Analysis of the periodic solutions of a smooth and discontinuous oscillator

Zhi-Xin Li, Qing-Jie Cao, Marian Wiercigroch, Alain Léger

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16 Citations (Scopus)


In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth (α > 0) and discontinuous (α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.

Original languageEnglish
Pages (from-to)575-582
Number of pages8
JournalActa Mechanica Sinica
Issue number4
Early online date13 Aug 2013
Publication statusPublished - Aug 2013

Bibliographical note

The project was supported by the National Natural Science Foundation of China (11072065).


  • Averaging method
  • Elliptic integral
  • Irrational nonlinearity
  • Periodic solution
  • SD oscillator


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