Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium.(English)Zbl 1276.34043

Authors’ abstract: By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attactor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors according to detailed numerical as well as theoretical analysis. The analytic results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov’s stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronization between the Sprott E system and the original one. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

MSC:

 34D06 Synchronization of solutions to ordinary differential equations 34H10 Chaos control for problems involving ordinary differential equations 34H20 Bifurcation control of ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 34D08 Characteristic and Lyapunov exponents of ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 93C40 Adaptive control/observation systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text:

References:

 [1] Chen, G. R., Ueta, T.: Yet another chaotic attractor. Internat. J. Bifur. Chaos 9 (1999), 1465-1466. · Zbl 0962.37013 [2] Lorenz, E. N.: Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130-141. DOI 10.1175/1520-0469(1963)0202.0.CO;2 [3] Lü, J. H., Chen, G. R.: A new chaotic attractor coined. Internat. J. Bifur. Chaos 12 (2002), 659-661. · Zbl 1063.34510 [4] Lü, J. H., Chen, G. R., Cheng, D. Z.: A new chaotic system and beyond: The generalized Lorenz-like system. Internat. J. Bifur. Chaos 14 (2004), 1507-1537. · Zbl 1129.37323 [5] Lü, J. H., Han, F. L., Yu, X. H., Chen, G. R.: Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method. Automatica 40 (2004), 1677-1687. · Zbl 1162.93353 [6] Lü, J. H., Yu, S. M., Leung, H., Chen, G. R.: Experimental verification of multidirectional multiscroll chaotic attractors. IEEE Trans. Circuits Systems C I: Regular Papers 53 (2006), 149-165. [7] Liu, Y. J., Pang, G. P.: The basin of attraction of the Liu system. Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 2065-2071. · Zbl 1221.34145 [8] Li, J. M.: Limit cycles bifurcated from a reversible quadratic center. Qualit. Theory Dynam. Systems 6 (2005), 205-215. · Zbl 1142.34019 [9] Rössler, O. E.: An equation for continuious chaos. Phys. Lett. A 57 (1976), 397-398. · Zbl 1371.37062 [10] Silva, C. P.: Shilnikov’s theorem - A tutorial. IEEE Trans. Circuits Syst. I 40 (1993), 657-682. · Zbl 0850.93352 [11] Sprott, J. C.: Some simple chaotic flows. Phys. Rev. E 50 (1994), 647-650. [12] Sprott, J. C.: A new class of chaotic circuit. Phys. Lett. A 266 (2000), 19-23. [13] Sprott, J. C.: Simplest dissipative chaotic flow. Phys. Lett. A 228 (1997), 271-274. · Zbl 1043.37504 [14] Shilnikov, L. P.: A case of the existence of a countable number of periodic motions. Soviet Math. Dokl. 6 (1965), 163-166. · Zbl 0136.08202 [15] Shilnikov, L. P.: A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type. Math. USSR-Sb. 10 (1970), 91-102. · Zbl 0216.11201 [16] Shaw, R.: Strange attractor, chaotic behaviour and information flow. Z. Naturforsch. 36A (1981), 80-112. · Zbl 0599.58033 [17] Schrier, G. V., Maas, L. R. M.: The difusionless Lorenz equations: Shilnikov bifurcations and reduction to an explicit map. Physica D 141 (2000), 19-36. · Zbl 0956.37038 [18] Vaněček, A., Čelikovský, S.: Control Systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, London 1996. · Zbl 0874.93006 [19] Wang, X., Chen, G. R.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1264-1272. · Zbl 06048895 [20] Wang, Z.: Existence of attractor and control of a 3D differential system. Nonlinear Dynamics 60(3) (2010), 369-373. · Zbl 1189.70103 [21] Wei, Z. C.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376 (2011), 248-253. · Zbl 1255.37013 [22] Yang, Q. G., Chen, G. R., Huang, K. F.: Chaotic attractors of the conjugate Lorenz-type system. Internat. J. Bifur. Chaos 17 (2007), 3929-3949. · Zbl 1149.37308 [23] Yang, Q. G., Chen, G. R.: A chaotic system with one saddle and two stable node-foci. Internat. J. Bifur. Chaos 18 (2008), 1393-1414. · Zbl 1147.34306 [24] Yang, Q. G., Wei, Z. C., Chen, G. R.: A unusual 3D autonomons quadratic chaotic system with two stable node-foci. Internat. J. Bifur. Chaos 20 (2010), 1061-1083. · Zbl 1193.34091 [25] Yu, S. M., Lü, J. H., Yu, X. H.: Design and implementation of grid multiwing hyperchaotic Lorenz system family via switching control and constructing super-heteroclinic loops. IEEE Trans. Circuits Systems C I: Regular Papers 59 (2012), 1015-1028. [26] Zhao, L. Q., Wang, Q.: Perturbations from a kind of quartic hamiltonians under general cubic polynomials. Sci. China Series A: Mathematics 52 (2009), 427-442. · Zbl 1192.34044
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.