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.

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.

Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call generating. © 2008 Cambridge University Press.

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

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 chaotic behavior of a fixed offshore structure

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

Hopf-zero bifurcations of reversible vector fields

We study the dynamics of a class of reversible vector fields having eigenvalues (0, alphai, -alphai) around their symmetric equilibria. We give a complete list of all normal forms for such vector fields, their versal unfoldings, and the corresponding bifurcation diagrams of the codimensional-one case. We also obtain some important conclusions on the existence of homoclinic and heteroclinic orbits, invariant tori and symmetric periodic orbits.

Relaxation-chaos phenomena in celestial mechanics. I. On wisdom's model for the 3/1 kirkwood gap

Sudden eccentricity increases of asteroidal motion in 3/1 resonance with Jupiter were discovered and explained by J. Wisdom through the occurrence of jumps in the action corresponding to the critical angle (resonant combination of the mean motions). We pursue some aspects of this mechanism, which could be termed relaxation-chaos: that is, an unconventional form of homoclinic behavior arising in perturbed integrable Hamiltonian systems for which the KAM theorem hypothesis do not hold. © 1987.

Intersecções homoclínicas

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

Obtenção de hibridos lapdesf fur-fd e atividade anti-inflamatória com potencial atividade neuroprotetora

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Variedade central para laços homoclínicos

Pós-graduação em Matemática - IBILCE

New treatment strategies for Alzheimer's disease: is there a hope?

Alzheimer's disease (AD) is a progressive and irreversible neurodegenerative disease, and corresponds to the most common cause of dementia worldwide. Although not fully understood, the pathophysiology of AD is largely represented by the neurotoxic events triggered by the beta-amyloid cascade and by cytoskeletal abnormalities subsequent to the hyperphosphorylation of microtubule-associated Tau protein in neurons. These processes lead respectively to the formation of neuritic plaques and neurofibrillary tangles, which are the pathological hallmarks of the disease. Clinical benefits of the available pharmacological treatment for AD with antidementia drugs (namely cholinesterase inhibitors and memantine) are unquestionable, although limited to a temporary, symptomatic support to cognitive and related functions. Over the past decade, substantial funding and research have been dedicated to the search and development of new pharmaceutical compounds with disease-modifying properties. The rationale of such approach is that by tackling key pathological processes in AD it may be possible to attenuate or even change its natural history. In the present review, we summarize the available evidence on the new therapeutic approaches that target amyloid and Tau pathology in AD, focusing on pharmaceutical compounds undergoing phase 2 and phase 3 randomized controlled trials.

Multifractality, stickiness, and recurrence-time statistics

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