José Ángel Rodríguez

Persistence of instanton connections in chemical reactions with time-dependent rates.

The evolution of a system of chemical reactions can be studied, in the eikonal approximation, by means of a Hamiltonian dynamical system. The fixed points of this dynamical system represent the different states in which the chemical system can be found, and the connections among them represent instantons or optimal paths linking these states. We study the relation between the phase portrait of the Hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time. We show that the topology of the phase space is robust for small time-dependent perturbations in concrete examples and state general results when possible. This robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time-dependent situation.

Chaotic dynamics for two-dimensional tent maps

For a two-dimensional extension of the classical one-dimensional family of tent maps, we prove the existence of an open set of parameters for which the respective transformation presents a strange attractor with two positive Lyapounov exponents. Moreover, periodic orbits are dense on this attractor and the attractor supports a unique ergodic invariant probability measure.

Piecewise Linear Bidimensional Maps as Models of Return Maps for 3D Diffeomorphisms

The goal of this paper is to study the dynamics of a simple family of piecewise linear maps in dimension two, that we call Expanding Baker Maps (EBM), which is a simplified model of a quadratic limit return map which appears in the study of certain homoclinic bifurcations of two-parameter families of three-dimensional dissipative diffeomorphisms. In spite of its simplicity the EBM capture some of the more relevant dynamics of the quadratic family, specially that related to the evolution of 2D strange attractors.

ON THE BOUNDARY OF TOPOLOGICAL CHAOS FOR THE MILNOR–THURSTON WORLD

In the present paper we characterize the boundary of topological chaos and the levels of the entropy for the Milnor–Thurston shift map.

Complexity and Dynamical Uncertainty

Uncertainty is usually linked to non-deterministic evolutions. Nevertheless, along the second half of the past century deterministic phenomena with unpredictable behaviour were discover and the notion of strange attractor emerged as the new paradigm to describe chaotic behaviours. The goal of this paper is to review all this story and to provide a perspective of the state of the art regarding this subject.

ON BIFURCATION SETS FOR SYMBOLIC DYNAMICS IN THE MILNOR–THURSTON WORLD

We show the continuity of the topological entropy for the Milnor–Thurston world of interval maps and we compute the minimum and the maximum values for the entropy of a maximal sequence of any given period. We also study (fractal) geometric properties of the bifurcation set in the parameter space and in the associated phase spaces Σ[a, b], and we compare these results with the previously known results about the lexicographic world of interval maps (related to Lorenz-like maps).