Existence and bifurcation of sub-harmonic solutions in reversible non-autonomous differential equations

Abstract

We study the existence of subharmonic solutions in the system ¨u(t) = f(t, u(t)) with u(t) ∈ R k , where f(t, u) is a continuous map that is p-periodic and even with respect to t and odd and Γ-equivariant with respect to u (with the linear action of a finite group Γ). The problem of finding mp-periodic solutions is reformulated in an appropriate functional space, as a nonlinear Γ × Z2 × Dm-equivariant equation. Under certain conditions on the linearization of f at zero and Nagumo growth condition on f at infinity, we prove the existence of an infinite number of subharmonic solutions by means of the Brouwer equivariant degree. In addition, we discuss the bifurcation of subharmonic solutions for the system depending on an extra parameter α.