Bifurcations in a family of Hamiltonian systems and associated nontwist cubic maps

The aim of the paper is to study systems with one-and-a-half degrees of freedom generated by a Hamiltonian with a quartic unperturbed part and broad perturbation spectrum. To this end, an approximate interpolating Hamiltonian system is firstly studied. Behaviour of the Poincaré-Birkhoff or dimerised chains in their routes to reconnection when the perturbation parameter varies is particularly presented. In the second step, a discrete system associated to the full Hamiltonian system is constructed and studied. We point out interesting properties of the dynamics of the Poincaré-Birkhoff or dimerised chains, such as pairs of homoclinic orbits to the same equilibrium point (sandglass) and triple reconnection. Then we use the scenario of reconnections to explain the destruction of transport barriers in the non-autonomous system.