# 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 are listed below
(the most recent come first). Please, note that this list is somewhat outdated,
and Web pages with materials to my most recent publications are only referred to
in the papers which one can download from the Publications page.

[ finresdyn ] |
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 Graphs ] |
database approach to the classification of global dynamics
using Conley-Morse graphs:
software and results of computations |

[ Parallelization Method for a Continuous Property ] |
parallelization method for a continuous property:
software, examples and applications |

[ 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 |

[ 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 ] |
*Computer Assisted Proofs
in Dynamics Group*, which I am a member of. |

[ CHomP ] |
*Computational Homology Project*,
which I actively participate in. |