A class of Hamiltonian impact systems exhibiting smooth near integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded energy level sets and a single straight wall which preserves the separable structure. Singularities in the system appear either
Tools for analyzing dynamics in a class of 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable structure are derived. Integrable, near-integrable and far-from integrable cases are considered. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to this class is constructed. The projection of Liouville leaves of the smooth integrable dynamics to the configuration space allows to extend these tools to impact surfaces which produce far from integrable dynamics. It is suggested that such representations classify dynamically different regions in phase space. For the integrable and near integrable cases these provide global information on the dynamics whereas for the far from integrable regimes (caused by finite deformations of the impact surface), these provide information on the singular set and on the non-impact orbits. The results are presented and demonstrated for the Duffing-center system with impacts from a slanted wall.
Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an additional rich class of non-smooth systems that can be analyzed. Such systems exhibit rich dynamics, as, in addition to the underlying integrable structure, some of the trajectories may undergo only transverse impact, others may undergo also tangential impacts, and some trajectories do not impact at all. Under perturbations, each of these classes of orbits behaves differently. Tools for classifying these different types of dynamics in 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable dynamics are presented. Moreover, some of these tools may be extended to far from integrable cases. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to impact systems is constructed. It is shown that such representations classify dynamically different regions in phase space. For the integrable and near integrable (small perturbations) cases these provide global information on the dynamics whereas for the far from integrable (non-perturbative) regimes, these provide rough classification of the first impact map. The interpretation of these results in terms of projections of solutions to the configuration space as well as the relations of these to the Hill region are presented.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.