Global recurrences of multi-time scaled systems

We develop here the rst steps of a long-term investigation on multi-time scaled systems with some spatial heterogeneity. They can be characterized by a slow-variation of \actions" interspaced by local fast variations of \angles". The principle of construction of these systems is presented on a rst example where the actions are periodic solutions of a planar Hamiltonian system. Once the fast perturbation of the angles is added, the whole system displays kind of bursting oscillations (characterized by the alternate of quiescent phases interspaced by fast oscillations) although it is completely integrable. In this rst example, the full analysis of the underlying recurrence is possible. We then discuss a second example which has been discovered by Rossler in the 70s and which inspired this study. This example looks paradigmatic of the complex global recurrences these systems may have.