Two approaches for evaluating the period function of some Hamiltonian systems

The theory of nonlinear oscillators is an extremely active field of contemporary research, given the vastness of the phenomena modelled through nonlinear differential equations with periodic solutions. In particular, the period function associated with Hamiltonian systems has been the subject of an outstanding piece of research. Our contribution, here, concerns oscillatory systems ruled by odd–degree polynomial restoring forces. We present two approaches for the symbolic approximation of the energy–period function: one based on asymptotic expansions of the period function, and the other through the approximation obtained using fifth–order Chebyshev polynomials of the restoring force and the consequent closed–form solution of the approximate equation. The obtained approximation quality is verified through different comparison methods illustrated in this chapter.