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Dynamics of a class of fractional-order nonautonomous Lorenz-type systems. (English) Zbl 1387.37028
Summary: The dynamical properties of a class of fractional-order Lorenz-type systems with quasi-periodic time-varying parameters are studied, where the fractional derivative is defined in the sense of Caputo. The effective non-integer dimension \(\beta\) is the sum of all the fractional orders. Deferring from the fractional-order autonomous Lorenz systems, the present nonautonomous systems have two critical values, \(\beta_*\) and \(\beta^*\), of the effective non-integer dimension, \(0 < \beta_* < \beta^* < 3\), under which there exist a transition from chaos to quasi-periodic dynamics for some \(\beta\) near \(\beta^*\) and a transition from quasi-periodic motion to regular dynamics (diverging to infinity) for some \(\beta\) near \(\beta_*\). The 0-1 test is applied to verify the existence of such strange dynamics.{
©2017 American Institute of Physics}

37C60 Nonautonomous smooth dynamical systems
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34D45 Attractors of solutions to ordinary differential equations
70K43 Quasi-periodic motions and invariant tori for nonlinear problems in mechanics
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[1] Abbas, S.; Benchohra, M.; N’Guérékata, G. M., Topics in Fractional Differential Equations, (2012), Springer: Springer, New York · Zbl 1273.35001
[2] Cafagna, D.; Grassi, G., An effective method for detecting chaos in fractional-order systems, Int. J. Bifurcation Chaos, 20, 669-678, (2010) · Zbl 1193.34086
[3] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. R. Astron. Soc., 13, 529-539, (1967)
[4] Cheban, D. N.; Duan, J., Recurrent motions and global attractors of non-autonomous Lorenz systems, Dyn. Syst., 19, 41-59, (2004) · Zbl 1060.37016
[5] Cheban, D. N.; Kloeden, P. E.; Schmalfuß, B., The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2, 125-144, (2002) · Zbl 1054.34087
[6] Chen, G.; Lü, J., Dynamics of the Lorenz System Family: Analysis, Control and Synchronization (in Chinese), (2003), Science Press: Science Press, Beijing
[7] Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9, 1465-1466, (1999) · Zbl 0962.37013
[8] Chepyzhov, V. V., Uniform attractors of dynamical processes and non-autonomous equations of mathematical physics, Russ. Math. Surv., 68, 349-382, (2013) · Zbl 1278.35024
[9] Chua, L. O.; Komuro, M.; Matsumoto, T., The double scroll family. I. Rigorous proof of chaos. II. Rigorous analysis of bifurcation phenomena, IEEE Trans. Circuits Syst., 33, 1072-1118, (1986) · Zbl 0634.58015
[10] Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J. Math. Anal. Appl., 293, 511-522, (2004) · Zbl 1058.34002
[11] Daftardar-Gejji, V.; Jafari, H., Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328, 1026-1033, (2007) · Zbl 1115.34006
[12] Deng, W., Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math., 206, 174-188, (2007) · Zbl 1121.65128
[13] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248, (2002) · Zbl 1014.34003
[14] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3-22, (2002) · Zbl 1009.65049
[15] Franz, M.; Zhang, M., Suppression and creation of chaos in a periodically forced Lorenz system, Phys. Rev. E, 52, 4, 3558-3565, (1995)
[16] Gopal, R.; Venkatesan, A.; Lakshmanan, M., Applicability of 0-1 test for strange nonchaotic attractors, Chaos, 23, 023123, (2013) · Zbl 1331.37041
[17] Gottwald, G. A.; Melbourne, I., A new test for chaos in deterministic systems, Proc. R. Soc. London, Ser. A, 460, 603-611, (2004) · Zbl 1042.37060
[18] Gottwald, G. A.; Melbourne, I., Testing for chaos in deterministic systems with noise, Physica D, 212, 100-110, (2005) · Zbl 1097.37024
[19] Gottwald, G. A.; Melbourne, I., On the implementation of the 0-1 test for chaos, SIAM J. Appl. Dyn. Syst., 8, 129-145, (2009) · Zbl 1161.37054
[20] Gottwald, G. A.; Melbourne, I., On the validity of the 0-1 test for chaos, Nonlinearity, 22, 1367-1382, (2009) · Zbl 1171.65091
[21] Gottwald, G. A.; Melbourne, I.; Skokos, C.; Gottwald, G. A.; Laskar, J., The 0-1 test for chaos: A review, Chaos Detection and Predictability, (2016), Springer
[22] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 034101, (2003)
[23] Grigorenko, I.; Grigorenko, E., Erratum: Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 96, 199902, (2006)
[24] Hartley, T. T.; Lorenzo, C. F., Dynamics and control of initialized fractional-order systems, Nonlinear Dyn., 29, 201-233, (2002) · Zbl 1021.93019
[25] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst., 42, 485-490, (1995)
[26] Hilfer, R., Applications of Fractional Calculus in Physics, (2000), World Scientific Publishing Company: World Scientific Publishing Company, Singapore · Zbl 0998.26002
[27] Koeller, R. C., Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., 51, 299-307, (1984) · Zbl 0544.73052
[28] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, Soliton Fractals, 22, 549-554, (2004) · Zbl 1069.37025
[29] Li, C.; Yi, Q.; Chen, A., Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys., 316, 614-631, (2016) · Zbl 1349.65246
[30] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, (1993), John Wiley & Sons, Inc.: John Wiley & Sons, Inc., New York · Zbl 0789.26002
[31] Mustapha, K., An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements, IMA J. Numer. Anal., 31, 719-739, (2011) · Zbl 1219.65091
[32] Mustapha, K.; AIMutawa, J., A finite difference method for an anomalous sub-diffusion equation, theory and applications, Numer. Algorithms, 61, 525-543, (2012) · Zbl 1263.65082
[33] Parker, T. S.; Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, (1989), Springer-Verlag: Springer-Verlag, New York · Zbl 0692.58001
[34] Podlubny, I., Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, (1999), Academic Press: Academic Press, New York · Zbl 0924.34008
[35] Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 397-398, (1976) · Zbl 1371.37062
[36] Saravanan, R.; Narayan, O.; Banerjee, K.; Bhattacharjee, J. K., Chaos in a periodically forced Lorenz system, Phys. Rev. A, 31, 520-522, (1985)
[37] Sataev, E. A., Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Sb. Math., 196, 561-594, (2005) · Zbl 1101.37022
[38] Schmalfuss, B., Attractors for nonautonomous and random dynamical systems perturbed by impulses, Discrete Contin. Dyn. Syst., 9, 727-744, (2003) · Zbl 1029.37030
[39] Sparrow, C., The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, (1982), Springer-Verlag: Springer-Verlag, New York · Zbl 0504.58001
[40] Čermák, J.; Nechvátal, L., The Routh-Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system, Nonlinear Dyn., 87, 939-954, (2017) · Zbl 1372.34010
[41] Yuan, L.; Yang, Q.; Zeng, C., Chaos detection and parameter identification in fractional-order chaotic systems with delay, Nonlinear Dyn., 73, 439-448, (2013) · Zbl 1281.93037
[42] Zhang, X., Dynamics of a class of nonautonomous Lorenz-type systems, Int. J. Bifurcation Chaos, 26, 1650208, (2016) · Zbl 1352.34017
[43] Zhang, Y.; Sun, Z.; Liao, H., Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys., 265, 195-210, (2014) · Zbl 1349.65359
[44] Zhao, Q.; Zhou, S.; Li, X., Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation, Commun. Nonlinear Sci. Numer. Simul., 13, 928-938, (2008) · Zbl 1221.37073
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