Zin Arai

A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems

A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decom- positions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the ex- istence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional density-dependent Leslie population model. An interactive vi- sualization of the results of computations discussed in the paper can be accessed at the Web site http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available.

On Hyperbolic Plateaus of the Hénon Map

We propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real Hénon family, we prove the existence of many regions of hyperbolic parameters in the parameter plane of the family.

Rigorous Computations of Homoclinic Tangencies

In this paper, we propose a rigorous computational method for detecting homoclinic tangencies and structurally unstable connecting orbits. It is a combination of several tools and algorithms, including the interval arithmetic, the subdivision algorithm, the Conley index theory, and the computational homology theory. As an example, we prove the existence of generic homoclinic tangencies in the Henon family.

Recent Development in Rigorous Computational Methods in Dynamical Systems

We highlight selected results of recent development in the area of rigorous computations which use interval arithmetic to analyse dynamical systems. We describe general ideas and selected details of different ways of approach and we provide specific sample applications to illustrate the effectiveness of these methods. The emphasis is put on a topological approach, which combined with rigorous calculations provides a broad range of new methods that yield mathematically reliable results.

On Parameter Loci of the Hénon Family

The purpose of the current article is to investigate the dynamics of the Hénon family fa,b : (x, y)

Decomposition and Clustering for the Visualization of Dynamical Systems

{\mapsto}

A Database Schema for the Analysis of Global Dynamics

↦ (x2−a−by, x), where (a, b)