×

zbMATH — the first resource for mathematics

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}

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.