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PubMed2005

Interplay between noise and boundary conditions in pattern formation in adsorbed substances.

We have studied the interplay between noise and boundary conditions on the possibility of noise induced pattern formation. With this aim, we have exploited a deterministic model for pattern formation in adsorbed substances--including the effect of lateral interactions--used to describe the phenomenon of adsorption in surfaces, where a multiplicative noise fulfilling a fluctuation-dissipation relation was added. We have found solutions for different boundary conditions, particularly corresponding to two stable and one unstable patterns, where one of the stable and the unstable one, are purely induced by the multiplicative noise. In the case of albedo boundary conditions we have found a transition from monostable to a noise induced bistable behavior as the albedo parameter is varied.

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PubMed2005

Bifurcations in reaction-diffusion systems in chaotic flows.

We study the behavior of reacting tracers in a chaotic flow. In particular, we look at an autocatalytic reaction and at a bistable system which are subjected to stirring by a chaotic flow. The impact of the chaotic advection is described by a one-dimensional phenomenological model. We use a nonperturbative technique to describe the behavior near a saddle node bifurcation. We also find an approximation of the solution far away from the bifurcation point. The results are confirmed by numerical simulations and show good agreement.

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PubMed2005

Fractional rotational diffusion of rigid dipoles in an asymmetrical double-well potential.

The longitudinal and transverse components of the complex dielectric susceptibility tensor of an assembly of dipolar molecules rotating in an asymmetric double-well potential are evaluated using a fractional rotational diffusion equation (based on the diffusion limit of a fractal time random walk) for the distribution function of orientations of the molecules on the surface of the unit sphere. The calculation is the fractional analog of the Debye theory of orientational relaxation in the presence of external and mean field potentials (excluding inertial effects). Exact and approximate (based on the exponential separation for normal diffusion of the time scales of the intrawell and overbarrier relaxation processes associated with the bistable potential) solutions for the dielectric dispersion and absorption spectra are obtained. It is shown that a knowledge of the characteristic relaxation times for normal rotational diffusion is sufficient to predict accurately the anomalous dielectric relaxation behavior of the system for all time scales of interest. The model explains the anomalous (Cole-Cole-like) relaxation of complex dipolar systems, where the anomalous exponent differs from unity (corresponding to the normal dielectric relaxation), i.e., the relaxation process is characterized by a broad distribution of relaxation times (e.g., in glass-forming liquids).

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PubMed2005

Phase behavior of a nematic liquid crystal in contact with a chemically and geometrically structured substrate.

A nematic liquid crystal in contact with a grating surface possessing an alternating stripe pattern of locally homeotropic and planar anchoring is studied within the Frank-Oseen model. The combination of both chemical and geometrical surface pattern leads to rich phase diagrams, involving a homeotropic, a planar, and a tilted nematic texture. The effect of the groove depth and the anchoring strengths on the location and the order of phase transitions between different nematic textures is studied. A zenithally bistable nematic device is investigated by confining a nematic liquid crystal between the patterned grating surface and a flat substrate with strong homeotropic anchoring.

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PubMed2005

Ghost stochastic resonance in vertical-cavity surface-emitting lasers: experiment and theory.

We study the polarization response of a vertical-cavity surface-emitting laser, driven simultaneously by noise and two (or more) weak periodic signals. In the bistable regime, we observe experimentally the occurrence of stochastic resonance at a frequency that is absent in the input driving signal. The presence of this so-called ghost resonance is confirmed theoretically.

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PubMed2005

Coupled-mode theory for photonic band-gap inhibition of spatial instabilities.

We study the inhibition of pattern formation in nonlinear optical systems using intracavity photonic crystals. We consider mean-field models for singly and doubly degenerate optical parametric oscillators. Analytical expressions for the new (higher) modulational thresholds and the size of the "band gap" as a function of the system and photonic crystal parameters are obtained via a coupled-mode theory. Then, by means of a nonlinear analysis, we derive amplitude equations for the unstable modes and find the stationary solutions above threshold. The form of the unstable mode is different in the lower and upper parts of the band gap. In each part there is bistability between two spatially shifted patterns. In large systems stable wall defects between the two solutions are formed and we provide analytical expressions for their shape. The analytical results are favorably compared with results obtained from the full system equations. Inhibition of pattern formation can be used to spatially control signal generation in the transverse plane.

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PubMed2005

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures.

Nonlinear fronts between spatially extended traveling wave (TW) convection and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated from below. A finite-difference method is used to solve the full hydrodynamic field equations in a vertical cross section of the layer perpendicular to the convection roll axes. Results are presented for ethanol-water parameters with several strongly negative separation ratios where TW solutions bifurcate subcritically. Fronts and LTWs are compared with each other and similarities and differences are elucidated. Phase propagation out of the quiescent fluid into the convective structure entails a unique selection of the latter while fronts and interfaces where the phase moves into the quiescent state behave differently. Interpretations of various experimental observations are suggested.

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PubMed2005

Core genetic module: the mixed feedback loop.

The so-called mixed feedback loop (MFL) is a small two-gene network where protein A regulates the transcription of protein B and the two proteins form a heterodimer. It has been found to be statistically over-represented in statistical analyses of gene and protein interaction databases and to lie at the core of several computer-generated genetic networks. Here, we propose and mathematically study a model of the MFL and show that, by itself, it can serve both as a bistable switch and as a clock (an oscillator) depending on kinetic parameters. The MFL phase diagram as well as a detailed description of the nonlinear oscillation regime are presented and some biological examples are discussed. The results emphasize the role of protein interactions in the function of genetic modules and the usefulness of modeling RNA dynamics explicitly.

Cell Physiological Phenomena↗
PubMed2005

Chaotic resonance: two-state model with chaos-induced escape over potential barrier.

We consider the resonant effects of chaotic fluctuations on a strongly damped particle in a bistable potential driven by weak sinusoidal perturbation. We derive analytical expressions of chaos-induced transition rate between the neighboring potential wells based on the inhomogeneous Smoluchowski equation. Our first-order analysis reveals that the transition rate has the form of the Kramers escape rate except for a perturbed prefactor. This modification to the prefactor is found to arise from the statistical asymmetry of the chaotic noise. By means of the two-state model and the chaos-induced transition rate, we arrive at an analytical expression of the signal-to-noise ratio (SNR). Our first-order SNR shows that chaotic resonance can correspond directly to stochastic resonance.

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PubMed2005

Nondegenerate two-beam coupling in Kerr nonlinear photonic crystals.

We show that the energy-transfer efficiency by nondegenerate two-beam coupling in a one-dimensional Kerr-nonlinear superlattice can be enhanced by several orders of magnitude as compared with that in a homogeneous medium of the same nonlinearity and length. This significant enhancement utilizes the strong localized field at the band-edge state, two-frequency localized state, or defect state. Due to the intensity-induced index modulation, the bistability is observed, and because of the energy transfer between different wavelength components, the tristability behavior is induced.

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PubMed2005

Probability densities of periodically driven noisy systems: an approximation scheme incorporating linear-response and adiabatic theory.

We consider periodically driven noisy systems in the limit of long times. To deduce their asymptotic time-periodic probability distributions, two approaches are commonly used: adiabatic theory, valid if driving is very slow, and linear-response theory, applicable when driving is weak. We introduce an approximation scheme that combines these two approaches to yield the driven probability distribution even when driving is strong and moderately fast, so that both linear-response and adiabatic approximations break down. The high accuracy of this scheme is demonstrated on a driven overdamped noisy oscillator in a bistable quartic potential.

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PubMed2005

Anomalous nonequilibrium Ising-Bloch bifurcation in discrete systems.

We investigate the self-organization of two-dimensional activator-inhibitor discrete bistable systems in the neighborhood of a nonequilibrium Ising-Bloch bifurcation. The system exhibits an anomalous transition--induced by discretization--whose signature is the coexistence of Ising and Bloch walls for selected values of the spatial coupling. After curvature reduction of Bloch walls, coexistence gives rise to a unique and striking spatiotemporal dynamics: Bloch walls drive the motion and Ising walls play the role of "extended defects" oriented along the background grid directions. Strong enough external noise asymptotically restores the scenario found in the continuum limit.

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PubMed2005

Nonlinear amplifier and frequency shifter using a tunable periodic drive.

We consider the superposition of a weak and a strong force acting on an overdamped particle moving on either an asymmetric-periodic or a double-well potential. The velocity of the particle has only harmonics of the strong force, when the particle either oscillates near a minimum or runs away from it. Near a threshold drive (bistable point) separating these two dynamical regimes, the weak force drastically changes the velocity spectrum, greatly amplifying the mixing harmonics. This effect can be used either to amplify or to shift the frequency of a weak signal and can be observed in a wide variety of systems, including domain walls in a ferromagnet, SQUIDs, and tiny particles in a ratchet potential.

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PubMed2005

Bona fide stochastic resonance and multimodality in the stochastic Hodgkin-Huxley neuron.

The phenomena of stochastic resonance (SR) has attracted much attention in the studies of the excitable systems, in particular, the nervous systems under noise. Recently, an alternative SR condition, called the bona fide SR, was proposed for the bistable system under noise, based on the notion of the residence time distribution. As the forcing frequency increases, there exists an optimal resonant frequency. We study the SR in a stochastic Hodgkin-Huxley neuron, which has an inherent natural frequency in addition to the stochastic time scale. We have observed two resonant conditions; one between periodic forcing and natural frequencies, and the other between the periodic forcing and the stochastic frequencies. These resonance conditions show the bona fide stochastic resonance with multimodality. For comparison, we have studied the bona fide SR in the stochastic FitzHugh-Nagumo neuron, where, the multimodality is not observed. The differences in the resonance structure of two neuron models are understood in terms of differences in the phase portraits.

Action Potentials↗
PubMed2006

Two-dimensional solitons in the Gross-Pitaevskii equation with spatially modulated nonlinearity.

We introduce a dynamical model of a Bose-Einstein condensate based on the two-dimensional Gross-Pitaevskii equation, in which the nonlinear coefficient is a function of radius. The model describes a situation with spatial modulation of the negative atomic scattering length, via the Feshbach resonance controlled by a properly shaped magnetic of optical field. We focus on the configuration with the nonlinear coefficient different from zero in a circle or annulus, including the case of a narrow ring. Two-dimensional axisymmetric solitons are found in a numerical form, and also by means of a variational approximation; for an infinitely narrow ring, the soliton is found in an exact form (in the latter case, exact solitons are also found in a two-component model). A stability region for the solitons is identified by means of numerical and analytical methods. In particular, if the nonlinearity is supported on the annulus, the upper stability border is determined by azimuthal perturbations; the stability region disappears if the ratio of the inner and outer radii of the annulus exceeds a critical value . The model gives rise to bistability, as the stationary solitons coexist with stable axisymmetric breathers, whose stability region extends to higher values of the norm than that of the static solitons. The collapse threshold strongly increases with the radius of the inner hole of the annulus. Vortex solitons are found too, but they are unstable.

Journal Article↗
PubMed2006

Existence of solitons in infinite lattice.

We consider necessary conditions for existence of optical solitons in one-dimensional nonlinear periodic layered array. We show analytically that in the array with the cubic-quintic nonlinearity bistable solitons are possible whereas for the Kerr nonlinearity they never exist. We investigate asymptotic behavior of the soliton amplitude at infinity. With help of the asymptotic a numerical algorithm for searching the solitons may be developed so that finding a soliton on finite interval is simultaneously the numerical proof of its existence on infinite interval.

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PubMed2006

Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser.

We analyze theoretically and experimentally the influence of current noise on the longitudinal mode hopping dynamics of a bulk semiconductor laser. It is shown that the mean residence times on each mode have different sensitivity to external noise added to the bias current. In particular, an increase of the noise level enhances the residence time on the longitudinal mode that dominates at low current, evidencing the multiplicative nature of the stochastic process. A two-mode rate equation model for a semiconductor laser is able to reproduce the experimental findings. Under a suitable separation of the involved time scales, the model can be reduced to a one-dimensional bistable potential system with a multiplicative stochastic term related to the current noise strength. The reduced model clarifies the influence of the different noise sources on the hopping dynamics.

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PubMed2006

Nonlinear evolution of the plasma beat wave: Compressing the laser beat notes via electromagnetic cascading.

The near-resonant beat wave excitation of an electron plasma wave (EPW) can be employed for generating the trains of few-femtosecond electromagnetic (EM) pulses in rarefied plasmas. The EPW produces a comoving index grating that induces a laser phase modulation at the difference frequency. As a result, the cascade of sidebands red and blue shifted by integer multiples of the beat frequency is generated in the laser spectrum. The bandwidth of the phase-modulated laser is proportional to the product of the plasma length, laser wavelength, and amplitude of the electron density perturbation. When the beat frequency is lower than the electron plasma frequency, the redshifted spectral components are advanced in time with respect to the blueshifted ones near the center of each laser beat note. The group velocity dispersion of plasma compresses so chirped beat notes to a few-laser-cycle duration thus creating a train of sharp EM spikes with the beat periodicity. Depending on the plasma and laser parameters, chirping and compression can be implemented either concurrently in the same, or sequentially in different plasmas. Evolution of the laser beat wave and electron density perturbations is described in time and one spatial dimension in a weakly relativistic approximation. Using the compression effect, we demonstrate that the relativistic bistability regime of the EPW excitation [G. Shvets, Phys. Rev. Lett. 93, 195004 (2004)] can be achieved with the initially subthreshold beat wave pulse.

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