Dynamical phenomena connected with stability loss of equilibria and periodic trajectories

This is a study of a dynamical system depending on a parameter $\kappa$. Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on $\kappa$, 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, $\kappa$ is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, $\kappa$ varies slowly with time (the case of a dynamic bifurcation). In the simplest situation $\kappa=\varepsilon t$, where $\varepsilon$ is a small parameter. More generally, $\kappa(t)$ 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.
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