The Conley index for fast–slow systems II: Multidimensional slow variable

We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authorsʹ earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast–slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427–470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast–slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast–slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast–slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427–470].