Kybernetika 53 no. 2, 354-369, 2017

Periodic parametric perturbation control for a 3D autonomous chaotic system and its dynamics at infinity

Zhen Wang, Wei Sun, Zhouchao Wei and Shanwen ZhangDOI: 10.14736/kyb-2017-2-0354

Abstract:

Periodic parametric perturbation control and dynamics at infinity for a 3D autonomous quadratic chaotic system are studied in this paper. Using the Melnikov's method, the existence of homoclinic orbits, oscillating periodic orbits and rotating periodic orbits are discussed after transferring the 3D autonomous chaotic system to a slowly varying oscillator. Moreover, the parameter bifurcation conditions of these orbits are obtained. In order to study the global structure, the dynamics at infinity of this system are analyzed through Poincaré compactification. The simulation results demonstrate feasibility of periodic parametric perturbation control technology and correctness of the theoretical results.

Keywords:

Hamiltonian system, homoclinic orbits, Melnikov's methods, periodic orbits, periodic parametric perturbation, dynamics at infinity

Classification:

34H10, 34D20, 34H20

References:

  1. 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
  2. S. Čelikovský and A. Vaněček: Bilinear systems and chaos. Kybernetika 30 (1994), 403-424.   CrossRef
  3. F. Dumortier, J. Llibre and J. C. Artes: Qualitative Theory of Planar Differential Systems. Springer, Berlin 2006.   CrossRef
  4. Y. Y. Fang, Z. Y. Xu and C. H. Cai: Melnikov analysis of feedback control of chaotic dynamics system. J. Wuxi University of Light Industry 20 (2001), 624-629.   CrossRef
  5. J. Guckenheimer and P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin 2002.   DOI:10.1007/978-1-4612-1140-2
  6. S. Jafari and J. C. Sprott: Simple chaotic flows with a line equilibrium. Chaos, Solitons and Fractals 57 (2013), 79-84.   DOI:10.1016/j.chaos.2013.08.018
  7. 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.   DOI:10.1016/j.cnsns.2009.06.027
  8. J. B. Li, X. H. Zhao and Z. R. Liu: Theory of Generalized Hamiltonian System and its Applications. Science Press, Beijing 2007.   CrossRef
  9. Y. Li, X. Q. Wu, J. A. Lu and J. H. Lü: Synchronizability of duplex networks. IEEE Trans. Circuits and Systems II 63 (2016), 206-210.   DOI:10.1109/tcsii.2015.2468924
  10. K. X. Liu, L. L. Wu, J. H. Lü and H. H. Zhu: Finite-time adaptive consensus of a class of multi-agent systems. Science China-Technological Sciences 59 (2016), 22-32.   DOI:10.1007/s11431-015-5989-7
  11. Y. J. Liu: Analysis of global dynamics in an unusual 3D chaotic system. Nonlinear Dyn. 70 (2012), 2203-2212.   DOI:10.1007/s11071-012-0610-0
  12. Z. R. Liu: Perturbation Criteria for Chaos. Shanghai Scientific and Technological Education Publishing House, Shanghai 1994.   CrossRef
  13. E. N. Lorenz: Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1963), 130-141.   DOI:10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2
  14. J. H. Lü and G. R. Chen: A new chaotic attractor coined. Int. J. Bifurcation and Chaos 12 (2002), 659-661.   DOI:10.1142/s0218127402004620
  15. K. A. Mirus and J. C. Sprott: Controlling chaos in a high dimensional systems with periodic parametric perturbations. Phys. Lett. A 254 (1999), 275-278.   DOI:10.1016/s0375-9601(99)00068-7
  16. K. A. Mirus and J. C. Sprott: Controlling chaos in low- and high-dimensional systems with periodic parametric perturbations. Phys. Rev. E 59 (1999), 5313-5324.   DOI:10.1103/physreve.59.5313
  17. C. W. Shen, S. M. Yu, and G. R. Chen: Constructing hyperchaotic systems at will. Int. J. Circuit Theory Appl. 43 (2015), 2039-2056.   DOI:10.1002/cta.2062
  18. J. C. Sprott: Some simple chaotic flows. Phys. Rev. E {\mi50} (1994), 647-650.   DOI:10.1103/physreve.50.r647
  19. S. L. Tan, J. H. Lü and D. J. Hill: Towards a theoretical framework for analysis and intervention of random drift on general networks. IEEE Trans. Automat. Control 60 (2015), 576-581.   DOI:10.1109/tac.2014.2329235
  20. G. Tigan: Analysis of a dynamical system derived from the Lorenz system. Scientific Bull. Politehnica University of Timisoara 50 (2005), 61-72.   CrossRef
  21. Q. X. Wang, S. M. Yu, C. Q. Li, J. H. Lü, X. L. Fang and J. M. Bahi: Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits and Systems I 63 (2016), 401-412.   DOI:10.1109/tcsi.2016.2515398
  22. X. Wang and G. R. Chen: Constructing a chaotic system with any number of equilibria. Nonlinear Dynamics 71 (2013), 429-436.   DOI:10.1007/s11071-012-0669-7
  23. Z. Wang: Existence of attractor and control of a 3D differential system. Nonlinear Dynmics 60 (2010), 369-373.   DOI:10.1007/s11071-009-9601-1
  24. Z. Wang: Passivity control of nonlinear electromechanical transducer chaotic system. Control Theory Appl. 28 (2011), 1036-1040.   CrossRef
  25. 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
  26. Z. Wang, W. Sun, Z. C. Wei and X. J. Xi: Dynamics analysis and robust modified function projective synchronization of Sprott E system with quadratic perturbation. Kybernetika 50 (2014), 616-631.   DOI:10.14736/kyb-2014-4-0616
  27. Z. Wang, Z. C. Wei, X. J. Xi and Y. X. Li: Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface. Nonlinear Dynamics 77 (2014), 1503-1518.   DOI:10.1007/s11071-014-1395-0
  28. Z. C. Wei and Q. G. Yang: Controlling the diffusionless Lorenz equations with periodic parametric perturbation. Comput. Math. Appl. 58 (2009), 1979-1987.   DOI:10.1016/j.camwa.2009.07.058
  29. Z. C. Wei, W. Zhang, Z. Wang and M. H. Yao: Hidden attractors and dynamical behaviors in an extended Rikitake system. Int. J. Bifurcation and Chaos 22 (2015), 1550028.   DOI:10.1142/s0218127415500285
  30. Z. M. Wu, J. Y. Xie, Y. Y. Fang and Z. Y. Xu: Controlling chaos with periodic parametric perturbations in Lorenz system. Chaos Solitons and Fractals 32 (2007), 104-112.   DOI:10.1016/j.chaos.2005.10.060
  31. S. Wiggins and P. Holmes: Homiclinic orbits in slowly varying oscillators. SIAM J. Math. Anal. 18 (1987), 612-629.   DOI:10.1137/0518047
  32. S. Wiggins and P. Holmes: Periodic orbits in slowly varying oscillators. SIAM J. Math. Anal. 18 (1987), 592-611.   DOI:10.1137/0518046
  33. 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.   DOI:10.1142/s0218127408021063