Electronic Journal of Differential Equations,
Monograph 05, 2004, (55 pages).
Title: Singularities of transition processes in dynamical systems:
Qualitative theory of critical delays
Author: Alexander N. Gorban (ETH-Zentrum, Switzerland)
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.
Submitted May 29, 2004. Published August 7, 2004.
Math Subject Classifications: 54H20, 58D30, 37B25.
Key Words: Dynamical system; transition process; relaxation time;
bifurcation; limit set; Smale order.