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At least 793 records · Page 44Linked to original sources
PubMed2003

Time evolution of the extremely diluted Blume-Emery-Griffiths neural network.

A study of the time evolution and a stability analysis of the phases in the extremely diluted Blume-Emery-Griffiths neural network model are shown to yield new phase diagrams in which fluctuation retrieval may drive pattern retrieval. It is shown that saddle-point solutions associated with fluctuation overlaps slow down the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.

Biophysical Phenomena↗
PubMed2004

Long-wavelength thermocapillary instability with the Soret effect.

We study the onset of Marangoni instability of the quiescent equilibrium in a binary liquid layer with a nondeformable interface in the presence of the Soret effect. Linear stability analysis shows that both monotonic and oscillatory long-wavelength instabilities are possible depending on the value of the Soret number chi. Sets of long-wavelength nonlinear evolution equations are derived for both types of instability. Bifurcation analyses reveal that in the regime of monotonic instability square patterns bifurcate supercritically and they are preferred in competition with roll patterns. Hexagonal patterns bifurcate transcritically and the condition for the emergence of steady stable hexagonal patterns is derived. In the case of oscillatory instability, traveling and standing waves are found to bifurcate supercritically in the narrow range of the Soret parameter and traveling waves are found to become the selected type of flow.

Journal Article↗
PubMed2004

Secondary bifurcations of hexagonal patterns in a nonlinear optical system: alkali metal vapor in a single-mirror arrangement.

Secondary bifurcations of hexagonal patterns are analyzed in a model of a single-mirror arrangement with an alkali metal vapor as the nonlinear medium. A stability analysis of the hexagonal structures is performed numerically. Different instabilities are predicted in dependency on the wave number of the hexagons. Some of the instabilities take place at a finite wave number and result in the formation of structures with 12 spatial modes. These structures are compared with those observed experimentally.

Journal Article↗
PubMed2004

Pattern formation in weakly forced Taylor-Couette flow.

Low-inertia vortex formation and pattern selection are examined for axisymmetric Taylor-Couette flow with spatially modulated cylinders. The forcing is arbitrary but remains periodic. The modulation amplitude is assumed to be small, and a regular perturbation expansion is used to determine the flow field at small to moderately large Taylor numbers (below the critical threshold). It is found that the presence of a weak modulation leads unambiguously to the emergence of steady Taylor-vortex flow even at vanishingly small Taylor number. This situation is closely reminiscent of the effect of end plates, and the consequent onset of imperfect bifurcation. The vortex structure is found to have the same periodicity as the forcing when only one of the cylinders is modulated, or when the modulations are commensurate. For incommensurate modulations, the vortex pattern is quasiperiodic, with regions of almost purely azimuthal flow. When the counter-rotation speed of the outer cylinder increases, the original vortices are gradually replaced by new ones that end up spanning the entire gap width, and in turn break up into two vortices resulting in two rows of vortices commensurate with each cylinder modulation. It is also shown that, for any modulation amplitude, the forcing wave number that generates the most intense vortex flow for a given Taylor number varies monotonically with Ta, but always reaches the critical value predicted by linear stability analysis for straight cylinders, regardless of which cylinder is modulated.

Journal Article↗
PubMed2004

Traveling waves in rotating Rayleigh-Bénard convection.

A combined analytical, numerical, and experimental study of the traveling-wave wall mode in rotating Rayleigh-Bénard convection is presented. No-slip top and bottom boundary conditions are used for the numerical computation of the linear stability, and the coefficients of the linear complex Ginzburg-Landau equation are then computed for various rotation rates. Numerical results for the no-slip boundary conditions are compared with free-slip calculations and with experimental data, and detailed comparison is made at a dimensionless rotation rate Omega=274. It is found that the inclusion of the more realistic no-slip boundary conditions for the top and bottom surfaces brings the numerical linear stability analysis into better agreement with the experimental data compared with results using free-slip top/bottom boundary conditions. Some remaining discrepancies may be accounted for by the finite conductivity of the sidewall boundaries.

Journal Article↗
PubMed2004

Falling films and the Marangoni effect.

The instability of a falling liquid film of an aqueous surfactant solution along a vertical slope with surfactant adsorption-desorption at its open surface originating surface stresses (Marangoni effect) is investigated. The diffusion of surfactant to the film surface from the bulk and desorption of surfactant to the gas phase are taken into account. The Navier-Stokes and Fick equations are reduced to a system of simpler hence, analytically and numerically, more tractable nonlinear evolution equations albeit with nine dimensionless parameters. The linear stability analysis yields a dispersion equation that is numerically solved and eigenvalues are obtained for various values of significant dimensionless parameters. A very rich picture of instabilities appears. In addition to the earlier known (Kapitza) hydrodynamic mode there are up to four new (Marangoni-driven) diffusion modes. Two modes travel with the liquid velocity on the film surface and the other two travel on their own downstream and upstream, respectively. One diffusion mode could be identified, in the reference frame moving with the liquid on the film surface, as a monotonic instability mode hence leading to a patterned film surface. All other modes are oscillatory ones. Resonance of modes is also predicted for suitable combinations of the parameters of the problem. The mode observed depends upon the surface stress (in terms of a dimensionless Marangoni number), the particular choice of the adsorption-desorption kinetics, and the surface tension state equation at the open surface of the film.

Journal Article↗
PubMed2004

Breather statics and dynamics in Klein-Gordon chains with a bend.

In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also examine dynamical effects including the scattering of mobile localized modes (discrete breathers) off of such a geometric structure. The potential outcomes of such numerical experiments (including transmission, trapping within the bend as well as reflection) are highlighted and qualitatively explained. Such models are of interest both theoretically in understanding the interplay of breathers with curvature, but also practically in simple models of photonic crystals or of bent chains of DNA.

Journal Article↗
PubMed2004

Clustering, advection, and patterns in a model of population dynamics with neighborhood-dependent rates.

We introduce a simple model of population dynamics which considers reproducing individuals or particles with birth and death rates depending on the number of other individuals in their neighborhood. The model shows an inhomogeneous quasistationary pattern with many different clusters of particles arranged periodically in space. We derive the equation for the macroscopic density of particles, perform a linear stability analysis on it, and show that there is a finite-wavelength instability leading to pattern formation. This is responsible for the approximate periodicity with which the clusters of particles arrange in the microscopic model. In addition, we consider the population when immersed in a fluid medium and analyze the influence of advection on global properties of the model, such as the average number of individuals.

Journal Article↗
PubMed2004

Pattern formation driven by nematic ordering of assembling biopolymers.

The biopolymers actin and microtubules are often in an ongoing assembling-disassembling state far from thermal equilibrium. Above a critical density this leads to spatially periodic patterns, as shown by a scaling argument and in terms of a phenomenological continuum model, which meets also Onsager's statistical theory of the nematic-to-isotropic transition in the absence of reaction kinetics. This pattern forming process depends much on nonlinear effects and a common linear stability analysis of the isotropic distribution of the filaments is often misleading. The wave number of the pattern decreases with the assembling-disassembling rate and there is an uncommon discontinuous transition between the nematic and periodic states.

Actins↗
PubMed2004

Spiral wave drift in an electric field and scroll wave instabilities.

Here, I present the numerical computation of speed and direction of the drift of a spiral wave in an excitable medium in the presence of an electric field. The drift speed presents a strong variation close to the parameter value where the drift-speed component along the field direction from parallel becomes antiparallel. Using a simple phenomenological model and results from a numerical linear stability analysis of scroll waves, I show that this behavior can be attributed to a resonance of the meander modes with the translation modes of the spiral wave. Extending this phenomenological model to scroll waves also clarifies the link between the drift and long wavelength instabilities of scroll waves.

Journal Article↗
PubMed2004

Stabilizing the absolutely or convectively unstable homogeneous solutions of reaction-convection-diffusion systems.

We study the problem of stabilization of a homogeneous solution in a two-variable reaction-convection-diffusion one-dimensional system with oscillatory kinetics, in which moving or stationary patterns emerge in the absence of control. We propose to use a formal spatially weighted feedback control to suppress patterns in an absolutely or convectively unstable system and pinning control for a convectively unstable system. The latter approach is very effective and may require only one actuator to adjust feed conditions. In the former approach, the positive diagonal elements of the appropriate dynamics matrix are shifted to the left-hand part of the complex plane to ensure linear (asymptotic) stability of the system according to Gershgorin criterion. Moreover, we construct a controller that (with many actuators) will approach the global stability of the solution, according to Liapunov's direct method. We apply two alternative approaches to reveal the unstable modes: an approximate one that is based on linear stability analysis of an unbounded system, and an exact one that uses a traditional eigenstructure analysis of bounded systems. The number of required actuators increases dramatically with system size and with the distance from the bifurcation point. The methodology is developed for a system with learning cubic kinetics and is tested on a more realistic cross-flow reactor model.

Journal Article↗
PubMed2004

Convective instability of magnetic fluids.

A theoretical investigation of the convective instability problem in the thin horizontal layer of a magnetic fluid heated from below is carried out. The effects of the relaxation time tau and the vortex (rotational) viscosity xi are considered and discussed. The Chebyshev pseudospectral method is employed to solve the eigenvalue problems and numerical calculations are carried out for a number of magnetic fluids and in full range of the magnetic field. A variety of results under gravity-free conditions are also presented and the critical temperature gradient are determined for a variety of situations. It is shown that the consideration of (xi,tau), in the stability analysis, is most effective in the thin layer of the fluid and at low values of Langevin parameter alpha(L).

Journal Article↗
PubMed2004

Strongly asymmetric soliton explosions.

We have observed, in numerical simulations, strongly asymmetric soliton explosions in dissipative systems modeled by the one-dimensional complex cubic-quintic Ginzburg-Landau equation. The explosions occur at one side of the soliton, in spite of the fact that the initial conditions and the equation itself are symmetric. From one explosion to the next, the side of the soliton where it occurs alternates, so that the left- and right-hand sides of the soliton explode successively. We give explanations for this effect based on a linear stability analysis of the unstable soliton. We also describe the transition from the stationary soliton into the exploding one as a three-stage process of soliton cooling.

Journal Article↗
PubMed2004

Cylindrically symmetric electrohydrodynamic patterning.

Cylindrically symmetric structures such as concentric rings and rosettes arise out of thin polymeric films subjected to strong electric fields. Experiments that formed concentric rings and theory capable of explaining these and other cylindrical structures are presented. These rings represent an additional member of a class of structures, including pillars and holes, formed by electrohydrodynamic patterning of thin films, occasionally referred to as lithographically induced self-assembly. Fabrication of a set of concentric rings begins by spin coating a thin poly(methyl methacrylate) film onto a silicon wafer. A mask is superimposed parallel to the film leaving a similarly thin air gap. Electric fields, acting in opposition to surface tension, destabilize the free interface when raised above the glass transition temperature. Central pillars nucleate under small cylindrical protrusions patterned on the mask. Rings then emerge sequentially, with larger systems having as many as 10 fully formed rings. Ring-to-ring spacings and annular widths, typically on the order of a micron, are approximately constant within a concentric cluster. The formation rate is proportional to the viscosity and, consequently, has the expected Williams-Landel-Ferry dependence on temperature. In light of these developments we have undertaken a linear stability analysis in cylindrical coordinates to describe these rings and ringlike structures. The salient feature of this analysis is the use of perturbations that incorporate their radial dependence in terms of Bessel functions as opposed to the traditional sinusoids of Cartesian coordinates. The theory predicts approximately constant ring-to-ring spacings, constant annular widths, and growth rates that agree with experiment. A secondary instability is observed at higher temperatures, which causes the rings to segment into arcs or pillar arrays. The cylindrical theory may be generalized to describe hexagonal pillar/hole packing, gratings, and rosettes with the first being of particular importance given the ubiquitous observation of hexagonal packing. The perturbation analysis presented here is relevant to any system with cylindrical symmetry, for which the radial dependence can be described in terms of Bessel functions.

Journal Article↗
PubMed2004

Square vortex solitons with a large angular momentum.

We show the existence of square-shaped optical vortices with a large value of the angular momentum hosted in finite-size laser beams which propagate in nonlinear media with a cubic-quintic nonlinearity. The light profiles take the form of rings with sharp boundaries and variable sizes depending on the power carried. Our stability analysis shows that these light distributions remain stable when they propagate, probably for unlimited values of the angular momentum, provided the hosting beam is wide enough. This happens if the peak amplitude approaches a critical value which only depends on the nonlinear refractive index of the material. A variational approach allows us to calculate the main parameters involved. Our results add extra support to the concept of surface tension of light beams that can be considered as a trace of the existence of a liquid of light.

Journal Article↗
PubMed2005

Multiplicity of nonlinear thermal convection in a spherical shell.

Linear and weakly nonlinear thermal convection in a moderately thin spherical shell in the presence of a spherically symmetric gravity subject to a spherically symmetric boundary condition is systematically investigated through fully three-dimensional numerical simulations. The convection problem is self-adjoint and the linear convective stability is characterized by l, the degree of a spherical harmonics Yml (theta,phi). While the radial structure of the linear convection is determined by the stability analysis, there exists a (2l + 1)-fold degeneracy in the horizontal structure of the spherical convection. When l = O(10) , i.e., in a moderately thin spherical shell, the removal or partial removal of the degeneracy represents a mathematically difficult, physically not well-understood problem. By starting with carefully chosen initial conditions, we are able to obtain a variety of nonlinear convective flows at exactly the same parameters near the onset of convection, including steady axisymmetric convection, steady azimuthally periodic convection, steady azimuthally nonperiodic convection, equatorially asymmetric convection, and steady convection in the form of a single giant spiral roll covering the whole spherical shell which is stable and robust for a wide range of the Prandtl number.

Journal Article↗
PubMed2005

Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscillatory behavior.

We study the amplification of information transfer in excitable systems. We show that excitable systems residing in a steady state near a bifurcation point to complex oscillatory behavior incorporate several frequencies that can be exploited for a resonant amplification of information transfer. In particular, for excitable neurons that reside in a steady state near a bifurcation point to elliptic bursting oscillations, we show that in addition to the resonant frequency of damped oscillations around the stable focus, another frequency exists that resonantly enhances large amplitude bursts and thus amplifies the information transfer in the system. This additional frequency cannot be found by the local stability analysis and has never been used for amplifying the information transfer in a system. The results obtained for elliptic bursting oscillations can be generalized also to other complex oscillators, such as parabolic or square-wave bursters. Additionally, the biological importance of presented results in the field of neuroscience is outlined.

Adaptation, Physiological↗
PubMed2005

Tilt grain boundary instabilities in three-dimensional lamellar patterns.

We identify a finite wave-number instability of a 90 degrees tilt grain boundary in three-dimensional lamellar phases which is absent in two-dimensional configurations. Both a stability analysis of the slowly varying amplitude or envelope equation for the boundary, and a direct numerical solution of an order parameter model equation are presented. The instability mode involves two-dimensional perturbations of the planar base boundary, and is suppressed for purely one-dimensional perturbations. We find that both the most unstable wave numbers and their growth rate increase with epsilon , the dimensionless distance away from threshold of the lamellar phase.

Journal Article↗