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.