Manchester Geometry Seminar 2008/2009

Thursday 30 October 2008. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4pm

On Stability Loss Delay for Dynamical Bifurcations

Anatoly Neishtadt (Loughborough University)

Stability loss delay is an interesting, important and not yet completely investigated phenomenon. It was discovered by L. S. Pontryagin and M. A. Shishkova in a model example in 1973. This phenomenon can be described as follows. In classical bifurcation theory the behavior of systems depending on a parameter is considered for the values of the parameter close to some critical, bifurcational, value. In the theory of dynamical bifurcations the parameter changes slowly in time and passes through the value which would be bifurcational in the classical static theory. Suppose that at the bifurcational value of the parameter, the equilibrium or the limit cycle lose their asymptotic linear stability but remain non-degenerate. It turns out, that in the analytic systems this stability loss is inevitably delayed: the phase points remain near the unstable equilibrium (cycle) for a long time after the bifurcation; during this time the parameter changes by a quantity of order 1. Such a delay is not in general found in the non-analytic systems (even infinitely smooth).

In the talk I shall present a review of the current state of the theory of stability loss delay for dynamical bifurcations.