Alexander N. Gorban
Singularities of Transition Processes in Dynamical Systems
(907K, pdf)
ABSTRACT. 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 $\varepsilon$ of relaxation. This system of times contains:
the time before the first entering of the motion into $\varepsilon$-neighbourhood of the limit set,
the time of final entering in this neighbourhood and the time of stay of the motion outside the
$\varepsilon$-neighbourhood of the limit set. The singularities of relaxation times as functions of
$(x_0,k)$ under fixed $\varepsilon$ are studied. A classification of different 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 finite 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 stable systems.