Geodesic
Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium
Kybernetika, Tome 49 (2013) no. 2, pp. 359-374 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis 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 stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.
By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis 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 stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.
Classification : 34D06 34H10 34H20 34C28 34D08 34C23 93C40 37D45, 34H10, 34H20, 93C40
Keywords: chaotic attractors; stable equilibrium; Shilnikov theorem; Lyapunov exponent; synchronization 
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     title = {Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium},
     journal = {Kybernetika},
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     url = {http://geodesic.mathdoc.fr/item/KYB_2013_49_2_a9/}
}
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Wei, Zhouchao; Wang, Zhen. Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium. Kybernetika, Tome 49 (2013) no. 2, pp. 359-374. http://geodesic.mathdoc.fr/item/KYB_2013_49_2_a9/

[1] Chen, G. R., Ueta, T.: Yet another chaotic attractor. Internat. J. Bifur. Chaos 9 (1999), 1465-1466. | DOI | MR | Zbl

[2] Lorenz, E. N.: Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130-141. | DOI

[3] Lü, J. H., Chen, G. R.: A new chaotic attractor coined. Internat. J. Bifur. Chaos 12 (2002), 659-661. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI

[7] Liu, Y. J., Pang, G. P.: The basin of attraction of the Liu system. Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 2065-2071. | DOI | MR | Zbl

[8] Li, J. M.: Limit cycles bifurcated from a reversible quadratic center. Qualit. Theory Dynam. Systems 6 (2005), 205-215. | DOI | MR | Zbl

[9] Rössler, O. E.: An equation for continuious chaos. Phys. Lett. A 57 (1976), 397-398. | DOI

[10] Silva, C. P.: Shilnikov's theorem - A tutorial. IEEE Trans. Circuits Syst. I 40 (1993), 657-682. | DOI | MR

[11] Sprott, J. C.: Some simple chaotic flows. Phys. Rev. E 50 (1994), 647-650. | DOI | MR

[12] Sprott, J. C.: A new class of chaotic circuit. Phys. Lett. A 266 (2000), 19-23. | DOI

[13] Sprott, J. C.: Simplest dissipative chaotic flow. Phys. Lett. A 228 (1997), 271-274. | DOI | MR | Zbl

[14] Shilnikov, L. P.: A case of the existence of a countable number of periodic motions. Soviet Math. Dokl. 6 (1965), 163-166.

[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. | DOI

[16] Shaw, R.: Strange attractor, chaotic behaviour and information flow. Z. Naturforsch. 36A (1981), 80-112. | MR

[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. | MR

[18] Vaněček, A., Čelikovský, S.: Control Systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, London 1996. | Zbl

[19] Wang, X., Chen, G. R.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1264-1272. | DOI | MR

[20] Wang, Z.: Existence of attractor and control of a 3D differential system. Nonlinear Dynamics 60(3) (2010), 369-373. | DOI | MR | Zbl

[21] Wei, Z. C.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376 (2011), 248-253. | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR

[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. | DOI | MR

[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. | DOI | MR | Zbl