Nonlinear Dynamics and Chaos : Where do we go from here?

In keeping with the spirit of the Colston conference on Nonlinear Dynam-ics and Chaos, this chapter emphasizes ideas more than details, describingmy vision of how the bifurcation theory of multiple time scale systemswill unfold. Multiple time scale dynamical systems are rife with compli-cated phenomena. The subject has a complicated history that interweavesthree di erent viewpoints: nonstandard analysis, classical asymptotics andgeometric singular perturbation theory. My perspective is decidedly geo-metric but draws upon asymptotic analysis, recognizing the fundamentalcontributions rst expressed in the language of nonstandard analysis. Thesuccess of dynamical systems theory in elucidating patterns of bifurcationin generic systems with a single time scale motivates the goal here, namelyto extend this bifurcation theory to systems with two time scales. Thereare substantial obstacles to realizing this objective, both theoretical andcomputational. Consequently, the nal shape that the theory will take isstill fuzzy.It may seem strange to talk about computational barriers to a math-ematical theory, so I give some explanation for this. Much of the progress

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