Interests | Theory | Analysis | Gallery of Computations

Theory

The following theories are the essentials for the numerical investigation methods I study.

RIM Method

The RIM method aims at solving the root finding problem. In contrast to most other approaches, the method is not based on the Newton scheme. Advantages are that no initial guess is required, and that the method is in general applicable to non-smooth mappings. It turned out that the method is especially useful in case a large number of roots exist. In the context of dynamical systems theory, the method can be applied for the localization of periodic points of a fixed period size as well as for the computation of stable manifolds. The method has successfully been used to solve both investigation tasks for piecewise-smooth and non-invertible 2D and 3D maps. The development of the RIM method in theory as well as the implementation and analysis of the concepts for practical application were subject of my Ph.D. thesis.

The core idea of the method is to subdivide parts of the phase space in a limited number of sets. For each of these sets, the image is approximated by a convex hull. In case 0 is inside the convex hull, the set (probably) contains at least one root. The set is subdivided into smaller parts and the process will be repeated for each of these subsets until the remaining sets are a small enough approximation of the solution.

Symbolic Analysis

Basically, Symbolic Analysis provides a unified framework for the acquisition of information about the flow of a dynamical system wihout any restrictions concerning the stability of specific invariant sets. The main idea is the construction of a directed graph, the so-called Symbolic Image, which represents the structure of the phase space for the investigated system. From the computational point of view, the usage of such a graph bears the advantage that, once it is constructed, all investigation methods are matters of graph analysis.

The concepts of Symbolic Analysis were developed by G.S. Osipenko and presented in a series of works. In my Ph.D. thesis, an efficient implementation of the graph construction as well as some investigation methods on the graph were discussed. It turned out that these tasks require an extension of the theoretical concepts as well as some tunings of the original methods. The implemented methods have been successfully applied to solve several investigation tasks for maps and ODEs.