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Bifurcation and Chaos in Synchronous Manifold of a Forest Model

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.

E-mail Address: huangex.am97g@g2.nctu.edu.tw

and

Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.

National Center for Theoretical Sciences, Hsinchu, Taiwan, R.O.C.

E-mail Address: jjuang@math.nctu.edu.tw

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https://doi.org/10.1142/S0218127415501576Cited by:0 (Source: Crossref)

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 $d$ $d$ is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map $f_{d, β}$ $f_{d, β}$ containing two parameters $d$ $d$ and $β$ $β$ . Here $d$ $d$ is the energy depletion quantity and $β$ $β$ is the coupling strength. In particular, we obtain the following results. First, we prove that $f_{d, 0}$ $f_{d, 0}$ has a chaotic dynamic in the sense of Devaney on an invariant set whenever $d > 1$ $d > 1$ , which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that $f_{d, β}$ $f_{d, β}$ exhibits the period adding bifurcation. Specifically, we show that for any $β > 0$ $β > 0$ , $f_{d, β}$ $f_{d, β}$ has a unique global attracting fixed point whenever $d \leq \frac{1}{(β + 1) {(\frac{β + 1}{β + 2})}^{β}}$ $d \leq \frac{1}{(β + 1) {(\frac{β + 1}{β + 2})}^{β}}$ ( $< 1$ $< 1$ ) and that for any $β > 0$ $β > 0$ , $f_{d, β}$ $f_{d, β}$ has a unique attracting period $k + 1$ $k + 1$ point whenever $d$ $d$ is less than and near any positive integer $k$ $k$ . Furthermore, the corresponding period $k + 1$ $k + 1$ point instantly becomes unstable as $d$ $d$ moves pass the integer $k$ $k$ . Finally, we demonstrate numerically that there are chaotic dynamics whenever $d$ $d$ is in between and away from two consecutive positive integers. We also observe the route to chaos as $d$ $d$ increases from one positive integer to the next through finite period doubling.

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