Recent advances in geometric singular perturbation theory have established a powerful set of tools for analyzing resonance phenomena and homoclinic orbits in systems of ordinary differential equations with two time or length scales. Among these new tools is the Exponentially Small Exchange Lemma (called EXSEL), which was developed by the author with C. Jones and N. Kopell, and which has been used successfully to prove the existence of multiple fast-pulse homoclinic orbits in perturbed multi-degree-of-freedom Hamiltonian systems, as well as in traveling wave problems for coupled reaction diffusion equations. It is proposed here to analyze four fundamental aspects of resonance phenomena in singularly-perturbed systems: (1) the existence of resonant sub- and super-harmonic orbits, (2) the geometry of islands of stability, which are obstructions to mixing in adiabatic chaos, (3) the resonant response of homoclinic structures in the forced, damped sine-Gordon equation which are conjectured to be sources of chaos, (4) the existence of periodic orbits in problems of passage throughresonance. Physical problems with multiple time scales arise in many branches of science and technology, including fluid mechanics, accelerator dynamics, plasma physics, neurophysiology, and biology. These problems are modeled mathematically as singularly-perturbed or adiabatic systems. Despite the long history of progress and the considerable continuing interest in this area, a wealth of open problems exists. This proposal concerns a group of four fundamental mathematical questions concerning resonance phenomena and homoclinic behavior that are of direct significance in the applications. Several new mathematical techniques, some developed by the author, have recently become available which offer substantial promise of solving these problems. While the main goal of the work proposed here is to use and extend the scope of these new tools, it is also endeavored to answer the above questions with an eye toward the technological applications. Toward this end, the proposer points to his successful development of enhanced fluid-mixing technology as an application of his earlier work on the geometry of homoclinic tangles in singularly-perturbed systems.
