Abstract
In this chapter, we go on into the methods for obtaining the analytical solutions of the SD oscillator. A series of irrational elliptic functions and hyperbolic functions is proposed for the unperturbed oscillator to provide the analytical solutions for both the smooth and discontinuous cases with periodic solutions and the homoclinic ones which could not be expressed using classical tools, the traditional methodologies being applicable only for rational or polynomial systems. It is found that the solutions of the discontinuous case can also be given by letting \(\alpha \rightarrow 0\). With the help of the defined elliptic functions and the hyperbolic functions for the periodic and homoclinic orbits, the chaotic behaviours of the perturbed system can be detected analytically.
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