Dynamical phenomena connected with stability loss of equilibria and periodic trajectories

Abstract

Abstract This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.