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, 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.

Materials referred to in some of my publications (the most recent come first):
finresdynfinresdyn ] a new approach to the perception of finite resolution representation of dynamics, with an application to the Hénon map: software and results of computations
Conley-Morse GraphsConley-Morse Graphs ] database approach to the classification of global dynamics using Conley-Morse graphs: software and results of computations
Parallelization Method for a Continuous PropertyParallelization Method for a Continuous Property ] parallelization method for a continuous property: software, examples and applications
The Uniform Expansion ProjectThe Uniform Expansion Project ] uniform expansion in one-dimensional maps: software and results of computations
Excision-Preserving Approach to the Conley IndexExcision-Preserving Approach to the Conley Index ] excision-preserving approach to the computation of the Conley index: software and additional materials
Homology SoftwareHomology Software ] algorithmic homology computation: programs, libraries, examples
The Rössler EquationsThe 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 ResultsComputation 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 GroupThe CAPD Group ] Computer Assisted Proofs in Dynamics Group, which I am a member of.
CHomPCHomP ] Computational Homology Project, which I actively participate in.