Expanding baker maps: A first tool to study homoclinic biffurcations of 3-D diffeomorphisms

Abstract

Under the title Expanding Baker Maps: A First Tool To Study Homoclinic Bifurcations Of 3-D Diffeomorphisms¿ this thesis explores an interesting part of the Dynamical Systems: the study of the dynamics emerging when a family of diffeomorphisms unfolds a homoclinic tangency in a three-dimensional manifold. Even though this problem has been deeply considered in the case of two-dimensional tangencies, short time until little or nothing was known in three-dimensional framework. In a paper of Professor Joan Carles Tatjer published in 2002, the expression for the family of limit return maps associated to 3-D homoclinic tangencies is given for the first time. Later, in two papers in collaboration between Professor Antonio Pumariño and Tatjer himself, the wide fauna of strange attractors appearing in the unfolding of these tangencies is numerically discovered. However, analytical results explaining this chaotic nature have had to wait until the publication of the following papers: [1] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, Progress and Challenges in Dynamical Systems, Springer, 54, 351-366 (2013). [2] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete and continuous dynamical systems series B. 19 - 2, 523-541 (2014). [3] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Chaotic dynamics for 2-D tent maps, Nonlinearity, 28, 407-434 (2015). [4] José F. Alves, A. Pumariño and E. Vigil, Statistical stability for multidimensional piecewise expanding maps, preprint (2014). Without doubt, the most important advancement in this line of research has been the definition of what we have called Expanding Baker Maps (EBMs). These maps, as the name tries to explain, reproduce the method used by a baker to knead the bread: a two-dimensional domain is constantly bending and stretching until the final product is obtained. In our terms, the final product is not more than the attractor that arises in the corresponding dynamics. Attractor understood as the final product obtained by iterating a dissipative dynamics. While this type of dynamics (fold and expansion) had been discovered and thoroughly studied in the sixties in dimension one, the case is a milestone in the field of two-dimensional dynamical systems. In the same way as these dynamics have been extremely important in the study of homoclinic tangencies in two dimensions, our two-dimensional EBMs are undoubtedly placed as a fundamental tool in the study of three-dimensional homoclinic bifurcations. In this spirit we have dared to baptize the different attractors that we have found as fairy cakes attractors, bread roll attractors and country bread attractors. While these terms seem somewhat bucolic, they have been well accepted in Mathematics Community, overall due to the shape of the numerically obtained strange attractors. Also, the contents of this thesis show a first example of two-dimensional family of non-skew-product maps displaying two dimensional persistent strange attractors (surviving in an open set of parameters).