Irrational elliptic functions and the analytical solutions of SD oscillator

Qingjie Cao, Dan Wang, Y. Chen, Marian Wiercigroch

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


The smooth and discontinuous (SD) oscillator is a strongly nonlinear system with an irrational
restoring force proposed in P.R.E (2006), which leads to barriers for the conventional
methods to investigate the dynamical behaviour directly. In this paper, two kinds of irrational
elliptic functions and a kind of hyperbolic functions are defined in the real domain to
formulate the analytical solutions of the system. The properties of the functions are obtained
including differentiability, periodicity and parity. As the application of the defined irrational
functions, the chaotic thresholds of the oscillator are also depicted by using the Melnikov
method. Numerical analysis shows the efficiency of the proposed procedure.
Original languageEnglish
Pages (from-to)701-715
Number of pages15
JournalJournal of Theoretical and Applied Mechanics
Issue number3
Publication statusPublished - 2012


  • SD oscillator
  • irrational nonlinearity
  • irrational elliptic functions
  • threshhold of chaos


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