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Rich dynamics in some generalized difference equations

  • * Corresponding author: Mingshu Peng

    * Corresponding author: Mingshu Peng

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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  • 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.

    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

    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

    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 $

  • [1] H. D. I. AbarbanelR. BrownJ. J. Sidorowich and L. S. Tsimring, The analysis of observed chaotic data in physical systems, Rev. Modern Phys., 65 (1993), 1331-1392.  doi: 10.1103/RevModPhys.65.1331.
    [2] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.
    [3] G. A. Anastassiou, Discrete fractional calculus and inequalities, preprint, arXiv: math/0911.3370.
    [4] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.
    [5] F. M. Atici and S. Senguel, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1-9.  doi: 10.1016/j.jmaa.2010.02.009.
    [6] N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29 (2011), 417-437.  doi: 10.3934/dcds.2011.29.417.
    [7] D. Cafagna and G. Grassi, An effective method for detecting chaos in fractional-order systems, Int. J. Bifurcation Chaos, 20 (2010), 669-678.  doi: 10.1142/S0218127410025958.
    [8] R. Caponetto and S. Fazzino, A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 22-27.  doi: 10.1016/j.cnsns.2012.06.013.
    [9] F. L. Chen, X. N. Luo and Y. Zhou, Existence results for nonlinear fractional difference equations, Adv. Differ. Equ., 2011 (2011), 12pp. doi: 10.1155/2011/713201.
    [10] J. F. ChenTheory of Fractional Difference Equations, Xiamen University Publishing Press, Xiamen, 2011. 
    [11] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Inc., Redwood City, CA, 1989.
    [12] J. P. EckmannS. O. KamphorstD. Ruelle and S. Ciliberto, Liapunov exponents from time series, Phys. Rev. A (3), 34 (1986), 4971-4979.  doi: 10.1103/PhysRevA.34.4971.
    [13] J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.
    [14] C. S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl, 61 (2011), 191-202.  doi: 10.1016/j.camwa.2010.10.041.
    [15] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.
    [16] M. T. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.  doi: 10.4067/S0719-06462011000300009.
    [17] M. T. Holm, The Laplace transform in discrete fractional calculus, Comput. Math. Appl., 62 (2011), 1591-1601.  doi: 10.1016/j.camwa.2011.04.019.
    [18] J. Kaplan and J. Yorke, Chaotic behaviour of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730, Springer, Berlin, 1979, 204–227.
    [19] N. V. KuznetsovT. A. Alexeeva and G. A. Leonov, Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dynam., 85 (2016), 195-201.  doi: 10.1007/s11071-016-2678-4.
    [20] N. V. Kuznetsov and G. A. Leonov et al., Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285.  doi: 10.1007/s11071-018-4054-z.
    [21] G. A. Leonov and N. V. Kuznetsov, Time-varying linearization and the Perron effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1079-1107.  doi: 10.1142/S0218127407017732.
    [22] T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.
    [23] Y. Liu, Discrete chaos in fractional Hénon maps, Int. J. Nonlinear Sci., 18 (2014), 170-175. 
    [24] R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.  doi: 10.1038/261459a0.
    [25] K. S. Miller and B. Ross, Fractional difference calculus, in Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989,139–152.
    [26] N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Lett., 86 (2001), 183-186.  doi: 10.1103/PhysRevLett.86.183.
    [27] L. Stone, Period-doubling reversals and chaos in simple ecological models, Nature, 365 (1993), 617-620.  doi: 10.1038/365617a0.
    [28] F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981, 366–381. doi: 10.1007/BFb0091924.
    [29] A. WolfJ. B. SwiftH. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.
    [30] G. C. Wu and D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283-287.  doi: 10.1007/s11071-013-1065-7.
    [31] G. C. Wu and D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Processing, 102 (2014), 96-99. 
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