Continue Reading
We introduce horseshoe-type mappings which are geometrically similar to Smale's horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the… Continue Reading
We show how to link topological tools with a local hyperbolic behaviour to prove the existence of homoclinic and heteroclinic trajectories for a map. We apply this technique for the Henon map h with… Continue Reading
We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the interval for some particular choices of viscosity and nonautonomous… Continue Reading