Irrational elliptic functions and the analytical solutions of SD oscillator

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.