How to get far with little effort
Rafael de la Llave
Abstract.In many problems, periodic perturbations average out and small perturbations lead to small changes. On the other hand in other cases, the small
perturbations accumulate and, over time lead to very large effects. For a harmonic oscillator these depends on whether the frequency of the perturbation is
rationally related to the frequency of the oscillator. In typical nonlinear mechanical systems, where the frequency depends on the energy, this leads to a very
complicated geography of resonant regions. Also, the celebrated KAM theorem
shows that there are sets of large measure where the periodic perturbations do
not accumulate. Nekhoroshev also showed that accumulations can only occur
in very long times. Nevertheless, V. Arnold constructed a very remarkable and
delicate example where accumulation of effects does take place and similar effects were observed by Chirikov and others in numerical experiments. In recent
times, there has been considerable efforts in identifying geometric features that
lead to similar effects in very general systems and using them in applications.
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