Marian Mrozek

Chaos in the Lorenz equations: a computer-assisted proof

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer- assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics.

Homology computation by reduction of chain complexes

Abstract A new algorithm for computing the homology module of a finitely generated chain complex is given. It is based on local one-step reductions of the size of the initial chain complex and it has a clear geometrical interpretation. The complexity of the algorithm is discussed in special cases.

Leray functor and cohomological Conley index for discrete dynamical systems

We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.

Isolating neighborhoods and chaos

We show that the map part of the discrete Conley index carries information which can be used to detect the existence of connections in the repeller-attractor decomposition of an isolated invariant set of a homeomorphism. We use this information to provide a characterization of invariant sets which admit a semi-conjugacy onto the space of sequences on K symbols with dynamics given by a subshift. These ideas are applied to the Henon map to prove the existence of chaotic dynamics on an open set of parameter values.

Coreduction Homology Algorithm

This paper presents a new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes. The algorithm—particularly strong for low-dimensional sets embedded in high dimensions—runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates.

Conley index for discrete multi-valued dynamical systems

Abstract The definitions of isolating block, index pair, and the Conley index, together with the proof of homotopy and additivity properties of the index are generalized for discrete multi-valued dynamical systems. That generalization provides a theoretical background of numerical computation used by Mischaikow and Mrozek in their computer assisted proof of chaos in the Lorenz equations, where finitely represented multi-valued mappings appear as a tool for discretization.

Chapter 9 – Conley Index

This chapter discusses the Conley index theory. The Conley index is an index of isolating neighborhoods. The applicability of the Conley index depends essentially on three properties. The first gives great freedom in the choice of regions in phase space on which one will perform the analysis. The second allows for passage from the isolating neighborhood to an understanding of the dynamics of the isolated invariant set. The Wazewski property, while the most fundamental, is the simplest result of this type. It contains a variety of more sophisticated theorems that can be used to prove the existence of connecting orbits, periodic orbits, and even chaotic dynamics in the sense of symbolic dynamics. The third property is important for the following reason. The Conley index is a purely topological index, and it is a very coarse measure of the dynamics. Typically, if it can be computed directly at a particular parameter value, then knowledge of the dynamics at that parameter value is reasonably complete. The power of the index (as in degree theory) comes from being able to continue it to a parameter value where understanding of the dynamics is much less complete.

Homology algorithm based on acyclic subspace

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.

Graph Approach to the Computation of the Homology of Continuous Maps

AbstractWe introduce an efficient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based on a cubical approximation of the map and the theory of multivalued maps. A software implementation of the algorithms introduced in this paper is available at [27].

Chaos in the Lorenz equators: a computer assisted proof. part II: details

Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.