Bifurcation and Chaos in Synchronous Manifold of a Forest Model

Abstract

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant d is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map f(d,beta) containing two parameters d and beta. Here d is the energy depletion quantity and beta is the coupling strength. In particular, we obtain the following results. First, we prove that f(d,0) has a chaotic dynamic in the sense of Devaney on an invariant set whenever d > 1, which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that f(d,beta) exhibits the period adding bifurcation. Specifically, we show that for any beta > 0, f(d,beta) has a unique global attracting fixed point whenever d <= 1/(beta+ 1)(beta+ 1/beta+ 2)beta (< 1) and that for any beta > 0, f(d,beta) has a unique attracting period k + 1 point whenever d is less than and near any positive integer k. Furthermore, the corresponding period k + 1 point instantly becomes unstable as d moves pass the integer k. Finally, we demonstrate numerically that there are chaotic dynamics whenever d is in between and away from two consecutive positive integers. We also observe the route to chaos as d increases from one positive integer to the next through finite period doubling.