Skip to main content
SpringerLink
Log in

Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

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

Abstract

In this paper, homoclinic bifurcations and chaotic dynamics of a piecewise linear system subjected to a periodic excitation and a viscous damping are investigated by the Melnikov analysis for nonsmooth systems in detail. The piecewise linear system can be seen as a simple linear feedback control system with dead zone and saturation constrains. The unperturbed system is a piecewise linear Hamiltonian system, which contains two parameters and exhibits quintuple well characteristic. The discontinuous unperturbed system, which is obtained by reducing the two parameters to zero, has saddle-like singularity and homoclinic-like orbit. Analytical expressions for the unperturbed homoclinic and heteroclinic orbits are derived by using Hamiltonian function for the piecewise linear system. The Melnikov analysis for nonsmooth planar systems is first described briefly, and the theorem for homoclinic bifurcations for the nonsmooth planar systems is also obtained and then employed to detect the homoclinic and heteroclinic tangency under the perturbation of a viscous damping and a periodic excitation. Finally, the chaotic attractors and the bifurcation diagrams are presented to show the bifurcations and chaotic dynamics of the piecewise linear 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

Similar content being viewed by others

References

  1. Brogliato, B.: Nonsmooth Mechanics. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  2. Feeny, B.F., Moon, F.C.: Empirical dry-friction modelling in a forced oscillator using chaos. Nonlinear Dyn. 47, 129–141 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Banerjee, S., Verghese, G.: Nonlinear Phenomena in Power Electronics. IEEE Press, New York (2001)

    Book  Google Scholar 

  4. Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model: stability, complexity and scaling. ASME J. Biomech. Eng. 120, 281–288 (1998)

    Article  Google Scholar 

  5. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides: Mathematics and Its Applications. Kluwer Academic, Dordrecht (1988)

    Book  MATH  Google Scholar 

  6. Feigin, M.I.: Forced Vibrations of Nonlinear Systems with Discontinuities. Nauka, Moscow (1994). (in Russian)

    Google Scholar 

  7. Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  8. Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Application. Springer, London (2008)

    Google Scholar 

  9. Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)

    Article  MATH  Google Scholar 

  10. Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)

    Article  MathSciNet  Google Scholar 

  11. Melnikov, V.K.: On the stability of the center for time periodic perturbations. Tans. Moscow Math. Soc. 12, 1–57 (1963)

    Google Scholar 

  12. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Du, Z., Zhang, W.: Melnikov method for homoclinic bifurcations in nonlinear impact oscillators. Comput. Math. Appl. 50, 445–458 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kukučka, P.: Melnikov method for discontinuous planar systems. Nonlinear Anal. 66, 2698–2719 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and Melnikov-type Method. World Scientific, Singapore (2007)

    Google Scholar 

  17. Battelli, F., Fečkan, M.: Homoclinic trajectories in discontinuous systems. J. Dyn. Differ. Equ. 20, 337–376 (2008)

    Article  MATH  Google Scholar 

  18. Battelli, F., Fečkan, M.: Bifurcation and chaos near sliding homoclinics. J. Differ. Equ. 248, 2227–2262 (2010)

    Article  MATH  Google Scholar 

  19. Battelli, F., Fečkan, M.: Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems. Phys. D 241, 1962–1975 (2012)

  20. Li, S.B., Zhang, W., Hao, Y.X.: Melnikov-type method for a class of discontinuous planar systems and applications. Int. J. Bifur. Chaos 24(1450022), 1–18 (2014)

  21. Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90, 121–155 (1983)

    Article  MathSciNet  Google Scholar 

  22. Thompson, J.M., Bokain, A.B., Ghaffari, R.: Subharmonic resonances and chaotic motions of the bilinear oscillator. IMA J. Appl. Maths. 31, 207–234 (1983)

    Article  MATH  Google Scholar 

  23. Foale, S., Bishop, S.R.: Dynamical complexities of forced impacting systems. Philos. Trans. R. Soc. Lond. A 338, 547–556 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Foale, S., Bishop, S.R.: Bifurcations in impact oscillators. Nonlinear Dynam. 6, 285–229 (1994)

    Article  Google Scholar 

  25. Chatterjee, S., Mallik, A.K., Ghosh, A.: Period response of piecewise nonlinear oscillators under harmonic excitation. J. Sound Vib. 191, 129–144 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Luo, A.C.J., Menon, S.: Global chaos in a periodically forced, linear system with a dead-zone restoring force. Chaos Solitons Fractals 19, 1189–1199 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Luo, A.C.J.: The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation. J. Sound Vib. 283, 723–748 (2005)

    Article  MATH  Google Scholar 

  28. Shen, J., Li, Y.R., Du, Z.D.: Subharmonic and grazing bifurcations for a simple bilinear oscillator. Int. J. Non-linear Mech. 60, 70–82 (2014)

    Article  Google Scholar 

  29. Cao, Q., Wiercigroch, M., Pavlovskaia, E.E., Thompson, J.M.T., Grebogi, C.: Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics. Phil. Trans. R. Soc. A 366, 635–652 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ji, J.C., Leung, A.Y.T.: Periodic and chaotic motions of a harmonically forced piecewise linear system. Int. J. Mech. Sci. 46, 1807–1825 (2004)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the support of the National Science Foundation of China (NNSFC) through Grant Nos. 11072008, 10732020, 11290152, 11272063, 11102226, 11472298, 11472056, 11427801 the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the Natural Science Foundation of Tianjin City through Grant No. 13JCQNJC04400, the Fundamental Research Funds for the Central Universities through Grant Nos. ZXH2012K004 and 3122013k005.

Author information

Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    S. B. Li & C. Shen

  2. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100022, China

    W. Zhang

  3. College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing, 100192, China

    Y. X. Hao

Authors
  1. S. B. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. X. Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. B. Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S.B., Shen, C., Zhang, W. et al. Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping. Nonlinear Dyn 79, 2395–2406 (2015). https://doi.org/10.1007/s11071-014-1820-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1820-4

Keywords

Search

Navigation