Periodic orbits and 10 cases of unbounded dynamics for one Hamiltonian system defined by the conformally coupled field

•Equations for periodic orbits are got for many level sets.•Domains with unbounded dynamics are localized.•Localizations for compact invariant sets are obtained.

In this paper we study invariant domains with unbounded dynamics for one cosmological Hamiltonian system which is formed by the conformally coupled field; this system was introduced by Maciejewski et al. (2007). We find a few groups of conditions imposed on parameters of this system for which all trajectories are unbounded in both of time directions. Further, we present a few groups of other conditions imposed on system parameters under which we localize the invariant domain with unbounded dynamics; this domain is defined with help of bounds for values of the Hamiltonian level surface parameter. We describe one group of conditions when our system possesses two periodic orbits found explicitly. In some of rest cases we get localization bounds for compact invariant sets.