On stability loss delay for a periodic trajectory

Abstract

Stability loss delay is an interesting, important and so far not completely clear phenomenon. Its essence is as follows. Consider a system of differential equations depending on a slowly varying parameter. Suppose that the system has an equilibrium position or a periodic trajectory for any fixed value of the parameter. Suppose also that the parameter passes through a bifurcational value: the equilibrium (periodic trajectory) loses stability but remains nondegenerate. In the case of an equilibrium a pair of conjugate eigenvalues leaves the left half-plane not passing through zero. For a periodic trajectory either a pair of conjugate multipliers leaves the unit circle not passing through the point 1, or one real multiplier goes away from the unit circle through the point —1. If the system is analytic, a delay of stability loss takes place: phase points attracted to the equilibrium (periodic trajectory) long before the moment of the bifurcation remain close to the unstable equilibrium (periodic trajectory) until the change of the parameter is of order one. The velocity of the parameter changing can be arbitrary small. In non-analytic systems (even in the C ^∞case) in general there is no such a delay of stability loss.