In classical bifurcation theory the behavior of systems, depending on a parameter, is considered for values of the parameter close to some critical, bifurcational value. In the theory of dynamical bifurcations the parameter is changing slowly in time and passes through the value, which would be critical in classical static theory. Some phenomena, arising here, are drastically different from predictions derived by a static approach. At a bifurcational value of the parameter the equilibrium or the limit cycle loses its asymptotic linear stability, but remains nondegenerate. In analytic systems the stability loss is inevitably delayed: the phase points remain near the unstable equilibrium (cycle) for a long time after bifurcation; during this time the parameter changes by a quantity of order 1. Such delay is not in general found in nonanalytic (even infinitely smooth) systems.
The talk is devoted to estimates of delay time. The delay time is controlled by behavior of solutions in the plane of complex time.