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PubMed2002

Measurements of anomalous particle and energy fluxes in a magnetized plasma.

Electrostatic probe measurements of low frequency plasma fluctuations and anomalous particle and energy flux densities in a magnetized plasma are presented. A method allowing the simultaneous recording of instantaneous electric field, electron density, and temperature is invoked. The method is applied to flux density measurements in a weakly ionized, low beta plasma created in a toroidal device without magnetic rotational transform. It is also used to identify modes belonging to different dispersion branches and to obtain the dispersion relations for these modes. For the plasma states studied, the phase velocities and the cross phase between the electron density and electric field agree with those predicted from a local, linear stability analysis for electrostatic flute modes and drift waves. The instability threshold, however, is one order of magnitude higher than predicted by theory for unsheared flow. The fluxes measured are consistent with the estimated ionization rate.

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PubMed2002

Mode-I fracture in a nonlinear lattice with viscoelastic forces.

We study mode-I fracture in a viscoelastic lattice model with a nonlinear force law, with a focus on the velocity and linear stability of the steady-state propagating solution. This study is a continuation both of the study of the piecewise-linear model in mode I, and the study of more general nonlinear force laws in mode-III fracture. At small driving, there is a strong dependency of the velocity curve on the dissipation and a strong sensitivity to the smoothness of the force law at large dissipation. At large driving we calculate, via a linear stability analysis, the critical velocity for the onset of instability as a function of the smoothness, the dissipation and the ratio of lattice spacing to critical extension. This critical velocity is seen to be very sensitive to these parameters. We confirm our calculations via direct numerical simulations of the initial value problem.

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PubMed2002

Random coupling of chaotic maps leads to spatiotemporal synchronization.

We investigate the spatiotemporal dynamics of a network of coupled chaotic maps, with varying degrees of randomness in coupling connections. While strictly nearest neighbor coupling never allows spatiotemporal synchronization in our system, randomly rewiring some of those connections stabilizes entire networks at x*, where x* is the strongly unstable fixed point solution of the local chaotic map. In fact, the smallest degree of randomness in spatial connections opens up a window of stability for the synchronized fixed point in coupling parameter space. Further, the coupling epsilon(bifr) at which the onset of spatiotemporal synchronization occurs, scales with the fraction of rewired sites p as a power law, for 0.1<p<1. We also show that the regularizing effect of random connections can be understood from stability analysis of the probabilistic evolution equation for the system, and approximate analytical expressions for the range and epsilon(bifr) are obtained.

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PubMed2002

Stability and optimal parameters for continuous feedback chaos control.

We investigate the conditions under which an optimal continuous feedback control can be achieved. Chaotic oscillations in the single-well Duffing model, with either a positive or a negative nonlinear stiffness term, are tuned to their related Ritz approximation. The Floquet theory enables the stability analysis of the control. Critical values of the feedback control coefficient fulfilling the optimization criteria are derived. The influence of the chosen target orbit, of the feedback coefficient, and of the onset time of control on its duration is discussed. The analytic approach is confirmed by numerical simulations.

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PubMed2002

Pinning stationary planar fronts in diffusion-convection-reaction systems.

This paper considers various strategies for controlling a stationary planar front solution, in a rectangular domain with a diffusion-reaction distributed system, by pinning the solution to one or few points and using actuators with the simplest possible spatial dependence. We review previous results obtained for one-dimensional diffusion-reaction (with or without convection) systems, for which we applied two approaches: an approximate model reduction to a form that follows the front position while approximating the front velocity, and linear stability analysis. We apply the same two approaches for the planar fronts. The approximate model reduction allows us to analyze qualitatively various control strategies and to predict the critical width below which the control mode of the one-dimensional system is sufficient. These results are corroborated by linear analysis of a truncated model with the spectral methods representation, using concepts of finite and infinite zeros of linear multidimensional systems.

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PubMed2003

Time-delay-induced stabilization of coupled discrete-time systems.

This paper shows that the time-delay-induced stabilization occurs in discrete-time systems on numerical simulations. The stability analysis proves that this phenomenon never occurs in the discrete-time systems that have an odd-number property. This property is well known as the weak point of the delayed feedback control of chaos. Furthermore, we show that the phenomenon never occurs in any one-dimensional discrete-time system.

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PubMed2003

Modulational instability arising from collective Rayleigh scattering.

It is shown that under certain conditions a collection of dielectric Rayleigh particles suspended in a viscous medium and enclosed in a bidirectional ring cavity pumped by a strong laser field can produce a new modulational instability transverse to the wave-propagation direction. The source of the instability is collective Rayleigh scattering i.e., the spontaneous formation of periodic longitudinal particle-density modulations and a backscattered optical field. Using a linear stability analysis a dispersion relation is derived which determines the region of parameter space in which modulational instability of the backscattered field and the particle distribution occurs. In the linear regime the pump is modulationally stable. A numerical analysis is carried out to observe the dynamics of the interaction in the nonlinear regime. In the nonlinear regime the pump field also becomes modulationally unstable and strong pump depletion occurs.

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PubMed2003

Vlasov model using kinetic phase point trajectories.

A method of solution of the collisionless Vlasov equation by following fixed collisionless phase point ("particle") trajectories (characteristics) in phase space is presented. It solves the coupled Vlasov Maxwell system self-consistently and employs the Leapfrog-Trapezoidal scheme to solve for the characteristics explicitly. It then uses the bilinear finite element interpolation scheme in phase space and maps vital instantaneous phase point information (distribution function) to a fixed background phase space mesh while retaining it at the phase point. The scheme is an enhanced second order one in time and fourth order in space. The code is then used to model a thermal plasma as well as two stream instability using mobile electrons and fixed background ions: the scheme being a momentum conserving one by construction allows energy conservation without assignment of particle shape functions; Langmuir waves are obtained with very good agreement with the Bohm-Gross dispersion relation; the two stream results do not show any numerically induced oscillations attributed to the initial well-ordered velocity distributions. Retention of the characteristics also minimized diffusion. Extensive numerical stability analysis deriving Courant condition for the scheme as well as behavior of computational modes are done in Appendix A, as well as estimating the impact of numerical diffusion in Appendix B. Two to five dimensional versions in phase space exist.

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PubMed2003

Delay-induced synchronization phenomena in an array of globally coupled logistic maps.

We study the synchronization of a linear array of globally coupled identical logistic maps. We consider a time-delayed coupling that takes into account the finite velocity of propagation of the interactions. We find globally synchronized states in which the elements of the array evolve along a periodic orbit of the uncoupled map, while the spatial correlation along the array is such that an individual map sees all other maps in his present, current, state. For values of the nonlinear parameter such that the uncoupled maps are chaotic, time-delayed mutual coupling suppresses the chaotic behavior by stabilizing a periodic orbit that is unstable for the uncoupled maps. The stability analysis of the synchronized state allows us to calculate the range of the coupling strength in which global synchronization can be obtained.

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PubMed2003

Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium.

We investigate the formation of stable spatiotemporal three-dimensional (3D) solitons ("light bullets") with internal vorticity ("spin") in a bimodal system described by coupled cubic-quintic nonlinear Schrödinger equations. Two relevant versions of the model, for the linear and circular polarizations, are considered. In the former case, an important ingredient of the model are four-wave-mixing terms, which give rise to a phase-sensitive nonlinear coupling between two polarization components. Thresholds for the formation of both spinning and nonspinning 3D solitons are found. Instability growth rates of perturbation eigenmodes with different azimuthal indices are calculated as functions of the solitons' propagation constant. As a result, stability domains in the model's parameter plane are identified for solitons with the values of the spins of their components s=0 and s=1, while all the solitons with s> or =2 are unstable. The solitons with s=1 are stable only if their energy exceeds a certain critical value, so that, in typical cases, the stability region occupies approximately 25% of their existence domain. Direct simulations of the full system produce results that are in perfect agreement with the linear-stability analysis: stable 3D spinning solitons readily self-trap from initial Gaussian pulses with embedded vorticity, and easily heal themselves if strong perturbations are imposed, while unstable spinning solitons quickly split into a set of separating zero-spin fragments whose number is exactly equal to the azimuthal index of the strongest unstable perturbation eigenmode.

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PubMed2003

Granular clustering in a hydrodynamic simulation.

We examine the hydrodynamics of a granular gas using numerical simulation. We demonstrate the appearance of shearing and clustering instabilities predicted by linear stability analysis, and show that their appearance is directly related to the inelasticity of collisions in the material. We discuss the rate at which these instabilities arise and the manner in which clusters grow and merge.

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PubMed2003

Patterns in a quasiconfocal optical parametric oscillator.

The formation of transverse patterns in a triply resonant optical parametric oscillator is studied both numerically and analytically for a spherical cavity close to confocality. While the pump profile is Gaussian, the signal and idler intensities may be made of many rings, either stationary or time dependent. The mode selection and the time dependence are understood with the help of the linear stability analysis. It might explain observations reported for a quasiconfocal cavity with a KTP crystal.

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PubMed2003

Tangent double Hopf bifurcation in a differentially rotating cylinder flow.

A tangent double Hopf bifurcation has been found in a rotating cylinder flow driven by the counter-rotation of the top endwall. The codimension-3 tangent double Hopf point has been located using linear stability analysis. The nonlinear analysis of the multiple solutions in a neighborhood of this bifurcation point is performed by numerical solutions of the three-dimensional Navier-Stokes equations. At the tangent double Hopf point, two rotating waves and an unstable modulated rotating wave bifurcate simultaneously. A center manifold/normal form analysis is also performed, which is in very good agreement with both the linear and nonlinear computations. By a detailed exploration of the nonlinear flow, we have unraveled the complex dynamics generated by the mode competition, which is organized by the codimension-3 tangent double Hopf bifurcation.

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PubMed2003

Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes.

The double barrier resonant tunneling diode exhibits complex spatiotemporal patterns including low-dimensional chaos when operated in an active external circuit. We demonstrate how autosynchronization by time-delayed feedback control can be used to select and stabilize specific current density patterns in a noninvasive way. We compare the efficiency of different control schemes involving feedback in either local spatial or global degrees of freedom. The numerically obtained Floquet exponents are explained by analytical results from linear stability analysis.

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PubMed2003

Complex Ginzburg-Landau equation with nonlocal coupling.

A Ginzburg-Landau-type equation with nonlocal coupling is derived systematically as a reduced form of a universal class of reaction-diffusion systems near the Hopf bifurcation point and in the presence of another small parameter. The reaction-diffusion systems to be reduced are such that the chemical components constituting local oscillators are nondiffusive or hardly diffusive, so that the oscillators are almost uncoupled, while there is an extra diffusive component which introduces effective nonlocal coupling over the oscillators. Linear stability analysis of the reduced equation about the uniform oscillation is also carried out. This revealed that new types of instability which can never arise in the ordinary complex Ginzburg-Landau equation are possible, and their physical implication is briefly discussed.

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PubMed2003

Kinetics of the shear banding instability in startup flows.

Motivated by recent light scattering experiments on semidilute wormlike micelles, we study the early stages of the shear banding instability using the nonlocal Johnson-Segalman model with a "two-fluid" coupling of flow to micellar concentration. We perform a linear stability analysis for coupled fluctuations in shear rate gamma;, micellar strain W, and concentration phi about an initially homogeneous state. This resembles the Cahn-Hilliard (CH) analysis of fluid-fluid demixing (although we discuss important differences). First, assuming the initial state to lie on the intrinsic constitutive curve, we calculate the "spinodal" onset of instability in sweeps along this curve. We then consider start-up "quenches" into the unstable region. Here the instability in general occurs before the intrinsic constitutive curve can be attained, so we analyze the fluctuations with respect to the time-dependent start-up flow. We calculate the selected length and time scales at which inhomogeneity first emerges. When the coupling between flow and concentration is switched off, fluctuations in the "mechanical variables" gamma; and W are independent of those in phi, and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. Coupling to the concentration enhances this instability at short length scales, thereby selecting a length scale, consistent with the recent light scattering experiments. The spinodal region is then broadened by an extent that increases with proximity to an underlying (zero-shear) CH fluid-fluid (phi) demixing instability. Far from demixing, the broadening is slight and the instability is still mechanically dominated (by deltagamma; and deltaW) with only small deltaphi. Close to demixing, instability sets in at a very low shear rate, where it is dominated instead by deltaphi. In this way, the model captures a smooth crossover from shear banding instabilities that are perturbed by concentration coupling to demixing instabilities that are induced by shear.

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PubMed2003

Metastability of dark snoidal-type waves in quadratic nonlinear media.

We report the existence and basic properties of dark snoidal-type waves self-sustained in quadratic nonlinear media. Using a stability analysis technique, we reveal that they are almost completely stable, or metastable, in suitable ranges of input energy flows and material parameters. This opens the way to the experimental observation of dark-type multicolor periodic wave patterns supported by quadratic nonlinearities.

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PubMed2003

Stimulated Raman scattering of intense laser pulses in air.

Stimulated rotational Raman scattering (SRRS) is known to be one of the processes limiting the propagation of high-power laser beams in the atmosphere. In this paper, SRRS, Kerr nonlinearity effects, and group velocity dispersion of short laser pulses and pulse trains are analyzed and simulated. Fully time-dependent, three-dimensional, nonlinear propagation equations describing the Raman interaction, optical Kerr nonlinearity due to bound electrons, and group velocity dispersion are presented and discussed. The effective time-dependent nonlinear refractive index containing both Kerr and Raman processes is derived. Linear stability analysis is used to obtain growth rates and phase matching conditions for the SRRS, modulational, and filamentation instabilities. Numerical solutions of the propagation equations in three dimensions show the detailed evolution of the Raman scattering instability for various pulse formats. The dependence of the growth rate of SRRS on pulse duration is examined and under certain conditions it is shown that short (approximately psec) laser pulses are stable to the SRRS instability. The interaction of pulses in a train through the Raman polarization field is also illustrated.

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