Global bifurcation of periodic solutions to some autonomous differential delay equations

摘要

We are concerned with slowly oscillating, periodic solutions of the parametrized differential delay equation x[xdot](t) =-λƒ(x(t-1)), λ > 0, where ƒ is an odd, continuous, and nonlinear function satisfying xƒ(x)>0 for all x≠0. Due to the oddness of ƒ we can identify odd-harmonic periodic solutions having the period T with other solutions of the periods 2T/(T-2). From this observation and a result of Nussbaum we obtain the existence of periodic solutions with a period close to 2. Moreover, we can reach conclusions about a secondary bifurcation of periodic solutions. We present a combination of a Galerkin scheme with a continuation method which yields a feasable numerical procedure for the effective computation and continuation of periodic solutions. We report the results of an extensive case study for the nonlinearities ƒ(x)=x/(1+¦x¦p), p⩾ 1. For p = 8 we show many different continua of periodic solutions which are either disjoint or bifurcating from continua of periodic solutions that contain a higher symmetry. For p=1 and λ>π/2 it is known that periodic solutions are unique. We discuss two homotopies linking the cases p=8 and p=1 which exhibit very different transitions from multiplicity to uniqueness of solutions.