Kybernetika 54 no. 4, 648-663, 2018

Dependence of hidden attractors on non-linearity and Hamilton energy in a class of chaotic system

Ge Zhang, Chunni Wang, Ahmed Alsaedi, Jun Ma and Guodong RenDOI: 10.14736/kyb-2018-4-0648


Non-linearity is essential for occurrence of chaos in dynamical system. The size of phase space and formation of attractors are much dependent on the setting of nonlinear function and parameters. In this paper, a three-variable dynamical system is controlled by different nonlinear function thus a class of chaotic system is presented, the Hamilton function is calculated to find the statistical dynamical property of the improved dynamical systems composed of hidden attractors. The standard dynamical analysis is confirmed in numerical studies, and the dependence of attractors and Hamilton energy on non-linearity selection is discussed. It is found that lower average Hamilton energy can be detected when intensity of nonlinear function is enhanced. It indicates that non-linearity can decrease the energy cost triggering for dynamical behaviors.


chaos, Helmholtz theorem, hidden attractor, bifurcation, Hamilton energy


37B25, 37L30


  1. W. M. Ahmad and J. C. Sprott: Chaos in fractional-order autonomous nonlinear systems. Chaos 6 (2003), 339-351.   DOI:10.1016/s0960-0779(02)00438-1
  2. K. Aihara, T. Takabe and M. Toyoda: Chaotic neural networks. Phys. Lett. A 144 (2001), 333-340.   DOI:10.1016/0375-9601(90)90136-c
  3. Z. Aram, S. Jafari and J. Ma et al.: Using chaotic artificial neural networks to model memory in the brain. Commun. Nonlinear Sci. Numer. Simulat. 44 (2017), 449-459.   DOI:10.1016/j.cnsns.2016.08.025
  4. B. C. Bao, J. P. Xu and Z. Liu: Initial state dependent dynamical behaviors in a Memristor based chaotic circuit. Chinese Phys. Lett. 27 (2010), 070504.   DOI:10.1088/0256-307x/27/7/070504
  5. K. Barati, S. Jafari and J. C. Sprott et al.: Simple chaotic flows with a curve of equilibria. Int. J. Bifurcat. Chaos 26 (2016), 1630034.   DOI:10.1142/s0218127416300342
  6. J. Barrow-Green: Poincaré and the three body problem. Amer. Math. Soc. 2 (1997).   CrossRef
  7. L. Chua: Memristor-the missing circuit element. IEEE Trans. Circ. Theory 18 (1971), 507-519.   DOI:10.1109/tct.1971.1083337
  8. D. Dantsev: A Novel type of chaotic attractor for quadratic systems without equilibriums. Int. J. Bifurcat. Chaos 12 (2002), 659-661.   DOI:10.1142/s0218127402004620
  9. W. L. Ditto, S. N. Rauseo and M. L. Spano: Experimental control of chaos. Phys. Rev. Lett. 65 (1991), 3211-3214.   DOI:10.1103/physrevlett.65.3211
  10. D. Dudkowski, S. Jafari and T. Kapitaniak et al.: Hidden attractors in dynamical systems. Phys. Rep. 637 (2016), 1-50.   DOI:10.1016/j.physrep.2016.05.002
  11. I. V. Ermakov, S. T. Kingni and V. Z. Tronciu et al.: Chaotic semiconductor ring lasers subject to optical feedback: Applications to chaos-based communications. Optics Commun. 286 (2013), 265-272.   DOI:10.1016/j.optcom.2012.08.063
  12. M. J. Feigenbaum: The onset spectrum of turbulence. Phys. Lett. A 74 (1979), 375-378.   DOI:10.1016/0375-9601(79)90227-5
  13. A. Garfinkel, M. L. Spano and W. L. Ditto et al.: Controlling cardiac chaos. Science 257 (1992), 1230-1235.   DOI:10.1126/science.1519060
  14. T. Gotthans and J. Petržela: New class of chaotic systems with circular equilibrium. Nonlinear Dyn. 81 (2015), 1141-1149.   DOI:10.1007/s11071-015-2056-7
  15. T. Gotthans, J. C. Sprott and J. Petržela: Simple Chaotic Flow with Circle and Square Equilibrium. Int. J. Bifurcat. Chaos 26 (2016), 1650137.   DOI:10.1142/s0218127416501376
  16. Y. L. Guo, G. Y. Qi and Y. Hamam: A multi-wing spherical chaotic system using fractal process. Nonlinear Dyn. 85 (2016), 2765-2775.   DOI:10.1007/s11071-016-2861-7
  17. X. Hu, C. Liu and L. Liu et al.: Multi-scroll hidden attractors in improved Sprott A system. Nonlinear Dyn. 86 (2016), 1725-1734.   DOI:10.1007/s11071-016-2989-5
  18. M. Itoh and L. O. Chua: Memristor oscillators. Int. J. Bifurcat. Chaos 8 (2008), 3183-3206.   DOI:10.1142/s0218127408022354
  19. M. A. Jafari, E. Mliki and A. Akgul et al.: Chameleon: the most hidden chaotic flow. Nonlinear Dyn. 88 (2017), 2303-2317.   DOI:10.1007/s11071-017-3378-4
  20. S. Jafari and J. C. Sprott: Simple chaotic flows with a line equilibrium. Chaos Solutons Fractals 57 (2013), 79-84.   DOI:10.1016/j.chaos.2013.08.018
  21. S. Jafari, J. C. Sprott and V. T. Pham et al.: Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dyn. 86 (2016), 1349-1358.   DOI:10.1007/s11071-016-2968-x
  22. B. Jia, H. G. Gu and L. Li et al.: Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns. Cogn. Neurodyn. 6 (2012), 89-106.   DOI:10.1007/s11571-011-9184-7
  23. M. P. Kennedy: Chaos in the Colpitts oscillator. IEEE Trans. Circ. Syst. I 41 (1994), 711-774.   DOI:10.1109/81.331536
  24. D. H. Kobe: Helmholtz's theorem revisited. Amer. J. Physics 54 (1986), 552-554.   DOI:10.1119/1.14562
  25. H. S. Kwok and W. K. S. Tang: A fast image encryption system based on chaotic maps with finite precision representation. Chaos Solitons Fractals 32 (2007), 1518-1529.   DOI:10.1016/j.chaos.2005.11.090
  26. G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev: Hidden attractor in smooth Chua systems. Physica D 241 (2012), 1482-1486.   DOI:10.1016/j.physd.2012.05.016
  27. G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev: Localization of hidden Chua${'}$s attractors. Phys. Lett. A 375 (2011), 2230-2233.   DOI:10.1016/j.physleta.2011.04.037
  28. G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev: Hidden attractor in smooth Chua systems. Physica D 241 (2012), 1482-1486.   DOI:10.1016/j.physd.2012.05.016
  29. G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev: Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity. Commun. Nonlinear Sci. Numer. Simulat. 28 (2015), 166-176.   DOI:10.1016/j.cnsns.2015.04.007
  30. C. Li, S. Li and M. Asim et al.: On the security defects of an image encryption scheme. Image Vision Computing 27 (2009), 1371-1381.   DOI:10.1016/j.imavis.2008.12.008
  31. Y. Y. Li, H. G. and Gu: The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns. Nonlinear Dyn. 87 (2017), 2541-2562.   DOI:10.1007/s11071-016-3210-6
  32. X. Li, C. Li and I. K. Lee: Chaotic image. 125 (2016), 48-63.    DOI:10.1016/j.sigpro.2015.11.017
  33. F. Li and C. G. Yao: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 84 (2016), 2305-2315.   DOI:10.1007/s11071-016-2646-z
  34. T. Y. Li and J. Y. Yorke: Period three implies Chaos. Amer. Math. Monthly 82 (1975), 985-992.   DOI:10.2307/2318254
  35. E. N. Lorenz: Deterministic nonperiodic flow. J. Atmospher. Sci. 20 (1963), 130-141.   DOI:10.1175/1520-0469(1963)020<0130:dnf>;2
  36. M. Lv and J. Ma: Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205 (2016), 375-381.   DOI:10.1016/j.neucom.2016.05.004
  37. M. Lv, C. Wang and G. Ren et al.: Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn. 85 (2016), 1479-1490.   DOI:10.1007/s11071-016-2773-6
  38. J. Ma, X. Y. Wu and R. T. Chu et al.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76 (2014), 1951-1962.   DOI:10.1007/s11071-014-1260-1
  39. J. Ma, A. B. Li and Z. S. Pu et al.: A time-varying hyperchaotic system and its realization in circuit. Nonlinear Dyn. 62 (2010), 535-541.   DOI:10.1007/s11071-010-9739-x
  40. J. Ma, L. Mi and P. Zhou et al.: Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl. Math. Comput. 307 (2017), 321-328.   DOI:10.1016/j.amc.2017.03.002
  41. J. Ma, X. L. Song and J. Tang et al.: Wave emitting and propagation induced by autapse in a forward feedback neuronal network. Neurocomputing 167 (2015), 378-389.   DOI:10.1016/j.neucom.2015.04.056
  42. J. Ma, F. Wu and W. Jin et al.: Calculation of Hamilton energy and control of dynamical systems with different types of attractors. Chaos 27 (2017), 481-495.   DOI:10.1063/1.4983469
  43. J. Ma, F. Q. Wu and G. D. Ren et al.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298 (2017), 65-76.   DOI:10.1016/j.amc.2016.11.004
  44. J. Ma, F. Wu and C. Wang: Synchronization behaviors of coupled neurons under electromagnetic radiation. Int. J. Mod Phys. B 31 (2017), 1650251.   DOI:10.1142/s0217979216502519
  45. J. Ma, A. H. Zhang and Y. F. Xia et al.: Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems. Appl. Math. Comput. 215 (2010), 3318-3326.   DOI:10.1016/j.amc.2009.10.020
  46. R. M. May: Simple mathematical models with very complicated dynamics. Nature 261 (1976), 459-467.   DOI:10.1038/261459a0
  47. M. Molaie, S. Jafari and J. C. Sprott et al.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurcat. Chaos (2013), 1350188.   DOI:10.1142/s0218127413501885
  48. B. Muthuswamy: Implementing memristor based chaotic circuits. Int. J. Bifurcat. Chaos 20 (2010), 1335-1350.   DOI:10.1142/s0218127410026514
  49. V- T. Pham, S. Jafari and X. Wang X et al.: A chaotic system with different shapes of equilibria. Int. J. Bifurcat. Chaos 26 (2016), 1650069.   DOI:10.1142/s0218127416500693
  50. V. T. Pham, C. Volos, S. Jafari et al. and : A Chaotic system with different families of hidden attractors. Int. J. Bifurcat. Chaos 26 (2016), 1650139.   DOI:10.1142/s021812741650139x
  51. J. R. Piper and J. C. Sprott: Simple autonomous chaotic circuit. IEEE Trans. Circ. Syst. II 57 (2010), 730-734.   DOI:10.1109/tcsii.2010.2058493
  52. G. Y. Qi and G. R. Chen: A spherical chaotic system. Nonlinear Dyn. 81 (2015), 1381-1392.   DOI:10.1007/s11071-015-2075-4
  53. G. D. Ren, Y. Xu and C. N. Wang: Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn. 88 (2017), 893-901.   DOI:10.1007/s11071-015-2075-4
  54. J. K. Ryeu, K. Aihara and I. Tsuda: Fractal encoding in a chaotic neural network. Phys. Rev. E 64 (2001), 046202.   DOI:10.1103/physreve.64.046202
  55. R. Shaw: The dripping faucet as a model chaotic system. Aerial Press, Santa Cruz 1984.   CrossRef
  56. X. L. Song, W. Y. Jin and J. Ma: Energy dependence on the electric activities of a neuron. Chinese Phys. B 24 (2015), 604-609.   DOI:10.1088/1674-1056/24/12/128710
  57. D. B. Strukov, G. S. Snider and D. R. Stewart et al.: The missing memristor found. Nature 453( 2008), 80-83.   DOI:10.1038/nature06932
  58. Z. H. Wang, S. J. Cang and E. O. Ochola et al.: A hyperchaotic system without equilibrium. Nonlinear Dyn. 69 (2012), 531-537.   DOI:10.1007/s11071-011-0284-z
  59. X. Wang and G. R. Chen: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71 (2013), 429-436.   DOI:10.1007/s11071-012-0669-7
  60. C. Wang, R. Chu and J. Ma: Controlling a chaotic resonator by means of dynamic track control. Complexity 21 (2015), 370-378.   DOI:10.1002/cplx.21572
  61. S. Wang, J. Kuang and J. Li et al.: Chaos-based secure communications in a large community. Phys. Rev. E 66 (2002), 065202.   DOI:10.1103/physreve.66.065202
  62. C. N. Wang, J. Ma and Y. Liu et al.: Chaos control, spiral wave formation, and the emergence of spatiotemporal chaos in networked Chua circuits. Nonlinear Dyn. 67 (2012), 139-146.   DOI:10.1007/s11071-011-9965-x
  63. C. N. Wang, Y. Wang and J. Ma: Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem. Acta Physica Sinica 65 (2016), 240501.   CrossRef
  64. A. Wolf, J. B. Swift and H. L. Swinney et al.: Determining Lyapunov exponents from a time series. Physica D 16 (1985), 285-317.   DOI:10.1016/0167-2789(85)90011-9
  65. C. W. Wu and L. O. Chua: A simple way to synchronize chaotic systems with applications to secure communication systems. Int. J. Bifurcat. Chaos 3 (1993), 1619-1627.   DOI:10.1142/s0218127493001288
  66. X. Y. Wu, J. Ma and L. H. Yuan et al.: Simulating electric activities of neurons by using PSPICE. Nonlinear Dyn. 75 (2014), 113-126.   DOI:10.1007/s11071-013-1053-y
  67. M. E Yalcin: Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions. Chaos Solutons Fractals 34 (2007), 1659-1666.   DOI:10.1016/j.chaos.2006.04.058
  68. T. Yang: A survey of chaotic secure communication systems. Int. J. Comput. Cogn. 2 (2004), 81-130.   CrossRef
  69. A. Zarei: Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors. Nonlinear Dyn. 81 (2015), 585-605.   DOI:10.1007/s11071-015-2013-5
  70. A. Zarei and S. Tavakoli: Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system. Appl. Math. Comput. 291 (2016), 323-339.   DOI:10.1016/j.amc.2016.07.023