Periodic and Non-Periodic Solutions of a Ricker-type Second-Order Equation with Periodic Parameters

TL;DR

This paper analyzes the dynamics of a second-order Ricker-type difference equation with periodic parameters, revealing different behaviors depending on whether the period is odd or even, including coexistence of solutions and convergence to stable limit cycles.

Contribution

It introduces a semiconjugate factorization approach to study the equation, highlighting how periodicity affects solution stability and coexistence.

Findings

Odd period p allows coexistence of solutions with varying amplitudes.

Even period p solutions tend to a stable limit cycle under certain conditions.

The analysis uses a novel semiconjugate factorization into first-order systems.

Abstract

We study the dynamics of the positive solutions of a second-order, Ricker-type exponential difference equation with periodic parameters. We find that qualitatively different dynamics occur depending on whether the period p of the main parameter is odd or even. If p is odd then periodic and non-periodic solutions may coexist (with different initial values) if the amplitude of the periodic parameter is allowed to vary over a sufficiently large range. But if p is even then all solutions converge to an asymptotically stable limit cycle of period p if either all the odd-indexed or all the even-indexed terms of the periodic parameter are less than 2, and the sum of the even terms does not equal the sum of the odd terms. The key idea in our analysis is a semiconjugate factorization of the above equation into a triangular system of two first-order equations.