Bifurcation and Chaos in Synchronous Manifold of a Forest Model

Chun Ming Huang, Juang Jonq

Research output: Contribution to journalArticlepeer-review


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 fd,β containing two parameters d and β. Here d is the energy depletion quantity and β is the coupling strength. In particular, we obtain the following results. First, we prove that fd,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 fd,β exhibits the period adding bifurcation. Specifically, we show that for any β > 0, fd,β has a unique global attracting fixed point whenever d 1 (;betabeta&+1;betabeta&(;lt&1) and that for any;beta&;gt&0, fd,;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.

Original languageEnglish
Article number1550157
JournalInternational Journal of Bifurcation and Chaos
Issue number12
StatePublished - 1 Nov 2015


  • Coupled map lattices
  • Schwarzian derivative
  • global synchronization


Dive into the research topics of 'Bifurcation and Chaos in Synchronous Manifold of a Forest Model'. Together they form a unique fingerprint.

Cite this