Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

We consider the 2:1 internal resonances (such that Ω1>0 and Ω2 ≃ 2Ω1 are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness Λ is such that 0<Λ<π (to avoid the Rayleigh instability) and only the first eight capillary modes are considered. A normal form is derived that gives the slow evolution (in the viscous time scale) of the complex amplitudes of the eigenmodes associated with Ω1 and Ω2, and consists of two complex ODEs that are balances of terms accounting for inertia, damping, detuning from resonance, quadratic nonlinearity, and forcing. In order to obtain quantitatively good results, a two-term approximation is used for the damping rate. The coefficients of quadratic terms are seen to be nonzero if and only if the eigenmode associated with Ω2 is even. In that case the quadratic normal form possesses steady states (which correspond to mono- or bichromatic oscillations of the liquid bridge) and more complex periodic or chaotic attractors (corresponding to periodically or chaotically modulated oscillations). For illustration, several bifurcation diagrams are analyzed in some detail for an internal resonance that appears at Λ ≃ 2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Ω2 is odd, and only one of the eigenmodes associated with Ω1 and Ω2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple

Weakly Nonuniform Thermal Effects in a Porous Catalyst: Asymptotic Models and Local Nonlinear Stability of the Steady States.

This paper considers a first-order, irreversible exothermic reaction in a bounded porous catalyst, with smooth boundary, in one, two, and three space dimensions. It is assumed that the characteristic reaction time is sufficiently small for the chemical reaction to be confined to a thin layer near the boundary of the catalyst, and that the thermal diffusivity is large enough for the temperature to be uniform in the reaction layer, but that it is not so large as to avoid significant thermal gradients inside the catalyst. For appropriate realistic limiting values of the several nondimensional parameters of the problem, several time-dependent asymptotic models are derived that account for the chemical reaction at the boundary (that becomes essentially impervious to the reactant), heat conduction inside the catalyst, and exchange of heat and reactant with the surrounding unreacted fluid. These models possess asymmetrical steady states for symmetric shapes of the catalyst, and some of them exhibit a rich dynamic behavior that includes quasi-periodic phenomena. In one case, the linear stability of the steady states, and also the local bifurcation to quasi-periodic solutions via center manifold theory and normal form reduction, are analyzed

Viscous Faraday waves in two-dimensional large-aspect-ratio containers

A weakly nonlinear analysis of one-dimensional viscous Faraday waves in two-dimensional large-aspect-ratio containers is presented. The surface wave is coupled to a viscous long-wave mean flow that is slaved to the free-surface deformation. The relevant Ginzburg–Landau-like amplitude equations are derived from first principles, and can be of three different types, depending on the ratio between wavelength, depth and the viscous length. These three equations are new in the context of Faraday waves. The coefficients of these equations are calculated for arbitrary viscosity and compared with their counterparts in the literature for small viscosity; a discrepancy in the cubic coefficient is due to a dramatic sensitivity of this coefficient on a small wavenumber shift due to interplay between viscous effects and parametric forcing

Standing wave description of nearly conservative, parametrically excited waves in extended systems

We consider the standing wavetrains that appear near threshold in a nearly conservative, parametrically excited, extended system that is invariant under space translations and reflection. Sufficiently close to threshold, the relevant equation is a Ginzburg-Landau equation whose cubic coefficient is extremely sensitive to wavenumber shifts, which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative. This latter equation is derived from the standard system of amplitude equations for counterpropagating waves, whose validity is well established today. The coefficients of the amplitude equation for standing waves are obtained for 1D Faraday waves in a deep container, to correct several gaps in former analyses in the literature. This application requires to also consider the effect of the viscous mean flow produced by the surface waves, which couples the dynamics of the surface waves themselves with the free surface deformation induced by the mean flow

The Asymptotic Justification of a Nonlocal 1-D Model Arising in Porous Catalyst Theory

An asymptotic model of isothermal catalyst is obtained from a well-known model of porous catalyst for appropriate, realistic limiting values of some nondimensional parameters. In this limit, the original model is a singularly perturbedm-D reaction–diffusion system. The asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi-infinite one-dimensional interval

Faraday instability threshold in large-aspect-ratio containers

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the values of the parameters. We classify all distinguished limits for varying values of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with (a) the numerical solution of the original exact problem, (b) some ad hoc approximations in the literature, and (c) some experimental results

Sensibilidad de una tarea de tiempo de reacción al procesamiento automático de estímulos neutros enmascarados

[Resumen] El objetivo de este experimento era estudiar si los estímulos neutros reciben procesamiento automático cuando son presentados eficazmente enmascarados en un paradigma de priming. Los primes consistían en las letras "o", "e", o "u". Las mismas letras "o" y "e" fueron utilizadas como target, dando lugar a tres condiciones de enmascaramiento, cada una de ellas con dos combinaciones prime/target (congruente: e/e y %; incongruente: o/e y e/o; neutra: u/e y u/o). Además, el target fue considerado como un estímulo imperativo para una tarea de tiempo de reacción (TR). Treinta y seis sujetos fueron expuestos a 20 ensayos de cada condición de enmascaramiento con una asincronía del estímulos de 23 milisegundos. Los resultados demuestran que la presentación eficazmente enmascarada de un estímulo neutro produce facilitación o interferencia con la tarea de TR, dependiendo de si el target es un estímulo congruente o incongruente con el prime.[Abstract] Sensitivity of a reaction time task to automatic processing of neutral masked stimu1i The purpose of this experiment was to study if the neutral stimuli receive automatic processing when presented effective1y masked in a priming paradigm. The primes consisted of the "o", "e", or "u" 1etters. The same "o" and "e" 1etters were used as target, giving three masking conditions, each with two prime/target combinations (congruent -e/e and 0/0-; incongruent -o/e and e/o-; neutral -u/e and u/o-). Furthennore, the target was considered an imperative stimu1us for a reaction time (RT) task. Thirty-six subjects were exposed to 20 tria1s of each masking conditions with a stimu1us- onset asynchrony of 23 milliseconds. The results demonstrate that an effective masking presentation of a neutral stimulus produces either facilitation or interference with the RT task, depending on whether the target is a congruent or an incongruent stimu1us with the prim

On the linearization of some singular, nonlinear elliptic problems and applications

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutions to some semilinear, singular elliptic problems

Circular strings, wormholes and minimum size

The quantization of circular strings in an anti-de Sitter background spacetime is performed, obtaining a discrete spectrum for the string mass. A comparison with a four-dimensional homogeneous and isotropic spacetime coupled to a conformal scalar field shows that the string radius and the scale factor have the same classical solutions and that the quantum theories of these two models are formally equivalent. However, the physically relevant observables of these two systems have different spectra, although they are related to each other by a specific one-to-one transformation. We finally obtain a discrete spectrum for the spacetime size of both systems, which presents a nonvanishing lower bound.Comment: 11 pages, LaTeX2e, minor change

Reactive-diffuse System with Arrhenius Kinetics: Peculiarities of the Spherical Goemetry

The steady reactive-diffusive problem for a non isothermal permeable pellet with first-order Arrhenius kinetics is studied. In the large activation-energy limit, asymptotic solutions are derived for the spherical geometry. The solutions exhibit multiplicity and it is shown that a suitable choice of parameters can lead to an arbitrarily large number of solutions, thereby confirming a conjecture based upon past computational experiments. Explicit analytical expressions are given for the multiplicity bounds (ignition and extinction limits). The asymptotic results compare very well with those obtained numerically, even for moderate values of the activation energy