20-58 R. Calleja, A. Celletti, R. de la Llave
KAM theory for some dissipative systems (8017K, PDF/A) Aug 3, 20
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Abstract. Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods. The point of view that we adopt is that we are dealing with real problems and that we will have to use a very wide variety of methods. From the applications, to numerical studies to rigorous mathematics. As we will see, all of these methods feed on each other. The rigorous mathematics leads to efficient algorithms (and allows us to believe the results), the numerical experiments lead to deep mathematical conjectures, the applications benefit from all these tools, and set meaningful goals that prevent from doing things just because they are easy. Of course, the road towards this lofty goal is not rosy and there are many false starts, complications, etc. After several years, we can erase the false starts from the story, but we hope to provide some flavor. Given the rather wide scope is unavoidable that some arguments have different standards (rigorous proofs, numerical efficiency, conjectures). We have strived to make all those very explicit, but may be it would be hard to keep this present. Of course, similar programs can be applied to many problems, but in this paper we will deal with a rather concrete set of problems. In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as conformally symplectic systems, which have the property that they transform the symplectic form into a multiple of itself. To give explicit examples of conformally symplectic systems, we will present two different models: a discrete system known as the standard map and a continuous system known as the spin-orbit problem. In both cases we will consider the conservative and dissipative versions, that will help to highlight the differences between the symplectic and conformally symplectic dynamics. For such dissipative systems we will present a KAM theorem in an a-posteriori format: assume we start with an approximate solution satisfy- ing a suitable non-degeneracy condition, then we can find a true solution nearby. The theorem does not assume that the system is close to integrable. The method of proof is based on extending geometric identities originally developed in [39] for the symplectic case. Besides leading to streamlined proofs of KAM theorem, this method provides a very efficient algorithm which has been implemented. Coupling an efficient numerical algorithm with an a-posteriori theorem, we have a very efficient way to provide rigorous estimates close to optimal. Indeed, the method gives a criterion (the Sobolev blow up criterion) that allows to compute numerically the breakdown. We will review this method as well as an extension of J. Greene s method and present the results in the conservative and dissipative standard maps. Computing close to the breakdown, allows to discover new mathematical phenomena such as the bundle collapse mechanism. We will also provide a short survey of the present state of KAM estimates for the existence of invariant tori in the conservative and dissipative standard maps and spin-orbit problems.

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