Time-periodic perturbation of a Lienard equation with an unbounded homoclinic loop

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Homoclinic bifurcations in reversible Hamiltonian systems

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u(iv) + au - u +f(u, b) = 0 as a model, where fis an analytic function and a, b real parameters. These equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations, but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others Hamiltonian systems and successfully employed in situations where standard methods fail. We reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations. We also plot the homoclinic values configuration in parameters space, giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations. (c) 2005 Published by Elsevier B.V.

Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On homoclinic chaos and a control chaos strategy applied to a free-joint nonholonomic manipulator

In this work, the occurrence of chaos (homoclinic scene) is verified in a robotic system with two degrees of freedom by using Poincare-Mel'nikov method. The studied problem was based on experimental results of a two-joint planar manipulator-first joint actuated and the second joint free-that resides in a horizontal plane. This is the simplest model of nonholonomic free-joint manipulators. The purpose of the present study is to verify analytically those results and to suggest a control strategy.

On stick-slip homoclinic chaos and bifurcations in a mechanical system with dry friction

In this paper we consider a self-excited mechanical system by dry friction in order to study the bifurcational behavior of the arisen vibrations. The oscillating system consists of a mass block-belt-system which is self-excited by static and Coulomb friction. We analyze the system behavior numerically through bifurcation diagrams, phase portraits, frequency spectra and Poincare maps, which show the existence of nonhomoclinic and homoclinic chaos and a route to homoclinic chaos. The homoclinic chaos is also analyzed analytically via the Melnikov prediction method. The system dynamic is characterized by the existence of two potential wells in the phase plane which exhibit rich bifurcational and chaotic behavior.

HOPF BIFURCATION FROM LINES of EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bifurcation analysis of a new Lorenz-like chaotic system

In this paper we study the local codimension one and two bifurcations which occur in a family of three-dimensional vector fields depending on three parameters. An equivalent family, depending on five parameters, was recently proposed as a new chaotic system with a Lorenz-like butterfly shaped attractor and was studied mainly from a numerical point of view, for particular values of the parameters, for which computational evidences of the chaotic attractor was shown. In order to contribute to the understand of this new system we present an analytical study and the bifurcation diagrams of an equivalent three parameter system, showing the qualitative changes in the dynamics of its solutions, for different values of the parameters. (C) 2007 Elsevier Ltd. All rights reserved.

Bifurcation of limit cycles from an n-dimensional linear center inside a class of piecewise linear differential systems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

NONDEGENERATE UMBILICS, THE PATH FORMULATION and GRADIENT BIFURCATION PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

IN VITRO INCIDENCE of ROOT CANAL BIFURCATION IN MANDIBULAR INCISORS BY RADIOVISIOGRAPHY

Objective: The aim of this study was to verify, in vivo and in vitro, the prevalence of root canal bifurcation in mandibular incisors by digital radiography. Material and Methods: Four hundred teeth were analyzed for the in vivo study. Digital radiographs were taken in an orthoradial direction from the mandibular incisor and canine regions. The digital radiographs of the canine region allowed visualizing the incisors in a distoradial direction using 20 degrees deviation. All individuals agreed to participate by signing an informed consent form. The in vitro study was conducted on 200 mandibular incisors positioned on a model, simulating the mandibular dental arch. Digital radiographs were taken from the mandibular incisors in both buccolingual and mesiodistal directions. Results: The digital radiography showed presence of bifurcation in 20% of teeth evaluated in vitro in the mesiodistal direction. In the buccolingual direction, 17.5% of teeth evaluated in vivo and 15% evaluated in vitro presented bifurcation or characteristics indicating bifurcation. Conclusions: Digital radiography associated with X-ray beam distally allowed detection of a larger number of cases of bifurcated root canals or characteristics of bifurcation.

A comment on a nonideal centrifugal vibrator machine behavior with soft and hard springs

This paper concerns a type of rotating machine (centrifugal vibrator), which is supported on a nonlinear spring. This is a nonideal kind of mechanical system. The goal of the present work is to show the striking differences between the cases where we take into account soft and hard spring types. For soft spring, we prove the existence of homoclinic chaos. By using the Melnikov's Method, we show the existence of an interval with the following property: if a certain parameter belongs to this interval, then we have chaotic behavior; otherwise, this does not happen. Furthermore, if we use an appropriate damping coefficient, the chaotic behavior can be avoided. For hard spring, we prove the existence of Hopf's Bifurcation, by using reduction to Center Manifolds and the Bezout Theorem (a classical result about algebraic plane curves).

On the appearance of a Hopf bifurcation in a non-ideal mechanical problem

In this paper we studied a non-ideal system with two degrees of freedom consisting of a dumped nonlinear oscillator coupled to a rotatory part. We investigated the stability of the equilibrium point of the system and we obtain, in the critical case, sufficient conditions in order to obtain an appropriate Normal Form. From this, we get conditions for the appearance of Hopf Bifurcation when the difference between the driving torque and the resisting torque is small. It was necessary to use the Bezout Theorem, a classical result of Algebraic Geometry, in the obtaining of the foregoing results. (C) 2003 Elsevier Ltd. All rights reserved.

On chaotic behavior of a fixed offshore structure

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Quantum entanglement and fixed point Hopf bifurcation

We present the qualitative differences in the phase transitions of the mono-mode Dicke model in its integrable and chaotic versions. These qualitative differences are shown to be connected to the degree of entanglement of the ground state correlations as measured by the linear entropy. We show that a first order phase transition occurs in the integrable case whereas a second order in the chaotic one. This difference is also reflected in the classical limit: for the integrable case the stable fixed point in phase space undergoes a Hopf type whereas the second one a pitchfork type bifurcation. The calculation of the atomic Wigner functions of the ground state follows the same trends. Moreover, strong correlations are evidenced by its negative parts. (c) 2006 Elsevier B.V. All rights reserved.