Dinâmica simbólica e ferradura de Smale

Descrevemos a “ferradura de Smale”, um sistema dinâmico bem conhecido que apresenta um conjunto de propriedades muito importantes em Sistemas Dinâmicos. O estudo da dinâmica da “ferradura de Smale” permitenos entender a importância do conceito de dinâmica simbólica.We describe the Smale horseshoe, a well-known dynamical system that presents a set of properties which are very important in Dynamical Systems. The study of the dynamics of the Smale horseshoe allows us to understand the importance of the notion of symbolic dynamics

Invariant manifolds of the Bonhoeffer-van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.Comment: 14 pages, 15 figure

A Note on the Existence of a Smale Horseshoe in the Planar Circular Restricted Three-Body Problem

It has been proved that, in the classical planar circular restricted three-body problem, the degenerate saddle point processes transverse homoclinic orbits. Since the standard Smale-Birkhoff theorem cannot be directly applied to indicate the chaotic dynamics of the Smale horseshoe type, we in this note alternatively apply the Conley-Moser conditions to analytically prove the existence of a Smale horseshoe in this classical restricted three-body problem

Chaotic solutions in non linear economic-financial models

Following Mulligan and Sala-i-Martin (1993) we study a general class of endogenous growth models formalized as a non linear autonomous three-dimensional differential system. We consider the abstract model. By using the Shilnikov Theorem statements, we determine the parameters space in which the condition for the existence of a homoclinic Shilnikov orbit and Smale horseshoe chaos are true. The Lucas model (1998) can be considered as an application of the general result. The series expression of the homoclinic orbit is derived by the undetermined coecient method. We show the optimality for the solutions path based on the Shilnikov Theorem. Some economic implications of this analysis are discussed. Keywords: homoclinic Shilnikov bifurcation, Smale horseshoe chaos.

The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids

Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics.Comment: 16 pages, 12 figure

Quantum and classical echoes in scattering systems described by simple Smale horseshoes

We explore the quantum scattering of systems classically described by binary and other low order Smale horseshoes, in a stage of development where the stable island associated with the inner periodic orbit is large, but chaos around this island is well developed. For short incoming pulses we find periodic echoes modulating an exponential decay over many periods. The period is directly related to the development stage of the horseshoe. We exemplify our studies with a one-dimensional system periodically kicked in time and we mention possible experiments.Comment: 7 pages with 6 reduced quality figures! Please contact the authors ([email protected]) for an original good quality pre-prin

Homoclinic crossing in open systems: Chaos in periodically perturbed monopole plus quadrupolelike potentials

The Melnikov method is applied to periodically perturbed open systems modeled by an inverse--square--law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced. The (modified) Smale-Birkhoff homoclinic theorem is used to study transversal homoclinic intersections. A larger class of open systems with degenerated (nonhyperbolic) unstable periodic orbits after regularization is also briefly considered.Comment: 19 pages, 15 figures, Revtex