Kybernetika 50 no. 4, 616-631, 2014

Dynamics analysis and robust modified function projective synchronization of Sprott E system with quadratic perturbation

Zhen Wang, Wei Sun, Zhouchao Wei and Xiaojian XiDOI: 10.14736/kyb-2014-4-0616


Hopf bifurcation, dynamics at infinity and robust modified function projective synchronization (RMFPS) problem for Sprott E system with quadratic perturbation were studied in this paper. By using the method of projection for center manifold computation, the subcritical and the supercritical Hopf bifurcation were analyzed and obtained. Then, in accordance with the Poincare compactification of polynomial vector field in $R^3$, the dynamical behaviors at infinity were described completely. Moreover, a RMFPS scheme of this special system was proposed and proved based on Lyapunov direct method. The simulation results demonstrate the correctness of the dynamics analysis and the effectiveness of the proposed synchronization strategy.


Hopf bifurcation, center manifold theorem, Poincare compactification, robust modified function projective synchronization, chaotic systems


34H10, 34H20, 93C15


  1. G. R. Chen and T. Ueta: Yet another chaotic attractor. Int. J. Bifur. Chaos 9 (1999), 1465-1466.   CrossRef
  2. Y. Chen, L. Cao and M. Sun: Robust midified function projective synchronization in network with unknown parameters and mismatch parameters. Int. J. Nonlinear Sci. 10 (2010), 17-23.   CrossRef
  3. Y. Chen, J. H. Lü and Z. L. Lin: Consensus of discrete-time multi-agent systems with transmission nonlinearity. Automatica 49 (2013), 1768-1775.   CrossRef
  4. Y. Chen, J. H. Lü, X. H. Yu and D. J. Hill: Multi-agent systems with dynamical topologies: Consensus and applications. IEEE Circuits Syst. Magazine 13 (2013), 21-34.   CrossRef
  5. Y. Chen, J. H. Lü, X. H. Yu and Z. L. Lin: Consensus of discrete-time second order multi-agent systems based on infinite products of general stochastic matrices. SIAM J. Control Optim. 51 (2013), 3274-3301.   CrossRef
  6. A. Cima and J. Llibre: Bounded polynomial vector field. T. Amer. Math. Soc. 318 (1990), 557-579.   CrossRef
  7. F. Dumortier, J. Llibre and J. C. Artes: Qualitative Theory of Planar Differential Systems. Springer, Berlin 2006.   CrossRef
  8. A. P. Kuznetsov, S. P. Kuznetsov and N. V. Stankevich: A simple autonomous quasiperiodic self-oscillator. Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 1676-1681.   CrossRef
  9. Y. A. Kuznetsov: Elements of Applied Bifurcation Theory. Springer-Verlag, New York 1998.   CrossRef
  10. T. H. Lee and J. H. Park: Adaptive functional projective lag synchronization of a hyperchaotic Rössler system. Chin. Phys. Lett. 26 (2009), 090507.   CrossRef
  11. G. A. Leonov and N. V. Kuznetsov: Prediction of hidden oscillations existence in nonlinear dynamical systems: analytics and simulation. Adv. Intelligent Syst. Comput. 210 (2013), 5-13.   CrossRef
  12. G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev: Localization of hidden Chua's attractors. Phys. Lett. A 375 (2011), 2230-2233.   CrossRef
  13. H. T. Liang, Z. Wang, Z. M. Yue and R. H. Lu: Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication. Kybernetika 48 (2012), 190-205.   CrossRef
  14. Y. J. Liu: Analysis of global dynamics in an unusual 3D chaotic system. Nonlinear Dyn. 70 (2012), 2203-2212.   CrossRef
  15. Y. J. Liu and Q. G. Yang: Dynamics of the Lü system on the invariant algebraic surface and at infinity. Int. J. Bifur. Chaos 21 (2011), 2559-2582.   CrossRef
  16. E. N. Lorenz: Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130-141.   CrossRef
  17. M. Messias and M. R. Gouveia: Dynamics at infinity and other global dynamical aspects of Shimizu-Morioka equations. Nonlinear Dyn. 69 (2012), 577-587.   CrossRef
  18. C. W. Shen, S. M. Yu, J. H. Lü and G. R. Chen: A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation. IEEE Trans. Circuits-I 61 (2014), 854-864.   CrossRef
  19. J. C. Sprott: Some simple chaotic flows. Phys. Rev. E {\mi50} (1994), 647-650.   CrossRef
  20. K. S. Sudheer and M. Sabir: Adaptive modiffied function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters. Phys. Lett. A 373 (2009), 3743-3748.   CrossRef
  21. A. Vaněček and S. Čelikovský: Control systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, London 1996.   CrossRef
  22. X. Wang and G. R. Chen: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17 (2012). 1264-1272.   CrossRef
  23. Z. Wang: Existence of attractor and control of a 3D differential system. Nonlinear Dyn. 60 (2010), 369-373.   CrossRef
  24. Z. Wang, Y. X. Li, X. J. Xi and L. Lü: Heteoclinic orbit and backstepping control of a 3D chaotic system. Acta Phys. Sin. 60 (2011), 010513.   CrossRef
  25. Z. C. Wei and Z. Wang: Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium. Kybernetika 49 (2013), 359-374.   CrossRef
  26. Q. G. Yang and G. R. Chen: A chaotic system with one saddle and two stable node-foci. Int. J. Bifur. Chaos {\mi18} (2008), 1393-1414.   CrossRef