Generic bifurcation of steady-state solutions

Abstract

Brunovsky and Chow [ 1 ] have recently proved that for a generic C2 function with the Whitney topology, the “time map” r(.,f) (see [4]) associated with the differential equation U” +f(u) = 0 with homogeneous Dirichlet or Neumann boundary conditions, is a Morse function. In this note we give a simpler proof of this result as well as some new applications. Our method of proof is quite elementary, uses only Sard’s theorem and the implicit function theorem for functions in C’(lF?‘, R), and avoids the use of transversality in function spaces. An annoying difficulty that one has to face is that the domain of T varies with f: We get around this by constructing a continuous function H(f) which, if positive, implies that T is a Morse function. Thus our task is to prove that the set off with H(f) > 0 is generic. Of course, openness is trivial. To prove the denseness, we take anyf, perturb it by a monomial CU”, and consider the map 8: (u, c) + T’(u,p), wherej(u) =f(u) + CU”. We show that 0 is a regular value of 0 by checking explicitly that the relevant derivative has the form ~ln + b, where a # 0, and b is bounded. Thus for large n, the linear term dominates and this yields the density statement. One consequence of this result is that iff(u) M, then there are a finite number of (positive) stationary solutions of the equation U, = u,, +f(u), with homogeneous Dirichlet boundary conditions and, generically, we can completely describe all solutions of this partial differential equation.