Periodic and non-periodic solutions in a Ricker-type second-order equation with periodic parameters

Abstract

We study the dynamics of the positive solutions of the exponential difference equationwhere the sequence is periodic. We find that qualitatively different dynamics occurs depending on whether the period p of is odd or even. If p is odd then periodic and non-periodic solutions coexist (with different initial values) if the amplitudes of the terms are 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 are less than 2, and the sum of the even terms of does not equal the sum of its odd terms. The key idea in our analysis that explains this behavioural dichotomy is a semiconjugate factorization of the above equation into a triangular system of two first-order equations.