Periodic behaviour of nonlinear second order discrete dynamical systems

TL;DR

This paper establishes conditions for the existence of periodic solutions in nonlinear second-order difference equations, using Lyapunov-Schmidt reduction, fixed point methods, and topological degree theory.

Contribution

It introduces new analytical conditions for periodic solutions in nonlinear second-order discrete dynamical systems, employing advanced mathematical techniques.

Findings

Conditions for periodic solutions are derived.

The methods apply to equations with periodic forcing.

The approach combines Lyapunov-Schmidt reduction with topological tools.

Abstract

In this work we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form \begin{equation*} y(t+2)+by(t+1)+cy(t)=g(t,y(t)) \end{equation*} where $c\neq 0$, and $g:\mathbb{Z}^+\times\mathbb{R}\to \mathbb{R}$ is continuous and periodic in $t$. Our analysis uses the Lyapunov-Schmidt reduction in combination with fixed point methods and topological degree theory.