Singularities of Transition Processes In Dynamical Systems

Abstract. The paper gives the systematic analysis of singularities of transition The paper gives the systematic analysis of singularities of transition processes in general dynamical systems. Dynamical systems depending on parameter are studied. A system of relaxation times is constructed. Each relaxation time depends on three variables: initial conditions, parameters k of the system and accuracy " of relaxation. This system of times contains: the time before the rst entering of the motion into "-neighbourhood of the limit set, the time of nal entering in this neighbourhood and the time of stay of the motion outside the "-neighbourhood of the limit set. The singularities of relaxation times as functions of (x0; k) under xed " are studied. A classi cation of di erent bifurcations (explosions) of limit sets is performed. The bifurcations fall into those with appearance of new limit points and bifurcations with appearance of new limit sets at nite distance from the existing ones. The relations between the singularities of relaxation times and bifurcations of limit sets are studied. The peculiarities of dynamics which entail singularities of transition processes without bifurcations are described as well. The peculiarities of transition processes under perturbations are studied. 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 smooth two-dimensional structural