Skip to main content
SpringerLink
Log in

Dynamic analysis for the hyperchaotic system with nonholonomic constraints

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper first formulates a new hyperchaotic system for particle motion and analyzes the equilibrium stability of the system and the hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane. We then investigate the influences of two nonlinear nonholonomic constraints on the particle motion. The numerical simulations, including bifurcation diagrams, computation of Lyapunov exponents, Poincaré maps, and phase portraits for systems, not only show the hyperchaotic phenomena, but also exhibit the different hyperchaotic behaviors in the constrained parameters space for the various regimes. Additionally, a holonomic constraint is introduced to control the hyperchaotic system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Hertz, H.R.: Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt. Barth, Leipzig (1894)

    MATH  Google Scholar 

  2. Peng, Z., Yang, S., Wen, G., et al.: Adaptive distributed formation control for multiple nonholonomic wheeled mobile robots. Neurocomputing 173, 1485–1494 (2016)

    Article  Google Scholar 

  3. Sinha, A., Ghose, D.: Generalization of nonlinear cyclic pursuit. Automatica 43, 1954–1960 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Dynamics of rolling disk. Regul. Chaotic Dyn. 8, 201–212 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chang, C.M., Ge, Z.M.: Complete identification of chaos of nonlinear nonholonomic systems. Nonlinear Dyn. 60, 551–559 (2010)

    Article  MATH  Google Scholar 

  6. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 7, 397–398 (1976)

    Article  MATH  Google Scholar 

  7. Yang, J., Chen, Y., Zhu, F.: Singular reduced-order observer-based synchronization for uncertain chaotic systems subject to channel disturbance and chaos-based secure communication. Appl. Math. Comput. 229, 227–238 (2014)

    MATH  Google Scholar 

  8. Rhouma, R., Belghith, S.: Cryptanalysis of a new image encryption algorithm based on hyperchaos. Phys. Lett. A 372, 5973–5978 (2008)

    Article  MATH  Google Scholar 

  9. Hammami, S.: State feedback-based secure image cryptosystem using hyperchaotic synchronization. ISA Trans. 54, 52–59 (2015)

    Article  Google Scholar 

  10. Buscarino, A., Fortuna, L., Frasca, M.: Experimental robust synchronization of hyperchaotic circuits. Phys. D 238, 1917–1922 (2009)

    Article  MATH  Google Scholar 

  11. Ma, J., Li, A., Pu, Z., et al.: A time-varying hyperchaotic system and its realization in circuit. Nonlinear Dyn. 62, 535–541 (2010)

    Article  MATH  Google Scholar 

  12. Chen, C.H., Shu, L.J., Chen, H.K.: A new hyper-chaotic system and its synchronization. Nonlinear Anal. Real 10, 2088–2096 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zheng, S., Dong, G., Bi, Q.: A new hyperchaotic system and its synchronization. Appl. Math. Comput. 215, 3192–3200 (2010)

    MATH  MathSciNet  Google Scholar 

  14. Sun, K., Liu, X., Zhu, C.: Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system. Nonlinear Dyn. 69, 1383–1391 (2012)

    Article  MathSciNet  Google Scholar 

  15. Chen, C., Davidson, R.C.: Chaotic particle dynamics in free-electron lasers. Phys. Rev. A 43, 5541 (1991)

    Article  Google Scholar 

  16. Abbott, N.L.: Colloid science collides with liquid crystals. Science 342, 1326–1327 (2013)

    Article  Google Scholar 

  17. Alzahrani, A.M., Frolov, V.P., Shoom, A.A.: Critical escape velocity for a charged particle moving around a weakly magnetized Schwarzschild black hole. Phys. Rev. D 87, 084043 (2013)

    Article  Google Scholar 

  18. Bonfim, O.F., David, J., Hinkley, S.: Chaotic and hyperchaotic motion of a charged particle in a magnetic dipole field. Int. J. Bifurc. Chaos 10, 265–271 (2000)

    Article  Google Scholar 

  19. Wittkowski, R., Löwen, H.: Self-propelled Brownian spinning top: dynamics of a biaxial swimmer at low Reynolds numbers. Phys. Rev. E 85, 021406 (2012)

    Article  Google Scholar 

  20. Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Springer, New York (1988)

    Book  MATH  Google Scholar 

  21. Gaspard, P., Briggs, M., Francis, M., et al.: Experimental evidence for microscopic chaos. Nature 6696, 865–868 (1998)

    Article  Google Scholar 

  22. Ahmad, T.A., Sundarapandian, V.: Chaos Modeling and Control Systems Design, p. 581. Springer, Berlin (2015)

    MATH  Google Scholar 

  23. Cang, S., Wu, A., Wang, Z., et al.: On a 3-D generalized Hamiltonian model with conservative and dissipative chaotic flows. Chaos Solitons Fractals 99, 45–51 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sarasola, C., Torrealdea, F., d’Anjou, A.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. E 69, 011606 (2004)

    Article  Google Scholar 

  25. Wang, C., Wang, Y., Ma, J.: Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem. Acta Phys. Sin. 24, 240501 (2016)

    Google Scholar 

  26. Greenwood, D.T.: Advanced Dynamics, p. 116. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  27. Ni, J., Liu, L., Liu, C., et al.: Fixed-time dynamic surface high-order sliding mode control for chaotic oscillation in power system. Nonlinear Dyn. 86, 401–420 (2016)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant Nos. 11272050, 11672032), the Youth Science Foundations of Education Department of Hebei Province (No. QN2016265), Hebei Special Foundation “333 talent project” (No. A2016001123), and Scientific Research Funds of Hebei Institute of Architecture and Civil Engineering (Nos. 2016XJJQN03, 2016XJJYB05).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Junhong Li & Huibin Wu

  2. Department of Mathematics and Sciences, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou, 075000, Hebei, China

    Junhong Li

  3. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, China

    Huibin Wu

  4. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Fengxiang Mei

Authors
  1. Junhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huibin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fengxiang Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junhong Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Wu, H. & Mei, F. Dynamic analysis for the hyperchaotic system with nonholonomic constraints. Nonlinear Dyn 90, 2557–2569 (2017). https://doi.org/10.1007/s11071-017-3823-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3823-4

Keywords

Search

Navigation