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de la Llave, Rafael

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An introduction to KAM theory II: The twist theorem

2019-03-30 , de la Llave, Rafael

The KAM (Kolmogorov Arnold and Moser) theory studies the persistence of quasi-periodic solutions under perturbations. It started from a basic set of theorems and it has grown into a systematic theory that settles many questions. The basic theorem is rather surprising since it involves delicate regularity properties of the functions considered, rather subtle number theoretic properties of the frequency as well as geometric properties of the dynamical systems considered. In these lectures, we plan to cover a complete proof of a particularly representative theorem in KAM theory. The first lecture covered all the prerequisites (analysis, number theory and geometry). In this second lecture we will present a complete proof of Moser's twist map theorem (indeed a generalization to more dimensions). The proof also lends itself to very efficient numerical algorithms. If there is interest and energy, we will devote a third lecture to numerical implementations.

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An introduction to KAM theory I: the basics

2019-03-29 , de la Llave, Rafael

The KAM (Kolmogorov Arnold and Moser) theory studies the persistence of quasi-periodic solutions under perturbations. It started from a basic set of theorems and it has grown into a systematic theory that settles many questions. The basic theorem is rather surprising since it involves delicate regularity properties of the functions considered, rather subtle number theoretic properties of the frequency as well as geometric properties of the dynamical systems considered. In these lectures, we plan to cover a complete proof of a particularly representative theorem in KAM theory. In the first lecture we will cover all the prerequisites (analysis, number theory and geometry). In the second lecture we will present a complete proof of Moser's twist map theorem (indeed a generalization to more dimensions). The proof also lends itself to very efficient numerical algorithms.

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Invariant Objects in Volume Preserving Maps and Flows

2014-11-21 , de la Llave, Rafael

We consider smooth volume preserving maps. Two important phenomena are transport and mixing. We present several geometric obstructions that prevent transport and mixing and numerical methods to compute them. These are quasiperiodic orbits of the maps and their treatment requires KAM (Kolmogorov-Arnold-Moser) techniques for an analytic treatment. The result we present has an a-posteriori format (an approximate solution with good condition numbers implies a true solution) and it also leads to very efficient algorithms (low storage requirements and low operation count). These algorithms have been implemented and run (by J. Meiss and A. Fox) and they formulated to conjectures about breakdown. A novelty of the method is that the topology also plays a role. Depending on the global topology the tori may be obstructions to mixing but not to transport or be obstructions to transport and mixing. This is joint work with T. Blass and with A. Fox.

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