Subharmonic Bifurcation of Forced T-Periodic Solutions

Abstract

In this chapter, and in Chapter X, we consider the bifurcation of forced T-periodic solutions. In thinking about the origin and structure of such problems it would benefit the reader to reread the explanations given in §I.2 and §I.3. Following our usual procedure we do the theory in ℝⁿ , n ≥ 2, and show how the analysis reduces ℝ¹ or ℝ² or using projections associated with the Fredholm alternative. There is a sense in which the problem in ℝⁿ with n finite is actually infinite-dimensional. Unlike steady problems which involve only constant vectors, we must work with vector-valued functions which depend periodically on time and hence take on infinitely many distinct values. So, in this chapter the computational simplifications which would result from considering ℝ² rather than ℝⁿ are not great. In ℝⁿ we use the same notation we would use for an evolution equation in a Banach space. So our results hold equally in ℝⁿ and, say, for evolution problems governed by partial differential equations, like the Navier--Stokes equations or equations governing reaction and diffusion in chemical systems, provided the writing of these partial differential equations as evolution problems in Banach space can be justified.