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Chaos and bifurcations in chaotic maps with parameter \(q\): numerical and analytical studies. (English) Zbl 1417.37176

Summary: In this paper, a class of chaotic maps with parameter \(q\) are introduced and bifurcations and chaos in proposed maps are numerical and analytical studied. Euler method is employed to get the continuous systems corresponding to chaotic maps and the fractional styles in Caputo’s definition. Based on that, we finally infer a class of chaotic maps with the Adams-Bashforth-Moulton predictor-corrector method. In the simulation and analysis, we discuss the Logistic map with \(q\) and Hénon map with \(q\), observe the route from period to chaos and do tests to analyze properties of maps with parameter \(q\).

MSC:

37G10 Bifurcations of singular points in dynamical systems
39A28 Bifurcation theory for difference equations
39A33 Chaotic behavior of solutions of difference equations
26A33 Fractional derivatives and integrals
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[1] M. Caputo, Linear models of dissipation whose q is almost frequency independent, Geophys. J. R. Astron. Soc., 13:529-539, 1967.
[2] L.P. Chen, Y. Chai, R.C. Wu, Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems, Chaos, 21, 043107, 2011. · Zbl 1317.34008
[3] D.Y. Chen, C.F. Liu, C. Wu, H.H.C. Lu, Y.J. Liu, X.Y. Ma, Y.J. You, A new fractional-order chaotic system and its synchronization with circuit simulation, Circuits Syst. Signal Process., 31:1599-1613, 2012.
[4] G.R. Chen, J.H. Lü, Dynamical Analyses, Control and Synchronization of the Lorenz System Family, Science Press, Beijing, 2003.
[5] G.R. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcat. Chaos, 19:632-635, 2002. http://www.mii.lt/NA Chaos and bifurcations in maps withq261 · Zbl 0962.37013
[6] D.Y. Chen, C. Wu, H.H.C. Lu, X.Y. Ma, Circuit simulation for synchronization of a fractionalorder and integer-order chaotic system, Nonlinear Dyn., 73:1671-1686, 2013.
[7] D.Y. Chen, R.F. Zhang, J.C. Sprott, H.T. Chen, X.Y. Ma, Synchronization between integerorder chaotic systems and a class of fractional-order chaotic systems via sliding mode control, Chaos, 22, 023130, 2012. · Zbl 1331.34129
[8] W.H. Deng, C.P. Li, Chaos synchronization of the fractional Lü system, Physica A, 353:61-72, 2005.
[9] K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms, 16:231-253, 1997. · Zbl 0926.65070
[10] M.J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19:25-52, 1978. · Zbl 0509.58037
[11] I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 034101, 2003. · Zbl 1234.49040
[12] B.L. Hao, Symbolic dynamics and characterization of complexity, Physica D, 51:161-176, 1991. · Zbl 0744.58014
[13] M. Hénon, 2-dimensional mapping with a strange attractor, Commun. Math. Phys., 50:69-77, 1976. · Zbl 0576.58018
[14] C.G. Li, G.R. Chen, Chaos in the fractional-order Chen system and its control, Chaos Solitons Fractals, 22:549-554, 2004. · Zbl 1069.37025
[15] C.P. Li, G.J. Peng, Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals, 22:443-450, 2004. · Zbl 1060.37026
[16] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20:130-141, 1963. · Zbl 1417.37129
[17] J.H. Lü, G.R. Chen, A new chaotic attractor coined, Int. J. Bifurcat. Chaos, 12:659-661, 2002. · Zbl 1063.34510
[18] NIST: A statistical test suite for random and pseudo random number generators for cryptographic applications, 2010, http://csrc.nist.gov/publications/ nistpubs/800-22-rev1a/SP800-22rev1a.pdf.
[19] O.E. Rössler, Equation for continuous chaos, Phys. Lett. A, 57:397-398, 1976. · Zbl 1371.37062
[20] K.H. Sun, J.C. Sprott, Bifurcations of fractional-order diffusionless Lorenz system, Int. J. Bifurcat. Chaos, 20:1209-1219, 2010. · Zbl 1193.34005
[21] M.S. Tavazoei, M. Haeri, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems, IET Signal Process., 1:171-181, 2007.
[22] M.S. Tavazoei, M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal., Theory, Methods Appl., 69:1299-1320, 2008. · Zbl 1148.65094
[23] X.Y. Wang, Chaos in the Complex Nonlinearity System, Electronics Industry Press, Beijing, 2003.
[24] X.Y. Wang, X. Qin, A new pseudo-random number generator based on CML and chaotic iteration, Nonlinear Dyn., 70:1589-1592, 2012. Nonlinear Anal. Model. Control, 20(2):249-262 262H. Zhang et al.
[25] G.R. Wang, X.L. Yu, S.G. Chen, Chaotic Control, Synchronization and Utilizing, National Defence Industry Press, Beijing, 2001.
[26] X.Y. Wang, H. Zhang, Chaotic synchronization of fractional-order spatiotemporal coupled Lorenz system, Int. J. Mod. Phys. C, 23, 1250067, 2012.
[27] Y. Xu, R.C. Gu, H.Q. Zhang, D.X. Li, Chaos in diffusionless lorenz system with a fractional order and its control, Int. J. Bifurcat. Chaos, 22, 1250088, 2012. · Zbl 1258.34020
[28] R.X. Zhang, S.P. Yang, Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations. Nonlinear Dyn., 69:983-992, 2012. · Zbl 1253.93071
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