# Artículos

Autor | Título | Revista |
---|---|---|

## The dynamics near invariant cantorian sets of perturbed twist maps

**Arturo Olvera y Cristobal Vargas**

Publicado en: Synergetics: Chaos and Order, M. G. Velarde editor, World Scientific, Singapoure-London. 648-705, 1988.

We consider a perturbed twist map when the pertubation in big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant cantorian set appears. From another point of view, the destruction of the IRC is associated to the appearition of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated to the existence of non birkhoff orbits. In this note we relate the different approaches.

## A continuation method to study periodic orbits of the Froeschlé map

**Arturo Olvera y Cristobal Vargas**

Publicado en: Physica-D, 72, No. 4, 351 -- 371, 1994.

The dynamics of many hamiltonian systems with three degrees of freedom is represented by the Froechlé map, which is symplectic and four dimensional. In this paper we study sequences of periodic orbits approaching the invariant tori in order to obtain information about its stability. A homotopic method is used to continue the branches of periodic orbits. We found that the existence of turning points is related to the linear stability of the periodic orbit and its rotation vector.

## Métodos de continuación. Teoría y práctica

**Arturo Olvera y Cristobal Vargas**

Publicado en: Memorias de la Escuela de Verano: Geometría Diferencial, Análisis Numérico y Ecuaciones Diferenciales Universidad Nacional de Colombia, Medellin, Editorial Ecograficas Medellin, 1994.

En este trabajo mostramos la manera de utilizar el método de continuación para calcular soluciones periódicas en mapeos y en ecuaciones diferenciales. Para este propósito tomamos como ejemplos el mapeo de Froeschlé y la ecuación diferencial de Mathieu. En el trabajo mostramos que los métodos homotópicos son herramientas muy poderozas pero aún esta lejano el poder automatizar completamente estos procedimientos. Un profundo conocimiento del problema específico es necesario para poder sacar provecho de los diferentes métodos para poder obtener la solución deseada.

## Study of the continuation methods for periodic orbits of the Froechlé Map

**Arturo Olvera y Cristobal Vargas**

Publicado en: Neural, Parallel & Scientific Computations. Vol. 2, No. 4, 421--430, 1994.

In this paper we compare different continuation methods for the searching of periodic orbits of the Froeschlé map, which is a four dimensional symplectic map. We find relations between the stability of the periodic orbits and the problem to solve the system given in the continuation method.

## New Chaotic behaviour in a Singular perturbed system

**Ignacio B. Vivancos y Antonmaria A. Minzoni**

Sometido a: Physica D

The aim of this paper is to present a new type of chaotic behavior observed in a four dimensional singular system. This model arises as a transverse perturbation of a heteroclinic cycle in R^3. The physical motivation for this study comes from the dynamo problem i.e. the generation of a magnetic field from a convective velocity field. It is shown that for an open parameter regime the system presents a "chaotic attractor". The study of this chaos is presented both numerically and analytically with the help of singular perturbation techniques.

## Waves in Piezoelectric Random Media

**Federico J. Sabina**

Sometido a: E. Inan & K.Z. Markov, eds. Continuum Models and Discrete Systems (CMDS9) World Scientific Pub. Co. (In press, 1998).

A new theory has been developed for the analysis of the overall properties and the propagation of waves though inhomogeneous piezoelectric materials with randomly distributed microstructure. The self-consistent method has been used by means of a very simple instrumentation, which allows the prediction of resonance phenomena for wavelengths comparable to the diameter of the inclusions. Here the problem of predicting the overall properties for 0-3 connectivity piezocomposites and wave propagation through a piezoelectric matrix containing piezoelectric ellipsoidal inclusions, randomly positioned, but with a fixed orientation are studied. From the overall elastic, piezoelectric and dielectric properties of the piezocomposite, an eigenvalue problem yields the phase speed and the attenuation of the waves as a function of the frequency of the waves in an equivalent homogeneous, but attenuative, piezoelectric composite. The results show a characteristic resonance phenomena which depends on the azimuthal angle and type of wave.

## Symmetry-Breaking Convective Dynamos in Spherical Shells

**Ignacio B. Vivancos, P. Chossat, P. Laure**

Publicado en: Journal of Nonlinear Science. Vol.9: pp 169-196 (1999).

The convective dynamo is the generation of a magnetic field by the convective motion of an electrically conducting fluid. We assume a spherical domain and spherically invariant basic equations and boundary conditions. The initial state of rest is then spherically symmetric. A first instability leads to purely convective flows, the pattern of which is selected according to the known classification of *O*(3)-symmetry breaking bifurcation theory. A second instability can then lead to the dynamo effect.

Computing this instability is now a purely numerical problem, because the convective flow is known only by its numerical approximation. However, since the convective flow can still possess a nontrivial symmetry group *G*_{0}, this is again a symmetry-breaking bifurcation problem. After having determined numerically the critical linear magnetic modes, we determine the action of *G*_{0} in the space of these critical modes. Applying methods of equivariant bifurcation theory, we can classify the pattern selection rules in the dynamo bifurcation.We consider various aspect ratios of the spherical fluid domain, corresponding to different convective patterns, and we are able to describe the symmetry and generic properties of the bifurcated magnetic fields.

## Hydrodynamics of an Oscillating Water Column Seawater Pump. Part I: Theoretical Aspects

**S.P.R. Czitrom, R. Godoy, E. Prado, P. Perez y R. Peralta-Fabi**

Sometido a: Ocean Technology.

A wave-driven seawater pump, composed of a resonant and an exhaust duct joined by a variable-volume air compression chamber, is studied. The time dependent form of Bernoulli’s equation, adapted to incorporate losses due to friction, vortex formation at the mouths and radiation damping, describes the pump behavior. A dimensional analysis of the pump equations shows that a proposed scale-model will perform similar to a full-scale seawater pump. Fluid oscillations in the ducts perform similar to a damped, two-mass spring system, excited by the waves. A resonant condition can be maintained, for different wave frequencies, by varying the volume of air in the compression chamber. The dimensional analysis shows that the basic behavior of the pump is linear and that its’ performance can be significantly increased by optimizing the design of the duct mouths. Linear estimates of the resonant air chamber volume and flow rate through the pump are derived.

## Hydrodynamics of an Oscillating Water Column Seawater Pump. Part II: Tuning to Monochromatic Waves

**S.P.R. Czitrom, R. Godoy, A. Olvera, E. Prado y C. Stern**

Sometido a: Ocean Technology.

A wave-driven seawater pump, composed of a resonant and an exhaust duct joined by a variable-volume air compression chamber, is studied. The time dependent form of Bernoulli’s equation, adapted to incorporate losses due to friction, vortex formation at the mouths and radiation damping, describes the pump behavior. A dimensional analysis of the pump equations shows that a proposed scale-model will perform similar to a full-scale seawater pump. Fluid oscillations in the ducts perform similar to a damped, two-mass spring system, excited by the waves. A resonant condition can be maintained, for different wave frequencies, by varying the volume of air in the compression chamber. The dimensional analysis shows that the basic behavior of the pump is linear and that its’ performance can be significantly increased by optimizing the design of the duct mouths. Linear estimates of the resonant air chamber volume and flow rate through the pump are derived.

## Estimation of the Amplitude of Resonance in the General Standard Map

**Arturo Olvera**

Sometido a: Experimental Mathematics

The goal of this paper is to formulate some conjectures about the amplitude of resonance in the General Standard Map. The main idea of this work is to expand the periodic perturbation function in Fourier series. Given any rational rotation number, we choose a finite number of harmonics of their Fourier expansion and we compute the amplitude of resonance of the reduced perturbation function of the map, using a suitable normal form around the resonance, which is valid for asymptotically small values of the perturbation parameter. For this map, we obtain a relation between the amplitude of resonance and the perturbation parameter: the amplitude is proportional to a rational power of the parameter hence can be represented as a straight line in a log-log graph. The convex hull of these straight lines gives a lower bound for the amplitude of resonance, which is valid even when the perturbation parameter is of order one. We find that some perturbation functions give rise a phenomenon that we call collapse of resonance, this means that the amplitude of resonance goes to zero for some value of the perturbation parameter. We find an empirical procedure to estimate this value of the parameter related to the collapse of resonance.

## Performance improvement of OWC systems by Parametric Resonance

**A. Olvera, E. Prado y S.P.R. Czitrom**

Publicado en: Proceedings of "Wave Energy for the New Millennium", Fourth European Wave Energy Conference, Aalborg, 2000.

Peak performance of most OWC systems occurs at resonance with the driving waves. At resonance, oscillations increase linearly in time until damping inhibits further growth. In parametric resonance, oscillations increase exponentially in time, possibly allowing for increasing the performance of OWC systems. This type of resonance occurs when one of the parameters in an oscillator varies periodically. Some ideas are presented to improve the performance of OWC systems using this phenomenon.