The paper extends and improves an existing parametric study with special focus on the onset of chaos and on chaotic states. The results show that the effect of density ratio responsible for two qualitatively different oblique oscillating states has a significant impact both on the onset of chaos and on the behavior of fully chaotic states. The observed difference between dense and light spheres is associated to the strength of coupling between fluid and solid degrees of freedom. While the low frequency mode of oblique oscillating state presents specific features due to a strong solid-fluid coupling, the dynamics of the high frequency mode is shown to be driven by the same vortex shedding as the wake of a fixed sphere. The different fluid-solid coupling also determines two different ways how chaos sets in. Two outstanding ordered regimes are evidenced and investigated in the chaotic domain. One of them, characteristic for its helical trajectories, might provide a link to the experimentally evidenced, but so far numerically unexplained, vibrating regime of ascension of light spheres. For fully chaotic states, it is shown that statistical averaging converges in a satisfactory manner. Several statistical characteristics are suggested and evaluated. (C) 2015 Elsevier Ltd. All rights reserved.