A BUNCH OF STATIONARY OR PERIODIC SOLUTIONS NEAR AN EQUILIBRIUM BY A SLOW EXCHANGE OF STABILITY

Abstract

This chapter highlights a bunch of stationary or periodic solutions near equilibrium by a slow exchange of stability. The nonlinear evolution equation depending on a real parameter A in some real Banach space E is considered. If E is finite dimensional, this equation represents an ordinary dynamical system, and if E is infinite dimensional it is the abstract version of some class of nonlinear parabolic partial differential equations. The assumptions imposed admit, for instance, the Navier Stokes system in its Hilbert space formulation, such that the class of possible applications is large enough. As is well known, the first case leads to stationary bifurcation, whereas the second case leads to Hopf bifurcation of periodic solutions of equation. Closely connected to bifurcation at a simple eigenvalue is a principle of exchange of stability, which means that the stability lost by the trivial solution is taken over by the bifurcating solution which may be stationary or periodic. Bifurcation at simple eigenvalues is not as simple as it seems to be.