doi: 10.3934/dcdss.2020191

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:
[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.  Google Scholar

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T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Inc., Redwood City, CA, 1989.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

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M. T. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.  doi: 10.4067/S0719-06462011000300009.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[23]

Y. Liu, Discrete chaos in fractional Hénon maps, Int. J. Nonlinear Sci., 18 (2014), 170-175.   Google Scholar

[24]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.  doi: 10.1038/261459a0.  Google Scholar

[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.  Google Scholar

[26]

N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Lett., 86 (2001), 183-186.  doi: 10.1103/PhysRevLett.86.183.  Google Scholar

[27]

L. Stone, Period-doubling reversals and chaos in simple ecological models, Nature, 365 (1993), 617-620.  doi: 10.1038/365617a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

G. C. Wu and D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Processing, 102 (2014), 96-99.   Google Scholar

show all references

References:
[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.  Google Scholar

[2]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar

[3]

G. A. Anastassiou, Discrete fractional calculus and inequalities, preprint, arXiv: math/0911.3370. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] J. F. Chen, Theory of Fractional Difference Equations, Xiamen University Publishing Press, Xiamen, 2011.   Google Scholar
[11]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Inc., Redwood City, CA, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[16]

M. T. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.  doi: 10.4067/S0719-06462011000300009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[23]

Y. Liu, Discrete chaos in fractional Hénon maps, Int. J. Nonlinear Sci., 18 (2014), 170-175.   Google Scholar

[24]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.  doi: 10.1038/261459a0.  Google Scholar

[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.  Google Scholar

[26]

N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Lett., 86 (2001), 183-186.  doi: 10.1103/PhysRevLett.86.183.  Google Scholar

[27]

L. Stone, Period-doubling reversals and chaos in simple ecological models, Nature, 365 (1993), 617-620.  doi: 10.1038/365617a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

G. C. Wu and D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Processing, 102 (2014), 96-99.   Google Scholar

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 $
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