TY - JOUR

T1 - Bifurcation and Chaos in Synchronous Manifold of a Forest Model

AU - Huang, Chun Ming

AU - Jonq, Juang

N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - 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.

AB - 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.

KW - Coupled map lattices

KW - Schwarzian derivative

KW - global synchronization

UR - http://www.scopus.com/inward/record.url?scp=84949309485&partnerID=8YFLogxK

U2 - 10.1142/S0218127415501576

DO - 10.1142/S0218127415501576

M3 - Article

AN - SCOPUS:84949309485

SN - 0218-1274

VL - 25

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

IS - 12

M1 - 1550157

ER -