# Periodic Orbits

The aim of this research is to develop an automated procedure
for proving the existence of periodic trajectories in dynamical systems.
The key idea is to construct an isolating neighborhood which would give
rise to an index pair for the Conley index. Algorithmic homology computation
of this Conley index together with some other rigorous numerical
verifications would lead to a mathematically accepted proof of the existence
of such a periodic trajectory. Some results of this research have already
been published, some other work is still in progress.

## Stable Periodic Trajectories

In the case of an attracting periodic trajectory, an isolating
neighborhood itself can be used as an index pair with the empty exit set.
Theory and applications in such a situation have been described in the
following papers:

- P. Pilarczyk,
*Computer assisted method for proving existence
of periodic orbits*, TMNA 1999 Vol. 13 No. 2, 365-377.
- P. Pilarczyk,
*Computer assisted proof of the existence
of a periodic orbit in the Rössler equations*,
Proceedings of the International Conference on Differential Equations
(Berlin, 1999), 228-230, World Sci. Publishing, Singapore, 2000.
- M. Mrozek, P. Pilarczyk,
*The Conley index and rigorous numerics
for attracting periodic orbits*,
Proceedings of the Conference on Variational and Topological Methods
in the Study of Nonlinear Phenomena (Pisa, 2000), 65-74,
Progr. Nonlinear Differential Equations Appl., 49,
Birkhäuser Boston, Boston, MA, 2002.

## Unstable Periodic Trajectories

The case of unstable periodic orbits is much more demanding,
not only because an index pair with a non-empty exit set must be constructed,
but also due to numerical issues: Since unstable directions are present
in the dynamical system, numerical bounds for rigorously computed
trajectories tend to grow exponentially, making it very difficult
or even impossible to integrate the underlying differential equation
for prolonged periods of time. Some specific periodic orbits have been
addressed with success, though, and these cases are described in the
following papers:

- P. Pilarczyk,
*Topological-numerical approach
to the existence of periodic trajectories in ODEs*,
Discrete and Continuous Dynamical Systems 2003,
A Supplement Volume: Dynamical Systems and Differential Equations, 701-708.
- P. Pilarczyk,
*The Conley index and rigorous numerics
for hyperbolic periodic trajectories in ODEs*, in preparation.

## Homology Computation

Since the method used in this research requires one to compute
the homology of an index map, suitable algorithms for homology computation
must have been developed. The algorithms available in the book

- T. Kaczynski, K. Mischaikow, M. Mrozek,
*Computational homology*,
Applied Mathematical Sciences 157, Springer-Verlag, New York, 2004.

turned out to be insufficient, mainly because they do not address
the issue of relative homology, obviously necessary for the computation
of the homological Conley index map. Therefore, in addition to the
reduction algorithms introduced in one of the above-mentioned papers,
additional research had to be carried out, which resulted in the following
publication:

- K. Mischaikow, M. Mrozek, P. Pilarczyk,
*Graph approach
to the computation of the homology of continuous maps*,
Foundations of Computational Mathematics (2005), Vol. 5, No. 2, 199-229.

## Excision Preserving Maps

The definition of a cubical index pair with respect to a discrete
dynamical system used in this research has been chosen in a way that seems
to be optimal from the point of view of rigorous numerical computations
which appear to be the most difficult part of the method. However,
due to delicate isolating properties of such an index pair, in some cases
difficulties emerge with the index map computation, because the cubical
enclosure of the underlying continuous map may loose the excision
property of that map. In order to overcome this difficluty, an alternative
approach to the index map is proposed in the following paper:

- P. Pilarczyk, K. Stolot,
*Excision-preserving cubical approach
to the algorithmic computation of the discrete Conley index*, submitted.

## Parallel Computing

In order to speed up the process of constructing an index pair,
a concurrent version of the algorithm introduced in one of the
previous papers is beeing developed. It will be described in the
following paper:

- P. Pilarczyk,
*A concurrent algorithm for the construction
of index pairs*, in preparation.