Perturbed invariant manifolds and chaos

Abstract

The goal of this thesis is to determine whether a given deterministic dynamical system can display chaotic behaviour, and if so, under which conditions. However, the complexity of the question forces us to reduce the problem to the study of two-dimensional $C^r$ diffeomorphisms. This work is structured in the following way; first, a preliminary chapter with the intention to familiarize the reader with the background needed. Then, two main blocks, which correspond to the third and forth chapters, where the answer to the question is provided in the first one, and whether these conditions can occur for Poincaré maps associated with periodically perturbed systems is treated in the second one. Last, there is an Appendix about the computation of improper integrals which typically occur in Melnikov’s theory by the residue theorem. In the first block, the Smale-Moser Theorem is the key point for seeing that a two-dimensional map, which possesses a homoclinic point at which the stable and unstable manifold of the hyperbolic fixed point intersect transversally, has chaotic behaviour. In the text, this result is clearly achieved in two parts. The first one, which corresponds to sections 3.1, 3.2 and 3.3 is the study of sufficient conditions for the existence of an invariant Cantor set topologically conjugate to a shift on N symbols. Here, Symbolic Dynamics, which is the method for characterizing the orbit structure through infinite sequences of symbols, takes an important role because it enables us to associate a point in a subset of the unit square with a bi-infinite sequence. The second one, which covers sections 3.4 and 3.5, is about re-writing the conditions needed, which are purely geometrical, to something more analytically approachable with the purpose of making them easier to be verified under the hypotheses of the Smale-Moser Theorem. In the second block, we study Hamiltonian systems that suffer periodic nonautonomous perturbations. The aim of this chapter is to provide criteria, which will let us conclude when the associated Poincaré map has a transversal homoclinic point. Therefore, on account of the results from the third chapter we are able to state, under suitable conditions, that there is a chaotic invariant set. Moreover, the research is generalized to Hamiltonian systems that present either a homoclinic orbit or a heteroclinic one, although no similar conclusions regarding its dynamics will be deduced for the latest. Furthermore, this criteria depends on whether the perturbed invariant manifold coincide, split completely or cross. Thus, the track of the distance between the manifolds is important. As a result, the Melnikov function is introduced with the intention to tell us when the distance between the two manifolds becomes nul. Seeing that, in sections 4.1, 4.2 and 4.3 we have the description of the phase space geometry for the unperturbed system, and its changes after the periodic perturbation. Later, in sections 4.4, 4.5 and 4.6 the Melnikov function is derived and its properties are discussed. Finally, section 4.7 enables the reader to see the applicability of the theory developed during the thesis with one heteroclinic case and one homoclinic case.