Normal form methods for symbolic creation of approximate solutions of nonlinear dynamical systems

We present a scheme for the computation (and further reduction) of normal forms of Hamiltonian systems, and we describe the scheme’s computer algebra application to some two-degrees-of-freedom mechanical models to build symbolic approximations for local families of their periodic solutions near the origin. More precisely, we consider one-parameter families of periodic solutions of real Hamiltonian systems, the parameter being the energy. We deal with some interesting cases in the generalized Hénon-Heiles system and another Hamiltonian considered by Churchill et al. The families of periodic solutions are represented as truncated Fourier series in the approximate frequencies and power series in the mechanical energy. It is also shown that some periods of the found families can be given closed expressions in terms of the energy constant in the form of very simple generalized hypergeometric functions.