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Dynamics of a Class of Nonautonomous Lorenz-Type Systems

Xu Zhang

Department of Mathematics, Shandong University, Weihai, Shandong 264209, P. R. China

https://doi.org/10.1142/S0218127416502084Cited by:12 (Source: Crossref)

Abstract

The dynamical properties of a kind of Lorenz-type systems with time-varying parameters are studied. The time-varying ultimate bounds are estimated, and a simple method is provided to generate strange attractors, including both the strange nonchaotic attractors (SNAs) and chaotic attractors. The approach is: (i) take an autonomous system with an ultimate bound from the Lorenz family; (ii) add at least two time-varying parameters with incommensurate frequencies satisfying certain conditions; (iii) choose the proper initial position and time by numerical simulations. Three interesting examples are given to illustrate this method with computer simulations. The first one is derived from the classical Lorenz model, and generates an SNA. The second one generates an SNA or a Lorenz-like attractor. The third one exhibits the coexistence of two SNAs and a Lorenz-like attractor. The nonautonomous Lorenz-type systems present more realistic models, which provide further understanding and applications of the numerical analysis in weather and climate predication, synchronization, and other fields.

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References

Anishchenko, V. S., Vadivasova, T. E. & Sosnovtseva, O. [1996] “ Strange nonchaotic attractors in autonomous and periodically driven systems,” Phys. Rev. E 54, 3231–3234. Crossref, Web of Science, Google Scholar
Anishchenko, V. S., Vadivasova, T. E. & Sosnovtseva, O. [1997] “ Reply to “comment on ‘strange nonchaotic attractors in autonomous and periodically driven systems,” Phys. Rev. E 56, 7322–7322. Crossref, Web of Science, Google Scholar
Cheban, D. N., Kloeden, P. E. & Schmalfuss, B. [2002] “ The relationship between pullback, forward and global attractors of nonautonomous dynamical systems,” Nonlin. Dyn. Syst. Th. 2, 125–144. Google Scholar
Cheban, D. & Duan, J. [2004] “ Recurrent motions and global attractors of nonautonomous Lorenz systems,” Dyn. Syst. 19, 41–59. Crossref, Web of Science, Google Scholar
Chen, G. & Ueta, T. [1999] “ Yet another chaotic attractor,” Int. J. Bifurcation and Chaos 9, 1465–1466. Link, Web of Science, Google Scholar
Chen, G. & Lü, J. [2003] Dynamics of the Lorenz System Family: Analysis, Control and Synchronization (in Chinese) (Science Press, Beijing). Google Scholar
Chepyzhov, V. V. [2013] “ Uniform attractors of dynamical processes and non-autonomous equations of mathematical physics,” Russ. Math. Surv. 68, 159–196. Crossref, Web of Science, Google Scholar
Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “ The double scroll family,” IEEE Trans. Circuits Syst. 33, 1072–1118. Crossref, Google Scholar
Feudel, U., Kurths, J. & Pikovsky, A. S. [1995] “ Strange non-chaotic attractor in a quasiperiodically forced circle map,” Physica D 88, 176–186. Crossref, Web of Science, Google Scholar
Franz, M. & Zhang, M. [1995] “ Suppression and creation of chaos in a periodically forced Lorenz system,” Phys. Rev. E 52, 3558–3565. Crossref, Web of Science, Google Scholar
Grebogi, C., Ott, E., Pelikan, S. & Yorke, J. A. [1984] “ Strange attractors that are not chaotic,” Physica D 13, 261–268. Crossref, Web of Science, Google Scholar
Heagy, J. F. & Ditto, W. L. [1991] “ Dynamics of a two-frequency parametrically driven Duffing oscillator,” J. Nonlin. Sci. 1, 423–455. Crossref, Google Scholar
Heagy, J. F. & Hammel, S. M. [1994] “ The birth of strange nonchaotic attractors,” Physica D 70, 140–153. Crossref, Web of Science, Google Scholar
Kuznetsov, S. P., Pikovsky, A. S. & Feudel, U. [1995] “ Birth of a strange nonchaotic attractor: A renormalization group analysis,” Phys. Rev. E 51, 1629–1632. Crossref, Web of Science, Google Scholar
Leonov, G., Bunin, A. & Koksch, N. [1987] “ Attractor localization of the Lorenz system,” Z. Angew. Math. Mech. 67, 649–656. Crossref, Web of Science, Google Scholar
Li, D., Lu, J., Wu, X. & Chen, G. [2005] “ Estimating the bounds for the Lorenz family of chaotic systems,” Chaos Solit. Fract. 23, 529–534. Crossref, Web of Science, Google Scholar
Li, D., Lu, J., Wu, X. & Chen, G. [2006] “ Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,” J. Math. Anal. Appl. 323, 844–853. Crossref, Web of Science, Google Scholar
Li, D., Wu, X. & Lu, J. [2009] “ Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system,” Chaos Solit. Fract. 39, 1290–1296. Crossref, Web of Science, Google Scholar
Liao, X. [2004] “ On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization,” Sci. China Ser. E 34, 1404–1419. Google Scholar
Lorenz, E. [1963] “ Deterministic non-periodic flow,” J. Atmos. Sci. 20, 130–141. Crossref, Web of Science, Google Scholar
Nishikawa, T. & Kaneko, K. [1996] “ Fractalization of torus revisited as a strange nonchaotic attractor,” Phys. Rev. E 54, 6114–6124. Crossref, Web of Science, Google Scholar
Parker, T. S. & Chua, L. O. [1989] Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, NY). Crossref, Google Scholar
Pikovsky, A. & Feudel, U. [1997] “ Comment on ‘Strange nonchaotic attractors in autonomous and periodically driven systems’,” Phys. Rev. E 56, 7320–7321. Crossref, Web of Science, Google Scholar
Pogromsky, A. Y., Santoboni, G. & Nijmeijer, H. [2003] “ An ultimate bound on the trajectories of the Lorenz system and its applications,” Nonlinearity 16, 1597–1605. Crossref, Web of Science, Google Scholar
Prasad, A., Negi, S. S. & Ramaswamy, R. [2001] “ Strange nonchaotic attractors,” Int. J. Bifurcation and Chaos 11, 291–309. Link, Web of Science, Google Scholar
Rössler, O. E. [1976] “ An equation for continuous chaos,” Phys. Lett. A 57, 397–398. Crossref, Web of Science, Google Scholar
Sandri, M. [1996] “ Numerical calculation of Lyapunov exponents,” Mathematica J. 6, 78–84. Google Scholar
Saravanan, R., Narayan, O., Banerjee, K. & Bhattacharjee, J. K. [1985] “ Chaos in a periodically forced Lorenz system,” Phys. Rev. A 31, 520–522. Crossref, Web of Science, Google Scholar
Sataev, E. A. [2005] “ Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type,” Sbornik Math. 196, 99–134. Crossref, Web of Science, Google Scholar
Schmalfuss, B. [2003] “ Attractors for nonautonomous and random dynamical systems perturbed by impulses,” Discr. Contin. Dyn. Syst. 9, 727–744. Crossref, Web of Science, Google Scholar
Shi, P., He, D., Fu, W., Kang, W. & Hu, G. [2001] “ Strange-nonchaotic-attractor-like behaviors in coupled map systems,” Commun. Theor. Phys. 35, 389–394. Crossref, Web of Science, Google Scholar
Sparrow, C. [1982] The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer-Verlag, NY). Crossref, Google Scholar
Wang, X., Zhan, M., Lai, C. & Lai, Y. [2004] “ Strange nonchaotic attractors in random dynamical systems,” Phys. Rev. Lett. 92, 741021–741024. Crossref, Web of Science, Google Scholar
Yalçınkaya, T. & Lai, Y. [1997] “ Bifurcation to strange nonchaotic attractors,” Phys. Rev. E 56, 1623–1630. Crossref, Web of Science, Google Scholar
Yang, T. & Bilimgut, K. [1997] “ Experimental results of strange nonchaotic phenomenon in a second order quasi-periodically forced electronic circuit,” Phys. Lett. A 236, 494–504. Crossref, Web of Science, Google Scholar
Zhang, F. & Zhang, G. [2015] “ Further results on ultimate bound on the trajectories of the Lorenz system,” Qual. Th. Dyn. Syst. 15, 221–235. Crossref, Web of Science, Google Scholar
Zhang, X. [2016] “A simple planar nonautonomous system with infinitely many transitive components,” preprint. Google Scholar
Zhao, Q., Zhou, S. & Li, X. [2008] “ Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation,” Commun. Nonlin. Sci. Numer. Simulat. 13, 928–938. Crossref, Web of Science, Google Scholar
Zhou, T., Moss, F. & Bulsara, A. [1992] “ Observation of strange nonchaotic attractor in a multistable potential,” Phys. Rev. A 45, 5394–5400. Crossref, Web of Science, Google Scholar