The European Physical Journal Special Topics

, Volume 228, Issue 6, pp 1385–1403 | Cite as

Complexity of dynamic system switching between two subsystems with cornered boundaries

  • Jianzhe HuangEmail author
  • Xilin FuEmail author
Regular Article Topical issue
Part of the following topical collections:
  1. Discontinuous Dynamical Systems and Synchronization


Switching between systems widely exists in real world applications, for instance morphing aerospace vehicle has different shapes at different flight conditions, which leads the system properties and external excitations differ and introduces discontinuity to make the system become singularity. In this paper, a single degree of freedom dynamical system which has two states to switch will be investigated. Domains and boundaries will be defined in the state space, and the switch mechanism on the discontinuous boundaries will be discussed based on the theory of discontinuity. The onset and vanish conditions for the sliding motion of such a dynamical system will be given based on G-function and its higher order. The mapping structure and periodic motions will be described. Through analytical bifurcation analysis, the motion evolution characteristics will be obtained, and one periodic motion switch to another periodic motion through graze bifurcation is observed when the parameter varies.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.P. Den Hartog, Trans. Am. Soc. Mech. Eng. 53, 107 (1931)Google Scholar
  2. 2.
    E.S. Levitan, J. Acoust. Soc. Am. 32, 1265 (1960)ADSCrossRefGoogle Scholar
  3. 3.
    A.F. Filippov, Differential equations with discontinuous right-hand sides (Kluwer Academic Publishers, Dordrecht, 1988)Google Scholar
  4. 4.
    M.A. Aizerman, E.S. Pyatnitskii, Autom. Remote Control 35, 1066 (1974)Google Scholar
  5. 5.
    M.A. Aizerman, E.S. Pyatnitskii, Autom. Remote Control 35, 1241 (1974)Google Scholar
  6. 6.
    V.I. Utkin, IEEE Trans. Autom. Control 22, 212 (1977)CrossRefGoogle Scholar
  7. 7.
    K. Narendra, J. Balakrishnan, IEEE Trans. Autom. Control 39, 2469 (1994)CrossRefGoogle Scholar
  8. 8.
    R.N. Shorten, K.S. Narendra, A sufficient condition for the existence of a common Lyapunov function for two second order linear systems, in Proceedings of the 36th Conference on Decision & Control (IEEE, San Diego, 1997)Google Scholar
  9. 9.
    M.S. Branicky, Stability of hybrid systems: state of the artin Proceedings of the 36th Conference on Decision & Control (IEEE, San Diego, 1997)Google Scholar
  10. 10.
    D. Liberzon, A.S. Morse, IEEE Control Syst. Mag. 19, 59 (1999)CrossRefGoogle Scholar
  11. 11.
    R.I. Leine, D.H. van Campen, B.L. van de Vrande, Nonlinear Dyn. 23, 105 (2000)CrossRefGoogle Scholar
  12. 12.
    M.F. Danca, Int. J. Bifurc. Chaos 12, 1813 (2002)CrossRefGoogle Scholar
  13. 13.
    Z. Zhusubaliyev, E. Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems (World Scientific, Singapore, 2003)Google Scholar
  14. 14.
    H. Oktem, Nonlinear Anal. 63, 336 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.C.J. Luo, Commun. Nonlinear Sci. Numer. Simul. 10, 1 (2005)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    A.C.J. Luo, J. Sound Vib. 285, 443 (2005)ADSCrossRefGoogle Scholar
  17. 17.
    Z. Sun, Automatica 43, 164 (2007)CrossRefGoogle Scholar
  18. 18.
    H. Lin, P.J. Antsaklis, IEEE Trans. Autom. Control 52, 633 (2007)CrossRefGoogle Scholar
  19. 19.
    A.C.J. Luo, J.Z. Huang, Nonlinear Anal.: Real World Appl. 13, 241 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    L.P. Li, L.H. Huang, J. Math. Anal. Appl. 411, 83 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    L.P. Li, Electron. J. Qual. Theor. Differ. Eq. 70, 1 (2014)Google Scholar
  22. 22.
    J.Z. Huang, A.C.J. Luo, J. Comput. Nonlinear Dyn. 12, 061014 (2017)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Aeronautics and Astronautics, Shanghai Jiao Tong UniversityShanghaiP.R. China
  2. 2.School of Mathematical Sciences, Shandong Normal UniversityJinan250014P.R. China

Personalised recommendations