Stability and Bifurcation

Abstract

As for ordinary differential equations, the primary objective in the qualitative theory of RFDE is to study the dependence of the flow Φ_t = Φ ^F_t on F. This implicitly requires the existence of a criterion for deciding when two RFDE and, more generally, two semigroups, are equivalent. A study of the dependence of the flow on changes of a certain parameter (in particular the semigroup itself) through the use of a notion of equivalence based on a comparison of all orbits is very difficult and is likely to give too small equivalence classes. The difficulty is associated with the infinite dimensionality of the phase space and the associated smoothing properties of the solution operator as well as with the appearance of special phenomena that do not make sense in the setting of flows on finite dimensional manifolds. For example, the case of two global trajectories, i.e. both defined for all t Î ℝ, with a collision point and, of course, after the collision they keep evolving together; that can happen even in a C ^∞ system and with two C ^∞ global solutions, as we saw in Chapter 3.