Low dimensional behavior in three-dimensional coupled map lattices

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Multifractal regime transition in a modified minority game model

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Dipolar Bose-Einstein condensate soliton on a two-dimensional optical lattice

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

Self-dual hopfions

We construct static and time-dependent exact soliton solutions with nontrivial Hopf topological charge for a field theory in 3 + 1 dimensions with the target space being the two dimensional sphere S(2). The model considered is a reduction of the so-called extended Skyrme-Faddeev theory by the removal of the quadratic term in derivatives of the fields. The solutions are constructed using an ansatz based on the conformal and target space symmetries. The solutions are said self-dual because they solve first order differential equations which together with some conditions on the coupling constants, imply the second order equations of motion. The solutions belong to a sub-sector of the theory with an infinite number of local conserved currents. The equation for the profile function of the ansatz corresponds to the Bogomolny equation for the sine-Gordon model.

Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Grassmannian and bosonic Thirring models with jump defects

In this paper we discuss the Lax formulation of the Grassmannian and Bosonic Thirring models in the presence of jump defects. For the Grassmannian case, the defect is described by Backlund transformation which is responsible for preserving the integrability of the model. We then propose an extension of the Backlund transformation for the Bosonic Thirring model which is verified by some Backlund transitions like vacuum-one soliton, one soliton-one soliton, one soliton-two solitons and two solitons-two solitons. The Lax formulation within the space split by the defect leads to the integrability of Bosonic Thirring model with jump defects.

Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices

The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.

Soliton dynamics at an interface between a uniform medium and a nonlinear optical lattice

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

Permutability of Backlund transformations for N=2 supersymmetric sine-Gordon

The permutability of two Backlund transformations is employed to construct a nonlinear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model. We present explicitly the one and two soliton solutions.

Negative Even Grade mKdV Hierarchy and its Soliton Solutions

In this paper we discuss the algebraic construction of the mKdV hierarchy in terms of an affine Lie algebra (s) over capl(2). An interesting novelty araises from the negative even grade sector of the affine algebra leading to nonlinear integro-differential equations admiting non-trivial vacuum configuration. These solitons solutions are constructed systematically from generalization of the dressing method based on non zero vacua. The sub-hierarchies admiting such class of solutions are classified.

Localized modes in chi((2)) media with PT-symmetric localized potential

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

Basins of attraction changes by amplitude constraining of oscillators with limited power supply

We investigate the dynamics of a Duffing oscillator driven by a limited power supply, such that the source of forcing is considered to be another oscillator, coupled to the first one. The resulting dynamics come from the interaction between both systems. Moreover, the Duffing oscillator is subjected to collisions with a rigid wall (amplitude constraint). Newtonian laws of impact are combined with the equations of motion of the two coupled oscillators. Their solutions in phase space display periodic (and chaotic) attractors, whose amplitudes, especially when they are too large, can be controlled by choosing the wall position in suitable ways. Moreover, their basins of attraction are significantly modified, with effects on the final state system sensitivity. (c) 2005 Elsevier Ltd. All rights reserved.

Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes

We consider a model for rattling in single-stage gearbox systems with some backlash consisting of two wheels with a sinusoidal driving; the equations of motions are analytically integrated between two impacts of the gear teeth. Just after each impact, a mapping is used to obtain the dynamical variables. We have observed a rich dynamical behavior in such system, by varying its control parameters, and we focus on intermittent switching between laminar oscillations and chaotic bursting, as well as crises, which are sudden changes in the chaotic behavior. The corresponding transient basins in phase space are found to be riddled-like, with a highly interwoven fractal structure. (C) 2004 Elsevier Ltd. All rights reserved.

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing. (C) 2007 Elsevier Ltd. All rights reserved.

Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterise the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent -2. (C) 2011 Elsevier Ltd. All rights reserved.