 2058 R. Calleja, A. Celletti, R. de la Llave
 KAM theory for some dissipative systems
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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 spinorbit
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
aposteriori format: assume we start with an approximate solution satisfy
ing a suitable nondegeneracy 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 aposteriori 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 spinorbit problems.
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