Subharmonic Solutions In Reversible Non-Autonomous Differential Equations

TL;DR

This paper proves the existence of infinitely many subharmonic solutions in reversible, non-autonomous differential equations with symmetric properties, using equivariant degree theory and bifurcation analysis.

Contribution

It introduces a novel application of Brouwer equivariant degree to establish subharmonic solutions in symmetric differential systems.

Findings

Existence of infinitely many subharmonic solutions under symmetry and growth conditions.

Application of equivariant degree theory to nonlinear differential equations.

Analysis of bifurcation phenomena with parameter dependence.

Abstract

We study the existence of subharmonic solutions in the system $\ddot {u}(t)=f(t,u(t))$, where $u(t)\in\mathbb{R}^{k}$ and $f$ is an even and $p$-periodic function in time. Under some additional symmetry conditions on the function $f$, the problem of finding $mp$-periodic solutions can be reformulated in a functional space as a $\Gamma\times\mathbb{Z}_{2}\times D_{m}% $-equivariant equation, where the group $\Gamma\times\mathbb{Z}_{2}$ acts on the space $\mathbb{R}^{k}$ and $D_{m}$ acts on $u(t)$ by time-shifts and reflection. We apply Brouwer equivariant degree to prove the existence of an infinite number of subharmonic solutions for the function $f$ that satisfies additional hypothesis on linear behavior near zero and the Nagumo condition at infinity. We also discuss the bifurcation of subharmonic solutions when the system depends on an extra parameter.