Rich dynamics in some generalized difference equations

Xiujuan Wang; Mingshu Peng

doi:10.3934/dcdss.2020191

Article Contents

2020, Volume 13, Issue 11: 3205-3212. Doi: 10.3934/dcdss.2020191

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Rich dynamics in some generalized difference equations

Xiujuan Wang^1,2, and
Mingshu Peng^1, ,

1.
Department of Mathematics, Beijing Jiao Tong University, Beijing 100044, China
2.
Weifang University, Weifang, Shandong 261061, China

^* Corresponding author: Mingshu Peng
^* Corresponding author: Mingshu Peng

Received: November 30, 2018

Revised: July 31, 2019

Early access: December 2019

Published: July 2020

The first author is supported by Natural Science foundation of Shandong Province (CN, ZR2015AL004) and the second author by NSFC grant 61977004

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Abstract

There has been an increasing interest in the study of fractional discrete difference since Miller and Ross introduced the $ v $-th fractional sum and the fractional integral was given as a fractional sum in 1989. It is known that fractional discrete difference equations hold discrete memory effects and can describe the long interaction of all the last states during evolution. Therefore the QR factorization algorithm described by Eckmann et al. in 1986 can not be directly applied to determine chaotic or nonchaotic behaviour in such a system, which becomes an interesting problem. Motivated by this, in this study, we propose a direct way to calculate the finite-time local largest Lyapunov exponent. Compared with those in the literature, we find that the test for determining the presence of chaos is reliable. Moreover, bifurcation diagrams which depends on the given fractional order parameter are given in Captuto like discrete Hénon maps and Logistic maps, which was not discussed in the literature. A transient behaviour in chaotic fractional Logistic maps is also discovered.

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Mathematics Subject Classification: Primary: 39A33, 39A28; Secondary: 37M25.

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Figure 1. Fractional Logistic maps ($ \mu = 2.18 $): (a) $ K $ (Chaotic ( = 1) or nonchaotic behaviour ($ = 0 $)); (b) the largest Lyapunov exponent (LLE) (Chaotic ($ >0 $) or nonchaotic behaviour ($ \leq 0 $)); (c) Bifurcation diagrams

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Figure 2. Fractional Hénon maps ($ \mu_1 = 0.8, \mu_2 = 0.3 $): (a) $ K $ (Chaotic ( = 1) or nonchaotic behaviour ($ = 0 $)); (b) the largest Lyapunov exponent (Chaotic ($ >0 $) or nonchaotic behaviour ($ \leq 0 $)); (c) Bifurcation diagrams

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Figure 3. Transient behaviour in fractional Logistic map with different initial conditions for (a) $ x(0) = 0.6 $ and (b) $ x(0) = 0.1 $ ($ v = 0.01, \mu = 2.18 $), (c) Bifurcation diagrams with last 50 iterations after hundreds of iterations under the initial condition $ x(0) = 0.1 $

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