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.

Velocity-dependent KAM islands

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

On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin-Huxley (HH) mathematical model

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

Enhanced multiwavelength holographic profilometry by laser mode selection

A method for improving the accuracy of surface shape measurement by multiwavelength holography is presented. In our holographic setup, a Bi12TiO20 photorefractive crystal was the holographic recording medium, and a multimode diode laser emitting in the red region was the light source in a two-wave mixing scheme. on employing such lasers the resulting holographic image appears covered with interference fringes corresponding to the object relief, and the interferogram spatial frequency is proportional to the diode laser's free spectral range (FSR). Our method consists in increasing the effective free spectral range of the laser by positioning a Fabry-Perot etalon at the laser output for mode selection. As larger effective values of the laser FSR were achieved, higher-spatial-frequency interferograms were obtained and therefore more sensitive and accurate measurements were performed. The quantitative evaluation of the interferograms was made through the phase-stepping technique, and the phase map unwrapping was carried out through the cellular-automata method. For a given surface, shape measurements with different interferogram spatial frequencies were performed and compared with respect to measurement noise and visual inspection. (c) 2007 Society of Photo-Optical Instrumentation Engineers.

Multi-wavelength holography in Bi12TiO20 crystals: Applications in refractometry

We present a recent development in holography with multimode, large free-spectral range (FSR) diode lasers in photorefractive sillenite crystals. A novel refractometry method based on this type of holographic recording in Bi12TiO20 (BTO) crystals is proposed. The holographic image of a prism-shaped transparent sample appears covered of interference fringes, and as the sample is properly translated, the fringes run along the holographic image. An expression providing the refractive index of the medium as a function of the sample displacement and the correspondent number of running fringes was derived. The refractive indexes of optical (BK7) glass, ethanol, hexan, cumene and aqueous solution of NaCl with different concentrations were measured in order to test the method. The obtained results are in good agreement with the ones reported in literature or measured by us using a commercial Abbe refractometre. (c) 2006 Elsevier B.V. All rights reserved.

Multiple active levels in Bi12TiO20 crystals

The existence of multiple active levels in a photorefractive Bi12TiO20 crystal is here investigated at 514.5nm wavelength. We carry out two-wave mixing experiments using symmetrically incident beams of equal intensities. A large amplitude fast phase modulation in one of the beams reduces the fringes visibility and improves the detection of the generated frequency modulated signals in both (R and S) output directions. Diffraction efficiencies of the phase (photorefractive) and the absorption (photochromic) gratings are quantitatively computed as functions of the grating period. Results show that the absorption grating has two distinct components: one associated to the photorefractive trap density modulation and another related to local light-induced effects between different levels. The photorefractive grating was also investigated at 633nm and 594nm (besides 514.5nm) and a significant quenching of the photorefractive effect was observed at these wavelengths.

Denisiuk-type reflection holography display with sillenite crystals for imaging and interferometry of small objects

A novel setup for imaging and interferometry through reflection holography with Bi12TiPO20(BTO) sillenite photorefractive crystals is proposed. A variation of the lensless Denisiuk arrangement was developed resulting in a compact, robust and simple interferometer. A red He-Ne laser was used as light source and the holographic recording occurred by diffusion with the grating vector parallel to the crystal [0 0 1]-axis. In order to enhance the holographic image quality and reduce noise a polarizing beam splitter (PBS) was positioned at the BTO input and the crystal was tilted around the [0 0 1]-axis. This enabled the orthogonally polarized transmission and diffracted beams to be separated by the PBS, providing the holographic image only. The possibility of performing deformation and strain analysis as well as vibration measurement of small objects was demonstrated. (C) 2007 Elsevier B.V. All rights reserved.

Measurement of low-derivative surface lenses by two-laser holography with Bi12TiO20 crystals

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Design, construction, and performance of a real-time holographic refractometry prototype for liquid analysis

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Affine Toda systems coupled to matter fields

We investigate higher grading integrable generalizations of the affine Toda systems, where the flat connections defining the models take values in eigensubspaces of an integral gradation of an affine Kac-Moody algebra, with grades varying from l to -l (l > 1). The corresponding target space possesses nontrivial vacua and soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. The models can also be formulated by a hamiltonian reduction procedure from the so-called two-loop WZNW models. We construct the general solution and show the classes corresponding to the solitons. Some of the particles and solitons become massive when the conformal symmetry is spontaneously broken by a mechanism with an intriguing topological character and leading to a very simple mass formula. The massive fields associated to nonzero grade generators obey field equations of the Dirac type and may be regarded as matter fields. A special class of models is remarkable. These theories possess a U(1 ) Noether current, which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one-dimensional bag model for QCD. These models are also relevant to the study of electron self-localization in (quasi-)one-dimensional electron-phonon systems.

Autosolitons in trapped Bose-Einstein condensates with two- and three-body inelastic processes

The conditions for the existence of autosolitons were considered in trapped Bose-Einstein condensates with attractive atomic interactions. The expression for the parameters of the autosoliton was derived using the time-dependent variational approach for the nonconservative 3-dimensional Gross-pitaevskii equation and their stability was checked. The results were in agreement with the exact numerical calculations. It was shown that the transition from unstable to stable point solely depends on the magnitude of the parameters.

Confinement and soliton solutions in the SL(3) Toda model coupled to matter fields

We consider an integrable conformally invariant two-dimensional model associated to the affine Kac-Moody algebra sl3(ℂ). It possesses four scalar fields and six Dirac spinors. The theory does not possesses a local Lagrangian since the spinor equations of motion present interaction terms which are bilinear in the spinors. There exists a submodel presenting an equivalence between a U(1) vector current and a topological current, which leads to a confinement of the spinors inside the solitons. We calculate the one-soliton and two-soliton solutions using a procedure which is a hybrid of the dressing and Hirota methods. The soliton masses and time delays due to the soliton interactions are also calculated. We give a computer program to calculate the soliton solutions. © 2002 Published by Elsevier Science B.V.

Soliton spectrum of integrable models with local symmetries

The soliton spectrum (massive and massless) of a family of integrable models with local U(1) and U(1) ⊗U(1) symmetries is studied. These models represent relevant integrable deformations of SL(2,ℝ) ⊗U(1) n-1-WZW and SL(2,ℝ) ⊗ SL(2,ℝ) ⊗U(1) n-2-WZW models. Their massless solitons appear as specific topological solutions of the U(1)(or U(1) ⊗ U(1)-) CFTs. The nonconformal analog of the GKO-coset formula is derived and used in the construction of the composite massive solitons of the ungauged integrable models. © SISSA/ISAS 2002.

Multicharged dyonic integrable models

We introduce and study new integrable models (IMs) of An (1)-nonabelian Toda type which admit U(1) ⊗ U(1) charged topological solitons. They correspond to the symmetry breaking SU(n + 1) → SU(2) ⊗ SU(2) ⊗ U(1)n-2 and are conjectured to describe charged dyonic domain walls of N = 1 SU(n + 1) SUSY gauge theory in large n limit. It is shown that this family of relativistic IMs corresponds to the first negative grade q = -1 member of a dyonic hierarchy of generalized cKP type. The explicit relation between the 1-soliton solutions (and the conserved charges as well) of the IMs of grades q = -1 and q = 2 is found. The properties of the IMs corresponding to more general symmetry breaking SU(n + 1) → SU(2)⊗p ⊗ U(1)n-p as well as IM with global SU(2) symmetries are discussed. © 2002 Elsevier Science B.V. All rights reserved.