# American Institute of Mathematical Sciences

## Rich dynamics in some generalized difference equations

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

* Corresponding author: Mingshu Peng

Received  December 2018 Revised  August 2019 Published  December 2019

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

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.

Citation: Xiujuan Wang, Mingshu Peng. Rich dynamics in some generalized difference equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020191
##### References:

show all references

##### References:
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
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
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 $x(0) = 0.1$
 [1] Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095 [2] Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573 [3] S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660 [4] Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633 [5] Olivier P. Le Maître, Lionel Mathelin, Omar M. Knio, M. Yousuff Hussaini. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199 [6] Juvencio Alberto Betancourt-Mar, José Manuel Nieto-Villar. Theoretical models for chronotherapy: Periodic perturbations in funnel chaos type. Mathematical Biosciences & Engineering, 2007, 4 (2) : 177-186. doi: 10.3934/mbe.2007.4.177 [7] Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861 [8] Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713 [9] Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080 [10] Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757 [11] Arsen R. Dzhanoev, Alexander Loskutov, Hongjun Cao, Miguel A.F. Sanjuán. A new mechanism of the chaos suppression. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 275-284. doi: 10.3934/dcdsb.2007.7.275 [12] Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic & Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85 [13] Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161 [14] Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383 [15] Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917 [16] Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861 [17] J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653 [18] Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575 [19] Bo Su. Doubling property of elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 143-147. doi: 10.3934/cpaa.2008.7.143 [20] Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013

2018 Impact Factor: 0.545

## Tools

Article outline

Figures and Tables