Periodic orbits and non-integrability in a cosmological scalar field

Abstract

We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing a universe filled with a scalar field which possesses three parameters. The main results are the following. First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level. Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville–Arnol'd, proving that there cannot exist any second first integral of class \documentclass[12pt]{minimal}\begin{document}$\mathcal {C}^1$\end{document}C1. It is important to mention that the tools (i.e., the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of these periodic orbits for studying the integrability of the Hamiltonian system) used here for proving our results on the cosmological scalar field can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.