Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 2995–3010 | Cite as

Dynamics analysis and Hamilton energy control of a generalized Lorenz system with hidden attractor

  • An Xin-lei
  • Zhang Li
Original Paper
  • 122 Downloads

Abstract

Hidden attractor can be found in some dynamic systems. More commonly, it can be excited by the stabilized equilibria, or be generated from the systems without equilibria. The generalized Lorenz system transformed from the Rabinovich system is researched by detecting the generating mechanism under different parameters and initial values, and then we have the good fortune to discover that the hidden attractor is coexisting with the states of stabilization, period, chaos, and even transient chaos. At the same time, the Hamilton energy function of the system is given to discuss the energy transform when the system undergoes a series of oscillations. The compositional principle can be used to design a new chaos control method, which is called Hamilton energy control. By numerical simulating, the feedback gain in the present control method is assigned and then controls the system with hidden attractor to expected states effectively. The feature of the control method can be indicated that the Hamilton energy can be detected during the oscillation control processes.

Keywords

Hidden attractor Lyapunov exponents Helmholtz’s theorem Hamilton Energy Energy control 

Notes

Acknowledgements

The authors gratefully acknowledge Prof. Jun Ma from Lanzhou University of Technology for the constructive suggestions. This work is supported from the National Natural Science Foundation (No. 51408288), China Postdoctoral Science Foundation(No. 2018M633649XB), Natural Science Foundation of Gansu Province, Government of China (No. 17JR5RA096), and Lzjtu (201606) EP support.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.

References

  1. 1.
    D’Onofrio, A., Manfredi, P.: Bifurcation thresholds in an SIR model with information-dependent vaccination. Math. Model. Nat. Phenom. 2, 26–43 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Healey, T.J., Dharmavaram, S.: Symmetry-breaking global bifurcation in a surface continuum phase-field model for lipid bilayer vesicles. Mathematics 11, 1554–1566 (2015)zbMATHGoogle Scholar
  3. 3.
    Ghergu, M., Ranulldulescu, V.: Bifurcation for a class of singular elliptic problems with quadratic convection term. Comptes R. Math. 338, 831–836 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ohno, W., Endo, T., Ueda, Y.: Extinction and intermittency of the chaotic attractor via crisis in phase-locked loop equation with periodic external forcing term. Electron. Commun. Jpn. 84, 52–61 (2015)CrossRefGoogle Scholar
  5. 5.
    Karsaklian, D.B.A., Akizawa, Y., Kanno, K.: Photonic integrated circuits unveil crisis-induced intermittency. Opt. Express 24, 22198–209 (2016)CrossRefGoogle Scholar
  6. 6.
    Qi, G.Y., Du, S.Z., Chen, G.R.: On a 4-dimensional chaotic system. Chaos Solitons Fractals 23, 1671–1682 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Qi, G.Y., Chen, G.R., Zhang, Y.H.: Analysis and circuit implementation of a new 4-D chaotic system. Phys. Lett. A 352, 386–397 (2006)CrossRefGoogle Scholar
  8. 8.
    Wei, J., Wei, S., Chu, Y.S.: Bifurcation and chaotic characteristics of helical gear system and parameter influences. J. Harbin Eng Univ. 34, 1301–1309 (2013)Google Scholar
  9. 9.
    Bouallegue, K., Chaari, A., Toumi, A.: Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal. Chaos Solitons Fractals 44, 79–85 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Yeniçeri, R., Yalçın, M.E.: Multi-scroll chaotic attractors from a generalized time-delay sampled-data system. Int. J. Circuit Theory Appl. 44, 1263–1276 (2016)CrossRefGoogle Scholar
  11. 11.
    Chen, L., Pan, W., Wu, R.: Design and implementation of grid multi-scroll fractional-order chaotic attractors. Chaos 26, 084303 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hu, X.Y., Liu, C.X., Liu, L.: Multi-scroll hidden attractors in improved Sprott A system. Nonlinear Dyn. 86, 1725–1734 (2016)CrossRefGoogle Scholar
  13. 13.
    Shen, S.Y., Ke, M.H., Zhou, P.: A 3D fractional-order chaos system with only one stable equilibrium and controlling chaos. Discrete Dyn. Nat. Soc. 2017, 1–5 (2017)zbMATHGoogle Scholar
  14. 14.
    Pham, V.T., Volos, C., Jafari, S.: Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 87, 2001–2010 (2017)CrossRefGoogle Scholar
  15. 15.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A 375, 2230–2233 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540, 167–218 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuznetsov, N.V., Leonov, G.A., Seledzhi, S.M.: Hidden oscillations in nonlinear control systems. World Congr. 18, 2506–2510 (2011)Google Scholar
  18. 18.
    Bragin, V.O., Vagaĭtsev, V.I., Kuznetsov, N.V.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50, 511–543 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua systems. Phys. D Nonlinear Phenom. 241, 1482–1486 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhao, H.T.: Bifurcating Periodic Orbits and Hidden Attractor of Nonlinear Dynamic Systems. Kunming University of Science and Technology, Kunming (2014)Google Scholar
  21. 21.
    Kuznetsov, N.V., Kuznetsova, O.A., Leonov, G.A.: Localization of hidden Chua attractors by the describing function method. Chaotic Dyn. (2017).  https://doi.org/10.1016/j.ifacol.2017.08.470 CrossRefGoogle Scholar
  22. 22.
    Dudkowski, D., Jafari, S., Kapitaniak, T.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chen, M., Li, M.Y., Yu, Q.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81, 215–226 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, G., Wu, F.Q., Wang, C.N.: Synchronization behaviors of coupled systems composed of hidden attractors. Int. J. Mod. Phys. B 31, 1750180-1-15 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Saha, P., Saha, D.C., Ray, A.: Memristive non-linear system and hidden attractor. Eur. Phys. J. Spec. Top. 224, 1563–1574 (2015)CrossRefGoogle Scholar
  26. 26.
    Danca, M.F., Kuznetsov, N.: Hidden chaotic sets in a Hopfield neural system. Chaos Solitons Fractals 103, 144–150 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sarasola, C., Torrealdea, F.J., D’Anjou, A.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. E 69, 011606 (2004)CrossRefGoogle Scholar
  28. 28.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)CrossRefGoogle Scholar
  29. 29.
    Li, J.B.: Generalized Hamiltonian Systems Theory and Its Applications. Science Press, Beijing (1994)Google Scholar
  30. 30.
    Sira-Ramirez, H., Cruz-Hernandez, C.: Synchronization of chaotic systems: a generalized Hamiltonian systems approach. Int. J. Bifurc. Chaos 11, 1381–1395 (2001)CrossRefGoogle Scholar
  31. 31.
    Torrealdea, F.J., D’Anjou, A., Graña, M.: Energy aspects of the synchronization of model neurons. Phys. Rev. E 74, 011905 (2006)Google Scholar
  32. 32.
    Torrealdea, F.J., Sarasola, C., D’Anjou, A.: Energy consumption and information transmission in model neurons. Chaos Solitons Fractals 40, 60–68 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Moujahid, A., D’Anjou, A., Torrealdea, F., et al.: Energy cost reduction in the synchronization of a pair of nonidentical coupled Hindmarsh–Rose neurons. Trends in Pract. Appl. Agents Multiagent Syst. 22(16), 657–664 (2012)Google Scholar
  34. 34.
    Ma, J., Wu, F.Q., Ren, G.D.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)MathSciNetGoogle Scholar
  35. 35.
    Wang, C.N., Wang, Y., Ma, J.: Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem. Acta Phys. Sin. 65, 30–35 (2016)Google Scholar
  36. 36.
    Song, X.L., Jin, W.Y., Ma, J.: Energy dependence on the electric activities of a neuron. Chin. Phys. B 24, 128710 (2015)CrossRefGoogle Scholar
  37. 37.
    Ma, J., Wu, F.Q., Jin, W.Y., et al.: Calculation of Hamilton energy and control of dynamical systems with different types of attractors. Chaos 27, 481–495 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Li, F., Yao, C.G.: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 84, 2305–2315 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Bilotta, E., Blasi, G.D., Stranges, F.: A gallery of Chua attractors. VI. Int J Bifurc Chaos 17, 49–51 (2015)zbMATHGoogle Scholar
  40. 40.
    Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224, 1421–1458 (2015)CrossRefGoogle Scholar
  41. 41.
    Rabinovich, M.: Stochastic autooscillations and turbulence. Uspekhi Fizicheskih Nauk 125, 123–168 (1978)CrossRefGoogle Scholar
  42. 42.
    Liu, B.Z.: Nonlinear Dynamics. Higher Education Press, Beijing (2001)Google Scholar
  43. 43.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  44. 44.
    Kengne, J., Chedjou, J.C., Kom, M.: Regular oscillations, chaos, and multi-stability in a system of two coupled van der Pol oscillators: numerical and experimental studies. Nonlinear Dyn. 76, 1119–1132 (2014)CrossRefGoogle Scholar
  45. 45.
    Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83, 751–765 (2016)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Zhou, P., Ke, M.H.: A new 3D autonomous continuous system with two isolated chaotic attractors and its topological horseshoes. Complexity 2017, 1–7 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of Electrical and Information EngineeringLanzhou University of TechnologyLanzhouChina
  2. 2.School of Mathematics and PhysicsLanzhou Jiaotong UniversityLanzhouChina
  3. 3.The Basic Courses Department of Lanzhou Institute of TechnologyLanzhouChina

Personalised recommendations