Polynomial maps with hidden complex dynamics

Xu Zhang; Guanrong Chen

doi:10.3934/dcdsb.2018293

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2019, Volume 24, Issue 6: 2941-2954. Doi: 10.3934/dcdsb.2018293

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Polynomial maps with hidden complex dynamics

Xu Zhang^1, , and
Guanrong Chen^2,

1.
Department of Mathematics, Shandong University, Weihai 264209, Shandong, China
2.
Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China

* Corresponding author
* Corresponding author

Received: October 31, 2017

Revised: February 28, 2018

Early access: October 2018

Published: June 2019

This research is partially supported by the National Natural Science Foundation of China (Grant 11701328), Shandong Provincial Natural Science Foundation, China (Grant ZR2017QA006), Young Scholars Program of Shandong University, Weihai (Grant 2017WHWLJH09), and China Postdoctoral Science Foundation (Grant 2016M602126)

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Abstract

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Hénon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

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Mathematics Subject Classification: Primary: 37C05, 37D45, 37B25; Secondary: 37E05, 37E30.

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Figure 1. Illustration diagram of the function $P(x) = ax^m(x+b)(x-b)$ with $a>0$, $x\in[-b,b]$, and odd $m$.

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Figure 2. Illustration diagram of the function $P(x) = ax^m(x+b)(x-b)$ with $a>0$, $x\in[-b,b]$, and even $m$.

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Figure 3. Bifurcation diagram of $P(x) = ax^2(x+1)(x-1)$ for $3\leq a\leq4$, where the initial value is $0.6$.

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Figure 4. Bifurcation diagram of $P(x) = ax^3(x+1)(x-1)$ for $4.6\leq a\leq5$, where the initial value is $0.6$.

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Figure 5. Illustration diagram of the horseshoe for a subshift of finite type for the matrix $A$.

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Figure 6. Illustration diagram for the map (4) with $a = 6$, $b = c = d = 1$, and $m = 3$.

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Figure 7. Simulation of the map (4) with $a = 5$, $b = 1$, $m = 3$, $d = 1$, and $c = 0.005$, where the initial value is $(0.8,0.8)$.

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Figure 8. Illustration diagram for the map (4) with $a = 5$, $b = c = d = 1$, and $m = 2$.

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