Resumen de Global instability in hamiltonian systems

Rodrigo Gonçalves Schaefer

• In Chapters 1 and 2 of this thesis, we prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps takes place.

  We separate the proof of the general system in two cases.

  The first one is studied in Chapter 1. There, a proof is given for the simplest perturbation function. Besides, we find out some very special diffusion orbits, called "highways", and we give estimates of the time of diffusion for these orbits. The second case is considered in Chapter 2, and the proof of diffusion is completed. In Chapter 2, the existence of piecewise smooth global scattering maps is also provided.

  In Chapter 3, we consider a similar Hamiltonian with 3 degrees of freedom.

  We prove the diffusion using a combination of scattering maps and inner dynamics with concrete diffusion paths. We also compare the results obtained in this case with the results in Chapter 1.

  Closing the thesis, we comment some open problems remained of the study that we have done along the three previous Chapters.