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 equationx n+1 = x n−1 e a n −x n−1 −x n where the sequence {a n } is periodic. We find that qualitatively different dynamics occurs depending on whether the period p of {a n } 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 a n 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 {a n } are less than 2, and the sum of the even terms of {a n } 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. ARTICLE HISTORY