Singularities of transition processes in dynamical systems: Qualitative theory of critical delays

Abstract

This monograph presents a systematic analysis of the singularities in the transition processes for dynamical systems. We study general dynamical systems, with dependence on a parameter, and construct relaxation times that depend on three variables: Initial conditions, parameters \(k\) of the system, and accuracy \(\varepsilon\) of the relaxation. We study the singularities of relaxation times as functions of \((x_0,k)\) under fixed \(\varepsilon\), and then classify the bifurcations (explosions) of limit sets. We study the relationship between singularities of relaxation times and bifurcations of limit sets. An analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems.