On Maximal Delay of Stability Loss for Dynamical Bifurcations

We consider a dynamical bifurcation caused by a slow passage through a static bifurcation point: in a system depending on a parameter, the parameter changes slowly in time and passes through the critical value corresponding to the loss of stability of an equilibrium via a Poincar\'e--Andronov--Hopf bifurcation in the frozen system. If the system is analytic, then the loss of stability is inevitably delayed: phase points attracted to the equilibrium in the stability region remain near the equilibrium for a long time after entering the instability region, so that the parameter changes by an amount of order ~1 independently of how slow the variation of the parameter is. Remarkably, there exists a {\it maximal delay}: all phase points attracted to the stable equilibrium before a certain threshold value of the parameter leave a neighbourhood of the unstable equilibrium almost simultaneously near another threshold value of the parameter, known as {\it a buffer point}. A delay of stability loss beyond the buffer point is impossible unless the initial data have a very special form. We assume that, although the equilibrium is non-degenerate for real values of the parameter, one of its eigenvalues vanishes generically for some complex value of the parameter (a complex analogue of a saddle-node bifurcation), and that this complex singularity is, in a suitable sense, the closest one to the real Poincar\'e--Andronov--Hopf bifurcation point. We show that the value of maximal delay is determined by this complex singularity: the threshold values defining the maximal delay are the intersection points of the Stokes lines associated with this singularity and the real axis. We study these phenomena in the framework of slow--fast dynamical systems.