GEOMETRY OF DYNAMICAL SYSTEMS AND TOPOLOGICAL STABILITY: FROM BIFURCATION, CHAOS AND FRACTALS TO DYNAMICS IN NATURAL AND LIFE SCIENCES

Abstract

The aim of this article is to review some basic concepts of the geometric theory of dynamical systems and stability. In this context, we also consider the related fundamental notions of broken symmetry, bifurcation and chaos. That of bifurcation is a very sophisticated mathematical concept, which displays a number of local and global behaviors of those spaces within which a large variety of natural forms unfold. The suited theoretical framework for understanding deeply the concept of bifurcation is the study of singularities of mappings, their topological structures and their classification into equivalence classes. Furthermore, we consider the fundamental role played by the phenomena of breaking symmetry and chaos in the evolution and organization of various natural and living systems. In the last part of the paper, we present some striking features and results of nonlinearity and stability in the framework of the geometrical theory of dynamical systems.