Research

In general, my research interests focus on the field of qualitative theory of dynamical systems, either with continuous time, or with discrete time. In particular, I am especially interested in using computer algorithms together with topological methods for investigating invariant sets of dynamical systems. A theoretical tool that proves to be useful for this purpose is the Conley index (a generalization of the Morse index that is based upon the notion of an index pair). With the use of rigorous numerical methods (interval arithmetic, etc.), it is possible to construct index pairs built of n-dimensional hypercubes, and then apply the homology functor to obtain easy to manipulate algebraic information. The desire to be able to compute the homological Conley index effectively is my main motivation for diving into computational aspects of cubical homology theory.

Current research:
[ periorbits ] a project aimed at developing an automated method for proving the existence of hyperbolic periodic orbits with the use of the Conley index theory and cubical homology algorithms

Materials referred to in my recent publications:

[ The Uniform Expansion Project ] uniform expansion in one-dimensional maps: software and results of computations
[ Excision-Preserving Approach to the Conley Index ] excision-preserving approach to the computation of the Conley index: software and additional materials
[ Parallelization Method for a Continuous Property ] parallelization method for a continuous property: software, examples and applications
[ Homology Software ] algorithmic homology computation: programs, libraries, examples
[ The Rössler Equations ] some data resulting from an application of a method for proving the existence of periodic trajectories to an attracting periodic orbit observed in the Rössler equations
[ Computation Results ] data obtained with a generalization of the above method applied to an unstable periodic trajectory observed in the Rössler equations

Projects closely related to my research:

[ The CAPD Group ] This is the website of the Computer Assisted Proofs in Dynamics Group which I am part of.
[ CHomP ] This is the website of the Computational Homology Project which I take part in.

Seminars held at the Jagiellonian University whose topics are related to my research (note that these pages are usually in Polish):

[ dso ] Symbolic Dynamics and Computations
[ rrp ] Differential Equations and Related Topics
[ dynsys ] Dynamical Systems