SEARCH · Search PubMed
Search indexed PubMed citations on genomics, clinical trials, systematic reviews and public health. Explore titles, authors and supplied subject terms, then open the PubMed record.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
We study numerically the Turing pattern in three dimensions in a FitzHugh-Nagumo-type reaction-diffusion system. We have found that interconnected periodic domain structures such as a gyroid, Fddd, and perforated lamellar structures appear in three dimensions, which never exist in lower dimensions. The stability analysis of these structures is also performed by means of a mode expansion.
Many theoretical analyses of feedback control of pattern-forming systems assume that feedback is applied at every spatial location, something that is often difficult to accomplish in experiments. This paper considers an experimentally more feasible scenario where feedback is applied at a sparse array of discrete spatial locations. We show how such feedback can be computed analytically for a class of reaction-diffusion systems and use generalized linear stability analysis to determine how dense the actuator array needs to be to select or maintain control of a given pattern state in the presence of noise. The one-dimensional Swift-Hohenberg equation is used to illustrate our theoretical results and explain earlier experimental observations on the control of the Rayleigh-Bénard convection.
We study the stability of the fixed-point solution of an array of mutually coupled logistic maps, focusing on the influence of the delay times, , of the interaction between the and maps. Two of us recently reported [Phys. Rev. Lett. 94, 134102 (2005)] that if are random enough, the array synchronizes in a spatially homogeneous steady state. Here we study this behavior by comparing the dynamics of a map of an array of delayed-coupled maps with the dynamics of a map with self-feedback delayed loops. If is sufficiently large, the dynamics of a map of the array is similar to the dynamics of a map with self-feedback loops with the same delay times. Several delayed loops stabilize the fixed point, when the delays are not the same; however, the distribution of delays plays a key role; if the delays are all odd a periodic orbit (and not the fixed point) is stabilized. We present a linear stability analysis and apply some mathematical theorems that explain the numerical results.
We generalize the Kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to Ornstein-Uhlenbeck dynamics. Such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work. Linear stability analysis of the Fokker-Planck equation for an infinite population is amenable to exact solution and we show that the incoherent state is unstable past a critical coupling strength K(c)(gamma,sigma(f)), where gamma is the inverse characteristic drifting time and sigma(f) the asymptotic frequency dispersion. Expectedly K(c)agrees with the noisy Kuramoto model in the large gamma (Schmolukowski) limit but increases slower as gamma decreases. Asymptotic expansion of the solution for gamma-->0 shows that the noiseless Kuramoto model with Gaussian frequency distribution is recovered in that limit. Thus varying a single parameter allows us to interpolate smoothly between two regimes: one dominated by the frequency dispersion and the other by phase diffusion.
The influence of time-delayed feedback upon the spatiotemporal current density patterns is investigated in a model of a semiconductor nanostructure, namely a double-barrier resonant tunneling diode. The parameters are chosen below the Hopf bifurcation, where the only stable state of the system is a spatially inhomogeneous "filamentary" steady state. The addition of weak Gaussian white noise to the system gives rise to spatially inhomogeneous self-sustained temporal oscillations that can be quite coherent. We show that applying a time-delayed feedback can either increase or decrease the regularity of the noise-induced dynamics in this spatially extended system. Using linear stability analysis, we can explain these effects, depending on the length of the delay interval. Furthermore, we study the influence of this additional control term upon the deterministic behavior of the system, which can change significantly depending on the choice of parameters.
In this paper we analyze experimentally and theoretically the competition between two pattern forming instabilities in a single mirror feedback scheme with sodium vapor as the nonlinear medium. Two types of structures with different transverse wave numbers are observed experimentally, if the spatial phase modulation of the light field is varied. This phenomenon results from the combination of a nonlinear self-lensing effect on the one hand and of the externally controlled phase front curvature of the light field on the other. A linear stability analysis yields two instabilities whose length scales match quite well the experimental findings. Further analysis reveals the mechanism of length-scale selection in this system and demonstrates the possibly crucial role of phase front curvature in optical pattern formation.
A rarefied gas between two coaxial circular cylinders of infinite length, rotating with different angular velocities and kept at a common temperature, is considered. The stability of the circumferentially as well as axially uniform flow (cylindrical Couette flow) for circumferentially uniform small disturbances is investigated on the basis of kinetic theory. The linear-stability analysis is performed using the Bhatnagar-Gross-Krook model of the Boltzmann equation and the diffuse reflection condition on the cylinders. The maximum growth rate of the disturbances is determined numerically by solving the initial and boundary value problem for the disturbances for relatively small Knudsen numbers and wide ranges of angular velocities of the cylinders. As a result, the parameter range where the cylindrical Couette flow is unstable is clarified. The result is compared with the corresponding result based on the continuum model of the compressible Navier-Stokes type. A comparison is also made with the result of a direct numerical analysis of the original Boltzmann system, obtained by the direct simulation Monte Carlo method in previous papers as well as in the present study.
We derive the second-order variation in the local static stress intensity factor of a tensile crack with a curved front. We then discuss the relevance of this result to the stability analysis of such fronts, and propose an equation of motion of planar crack fronts in heterogeneous media that contains two main ingredients--irreversibility of the propagation of the crack front and nonlinear effects.
A linear stability analysis is derived in self-imaging cavities for which the conditions for large Fresnel number are stated. In cases of both Fabry-Pérot and ring cavities a Hopf bifurcation is predicted at finite transverse wave number. The self-imaging Fabry-Pérot resonator operates as a longitudinal multimode cavity that invalidates the mean-field model. Above the bifurcation threshold, either vortex lattices, spirals, or targets occur, depending on the Fresnel number, the input intensity, and the mistunings. The time and spatial characteristics have different scales in the case of a self-imaging ring cavity, but the same sort of patterns are reported.
We investigate modulation instability (MI) in negative-index material (NIM) with a Kerr nonlinear polarization based on a derived (3+1)-dimensional nonlinear Schrödinger equation for ultrashort pulse propagation. By a standard linear stability analysis, we obtain the expression for instability gain, which unifies the temporal, spatial, and spatiotemporal MI. It is shown that negative refraction not only brings some new features to MI, but also makes MI possible in ordinary material in which it is otherwise impossible. For example, spatial MI can occur in the defocusing regime, while it only occurs in the focusing regime in ordinary material. Spatiotemporal MI can appear in NIM in the case of anomalous dispersion and defocusing nonlinearity, while it cannot appear in ordinary material in the same case. We believe that the difference between the MI in NIM and in ordinary material is due to the fact that negative refraction reverses the sign of the diffraction term, with the signs of dispersion and nonlinearity unchanged. The most notable property of MI in NIM is that it can be manipulated by engineering the self-steepening effect by choosing the size of split-ring resonator circuit elements. To sum up the MI in ordinary material and in NIM, MI may occur for all the combinations of dispersion and nonlinearity.
We present a stability analysis of a modified nonlinear Schrödinger equation describing the propagation of ultrashort pulses in negative refractive index media. Moreover, using methods of quantum statistics, we derive a kinetic equation for the pulses, making it possible to analyze and describe partial coherence in metamaterials. It is shown that a unique short pulse soliton, which is found analytically, can propagate in the medium.
We report in this paper the study of modulated wave trains in the one-dimensional (1D) discrete Ginzburg-Landau model. The full linear stability analysis of the nonlinear plane wave solutions is performed by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. In particular, it is shown that a threshold exists for the amplitude and above this threshold, the induced modulational instability leads to the formation of ordered and disordered patterns. The theoretical findings have been numerically tested through direct simulations and have been found to be in agreement with the theoretical prediction. We show numerically that modulational instability is also an indicator of the presence of discrete solitons as were early predicted to exist in Ginzburg-Landau lattices.
We report on and analyze the textural transformations in islands, thicker circular domains, floating in very thin free standing chiral Smectic-C* liquid crystal films. As an island is growing, an initial pure bend texture of the director changes into a reversing spiral at a critical size. Another distinct spiral texture is induced by changing the boundary condition at the central point defect in the island. To understand these transformations from a pure bend island, a linear stability analysis of the c-director free energy is developed, which predicts a state diagram for the island. Our observations are consistent with the theoretical phase diagram.
We investigate the connection between local and global dynamics in the Fermi-Pasta-Ulam (FPU) beta model from the point of view of stability of its simplest periodic orbits (SPO's). In particular, we show that there is a relatively high-q mode [q = 2(N + 1)/3] of the linear lattice, having one particle fixed every two oppositely moving ones (called SPO2 here), which can be exactly continued to the nonlinear case for N = 5 + 3m, m = 0,1,2,, and whose first destabilization E(2u), as the energy (or beta) increases for any fixed N, practically coincides with the onset of a "weak" form of chaos preceding the breakdown of FPU recurrences, as predicted recently in a similar study of the continuation of a very low (q = 3) mode of the corresponding linear chain. This energy threshold per particle behaves like E(2u)/N alpha N(-2). We also follow exactly the properties of another SPO [with q = (N + 1)/2] in which fixed and moving particles are interchanged (called SPO1 here) and which destabilizes at higher energies than SPO2, since E(1u)/N alpha N(-1). We find that, immediately after their first destabilization, these SPO's have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed N), these spectra converge to the same exponentially decreasing function, thus providing strong evidence that the chaotic regions around SPO1 and SPO2 have "merged" and large-scale chaos has spread throughout the lattice. Since these results hold for N arbitrarily large, they suggest a direct approach by which one can use local stability analysis of SPO's to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as Kolmogorov-Sinai entropies per particle can be computed for similar one-dimensional lattices.
We investigate shear-induced crystallization in a very dense flow of monodisperse inelastic hard spheres. We consider a steady plane Couette flow under constant pressure and neglect gravity. We assume that the granular density is greater than the melting point of the equilibrium phase diagram of elastic hard spheres. We employ a Navier-Stokes hydrodynamics with constitutive relations all of which (except the shear viscosity) diverge at the crystal-packing density, while the shear viscosity diverges at a smaller density. The phase diagram of the steady flow is described by three parameters: an effective Mach number, a scaled energy loss parameter, and an integer number m: the number of half-oscillations in a mechanical analogy that appears in this problem. In a steady shear flow the viscous heating is balanced by energy dissipation via inelastic collisions. This balance can have different forms, producing either a uniform shear flow or a variety of more complicated, nonlinear density, velocity, and temperature profiles. In particular, the model predicts a variety of multilayer two-phase steady shear flows with sharp interphase boundaries. Such a flow may include a few zero-shear (solidlike) layers, each of which moving as a whole, separated by fluidlike regions. As we are dealing with a hard sphere model, the granulate is fluidized within the "solid" layers: the granular temperature is nonzero there, and there is energy flow through the boundaries of the solid layers. A linear stability analysis of the uniform steady shear flow is performed, and a plausible bifurcation diagram of the system, for a fixed m, is suggested. The problem of selection of m remains open.
We propose a self-propelled particle model for the swarming of Daphnia that takes into account mutual repulsion and attraction to a center. Surprisingly, a vortex is formed only for an intermediate strength of propulsion. The phase diagram and the transitions between states with and without a vortex are analyzed, and the nature of the phase boundaries is discussed based on a linear stability analysis of the motion of individual swimmers. This allows us to identify various key parameters determining the characteristic features of the collective motion.
We study spatiotemporal patterns resulting from instabilities induced by nonlocal spatial coupling in the Oregonator model of the light-sensitive Belousov-Zhabotinsky reaction. In this system, nonlocal coupling can be externally imposed by means of an optical feedback loop which links the intensity of locally applied illumination with the activity in a certain vicinity of a particular point weighted by a given coupling function. This effect is included in the three-variable Oregonator model by an additional integral term in the photochemically induced bromide flow. A linear stability analysis of this modified Oregonator model predicts that wave and Turing instabilities of the homogeneous steady state can be induced for experimentally realistic parameter values. In particular, we find that a long-range inhibition in the optical feedback leads to a Turing instability, while a long-range activation induces wave patterns. Using a weakly nonlinear analysis, we derive amplitude equations for the wave instability which are valid close to the instability threshold. Therein, we find that the wave instability occurs supercritically or subcritically and that traveling waves are preferred over standing waves. The results of the theoretical analysis are in good agreement with numerical simulations of the model near the wave instability threshold. For larger distances from threshold, a secondary breathing instability is found for traveling waves.
In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted with the Ito calculus has pattern solutions that never appear for the kinetic interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.