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The global linear stability of a water drop on hot nonwetting surfaces is studied. The droplet is assumed to have a static shape and the surface tension gradient is neglected. First, the nonlinear steady Boussinesq equation is solved to obtain the axisymmetric toroidal base flow. Then, the linear stability analysis is conducted for different contact angles $\ensuremath{\beta}={110}^{\ensuremath{\circ}}$ (hydrophobic) and $\ensuremath{\beta}={160}^{\ensuremath{\circ}}$ (superhydrophobic) which correspond to the experimental study of Dash et al. [Phys. Rev. E 90, 062407 (2014)]. The droplet with $\ensuremath{\beta}={110}^{\ensuremath{\circ}}$ is stable while the one with $\ensuremath{\beta}={160}^{\ensuremath{\circ}}$ is unstable to the azimuthal wave number $m=1$ mode. This suggests that the experimental observation for a droplet with $\ensuremath{\beta}={110}^{\ensuremath{\circ}}$ corresponds to the steady toroidal base flow, while for $\ensuremath{\beta}={160}^{\ensuremath{\circ}}$, the $m=1$ instability promotes the rigid body rotation motion. A marginal stability analysis for different $\ensuremath{\beta}$ shows that a $3\text{\ensuremath{-}}\ensuremath{\mu}\mathrm{L}$ water droplet is unstable to the $m=1$ mode when the contact angle $\ensuremath{\beta}$ is larger than ${130}^{\ensuremath{\circ}}$. A marginal stability analysis for different volumes is also conducted for the two contact angles $\ensuremath{\beta}={110}^{\ensuremath{\circ}}$ and ${160}^{\ensuremath{\circ}}$. The droplet with $\ensuremath{\beta}={110}^{\ensuremath{\circ}}$ becomes unstable when the volume is larger than $3.5\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{L}$ while the one with $\ensuremath{\beta}={160}^{\ensuremath{\circ}}$ is always unstable to $m=1$ mode for the considered volume range ($2--5\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{L}$). In contrast to classical buoyancy driven (Rayleigh-B\'enard) problems whose instability is controlled independently by the geometrical aspect ratio and the Rayleigh number, in this problem, these parameters are all linked together with the volume and contact angles.

We study the dynamics of a thin liquid film on a compliant substrate in the presence of thermocapillary effect. A set of long-wave equations are derived to investigate the effects of fluid gravity $(G)$, fluid inertia (Re), and Marangoni stresses (Ma) on the dynamics of the liquid film and the compliant substrate. By performing linear stability analysis and time-dependent computations of the long-wave equations, we examine two different cases: thin-film flows on a horizontally compliant substrate $(\ensuremath{\beta}=0$, where $\ensuremath{\beta}$ is the inclined angle) and down a vertically compliant substrate $(\ensuremath{\beta}=\ensuremath{\pi}/2)$, respectively. For $\ensuremath{\beta}=0$, we neglect fluid inertia and identify two different modes: (1) sinuous mode, where the deformations of liquid-air and liquid-substrate interfaces are in phase, which is induced by the fluid gravity, and (2) varicose mode, where the deformations of two interfaces are in phase opposition, which is induced by the Marangoni stresses. For $\ensuremath{\beta}=\ensuremath{\pi}/2$, we consider a weak fluid inertia and only observe the varicose mode driven by fluid inertia and Marangoni stresses. However, because the gravity direction is parallel to the substrate, the fluid gravity modifies the varicose mode, making the deformations of two interfaces out of phase. In particular, we also seek the nonlinear traveling-wave solutions in the case of $\ensuremath{\beta}=\ensuremath{\pi}/2$, revealing that fluid inertia and/or heating effect enhance the height and speed of the traveling waves. In both cases, the introduction of a strong wall heating gives rise to large deformations of both the thin liquid film and the compliant substrate.

When a more mobile fluid displaces another immiscible one in a porous medium, viscous fingering propagates with a partial sweep, which hinders oil recovery and soil remedy. We experimentally investigate the feasibility of tuning such fingering propagation in a non-uniform narrow passage with a radial injection, which is widely used in various applications. We show that a radially converging cell can suppress the common viscous fingering observed in a uniform passage, and a full sweep of the displaced fluid is then achieved. The injection flow rate, $Q$, can be further exploited to manipulate the viscous fingering instability. For a fixed gap gradient $\alpha$, our experimental results show a full sweep at a small $Q$ but partial displacement with fingering at a sufficient $Q$. Finally, by varying $\alpha$, we identify and characterize the variation of the critical threshold between stable and unstable displacements. Our experimental results reveal good agreement with theoretical predictions by a linear stability analysis., Comment: 4 figures, Physical Review E, Rapid Communications (in press)

Microorganisms and plants very commonly release toxic secondary chemical compounds (allelochemicals) that inhibit or kill sensitive strains or individuals from their own or other species. In this work we study a model that describes two species interacting through allelopathic suppression and competing for resources. Employing linear stability analysis, the conditions for coexistence or extinction of species in spatially homogeneous systems were determined. We found that the borders between the regimes of bistability, coexistence, and the extinction of the weaker by the stronger competitor, are altered by allelopathic interactions. In addition, traveling wave solutions for one species invasion were obtained considering the spatially explicit nature of the model. Our findings indicate that the minimum speed of the invasion wavefronts depends primarily on the competition coefficients and the parameters characterizing the species' functional responses to their allelochemicals. As a general rule, the species provided with the most effective chemical weapons dominates the population dynamics. Finally, we found a tristability at the coexistence region due to the combination of allelopathy and patchy population distributions in space. So, our model provides a distinct mechanism, independent of social behaviors, that produces such unexpected tristability impossible in classical competition models involving one-to-one individual interactions.

We consider a reaction-diffusion system undergoing Turing instability and augment it by an additional unilateral source term. We investigate its influence on the Turing instability and on the character of resulting patterns. The nonsmooth positively homogeneous unilateral term $\tau v^-$ has favourable properties, but the standard linear stability analysis cannot be performed. We illustrate the importance of the nonsmoothness by a numerical case study, which shows that the Turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation. However, the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state. Further, we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns. Moreover, a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term.

We describe the physical mechanism responsible for electrohydrodynamic (EHD) instability of a fluid layer with collinear conductivity gradient and electric field. In particular, we resolve the ambiguity in literature regarding the cause for switching between stationary and oscillatory modes of EHD instability. Using linear stability analysis, we show that a small perturbation in conductivity field perturbs the local electric field and also induces a perturbation charge. The coupling of base-state electric field with the perturbation charge leads to a force which causes overstability. Whereas, the coupling of base-state free charge with perturbation electric field leads to a force which causes EHD instability via a stationary mode. The proposed mechanism correctly explains the existence of stationary and oscillatory modes for varying conductivity gradients and wave number of disturbances, depending upon the relative magnitude of these two forces.

The statistics and origin of the first discrete plastic event in a one dislocation dynamics simulation are studied. This is done via a linear stability analysis of the evolving dislocation configuration up to the onset of irreversible plasticity. It is found, via a fold catastrophe, the dislocation configuration prior to loading directly determines the stress at which the plastic event occurs and that between one and two trigger dislocations are involved. The resulting irreversible plastic strain arising from the instability is found to be highly correlated with these triggering dislocations., 11 Pages, 8 figures

A linear stability analysis and an experimental study of a viscoelastic Taylor-Couette flow corotating in the Keplerian ratio allow us to elucidate the analogy between the viscoelastic instability and the magnetorotational instability (MRI). A generalized Rayleigh criterion allows us to determine the potentially unstable zone to pure-elasticity-driven perturbations. Experiments with a viscoelastic polymer solution yield four modes: one pure-elasticity mode and three elastorotational instability (ERI) modes that represent the MRI-analog modes. The destabilization by the polymer viscosity is evidenced for the ERI modes.

We study the phenomena of oscillation quenching in a system of limit cycle oscillators which are coupled indirectly via a dynamic environment. The dynamics of the environment is assumed to decay exponentially with some decay parameter. We show that for appropriate coupling strength, the decay parameter of the environment plays a crucial role in the emergent dynamics such as amplitude death (AD) and oscillation death (OD). The critical curves for the regions of oscillation quenching as a function of coupling strength and decay parameter of the environment are obtained analytically using linear stability analysis and are found to be consistent with the numerics.

We investigate the modulation instability in oppositely directed coupler in the presence of higher-order effects. Using linear stability analysis, we obtain an expression for instability gain. Special attention is paid to find out the influence of self-steepening effect and intrapulse Raman scattering on modulation instability. The study shows that in normal dispersion, regime instability gain exists even if perturbation frequency $(\ensuremath{\Omega})$ is zero. But the instability gain at $\ensuremath{\Omega}=0$ is zero, when the dispersion is anomalous. Moreover, self-steepening effect and intrapulse Raman scattering form new instability regions and, hence, provide a new way to generate solitons or ultrashort pulses. Further, efficient control of modulation instability by adjusting self-steepening effect and intrapulse Raman scattering also successfully demonstrated.

Chimera states consisting of domains of coherently and incoherently oscillating identical oscillators with nonlocal coupling are studied. These states usually coexist with the fully synchronized state and have a small basin of attraction. We propose a nonlocal phase-coupled model in which chimera states develop from random initial conditions. Several classes of chimera states have been found: (a) stationary multicluster states with evenly distributed coherent clusters, (b) stationary multicluster states with unevenly distributed clusters, and (c) a single cluster state traveling with a constant speed across the system. Traveling coherent states are also identified. A self-consistent continuum description of these states is provided and their stability properties analyzed through a combination of linear stability analysis and numerical simulation.

We study localized nonlinear excitations in diffusive Hindmarsh-Rose neural networks. We show that the Hindmarsh-Rose model can be reduced to a modified Complex Ginzburg-Landau equation through the application of a perturbation technique. We equally report on the presence of envelop solitons of the nerve impulse in this neural network. From the biological point of view, this result suggests that neurons can participate in a collective processing of information, a relevant part of which is shared over all neurons but not concentrated at the single neuron level. By employing the standard linear stability analysis, the growth rate of the modulational instability is derived as a function of the wave number and systems parameters.

A modified Hele-Shaw cell in which the plate gap can be modulated in time was constructed. Highly nonlinear fingers on the interface between air and water in the cell were observed as the plate gap was driven at a variety of frequencies, but typically near 60 Hz. Modified equations to describe the flow in a periodically driven cell were derived and the linear stability analysis of waves on a circular fluid-fluid interface was performed.

In 1982 Bobylev [A.V. Bobylev, Sov. Phys. Dokl. 27, 29 (1982)] made a linear stability analysis of the Burnett equations and showed that beyond a certain critical reduced wave number there exist normal modes that grow exponentially, concluding that the Burnett equations are linearly unstable. We have partially extended his analysis, originally made for Maxwellian molecules, for any interaction potential and argue that his results can be reinterpreted as to give a bound for the Knudsen number above which the Burnett equations are not valid.

Using an example of cavity solitons in optical parametric oscillators it is demonstrated that Hopf instability of these dissipative structures can be directly associated with internal modes of their conservative counterparts. The latter ones are free propagating quadratic solitons in this case. Linear stability analysis and numerical simulation also reveal multistability and complex instability induced spatiotemporal dynamics of single and multihump cavity solitons.

The lattice hydrodynamic model is presented to take into account the passing effect in one-dimensional traffic flow. When the passing constant gamma is small, the conventional jamming transition occurs between the uniform traffic and kink density wave flows. When passing constant gamma is larger than the critical value, the jamming transitions occur from the uniform traffic flow, through the chaotic density wave flow, to the kink density wave flow, with an increasing delay time. The chaotic region increases with passing constant gamma. The neutral stability line is derived from the linear stability analysis. The neutral stability line coincides with the transition line between the uniform traffic and density wave flows. The modified Korteweg-de Vries equation describing the kink jam is derived for small values of gamma by use of a nonlinear analysis.

The stability of the recently discovered compacton solutions is studied by means of both linear stability analysis as well as Lyapunov stability criteria. From the results obtained it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The results are shown to be true even for the higher order nonlinear dispersion equations for compactons. Some conservation laws for the higher order nonlinear dispersion equations are also presented.

In this paper we investigate the stability of a straight vortex filament with phase twist described by the three-dimensional complex Ginzburg-Landau equation (CGLE). The results of the linear stability analysis show that the straight filament is stable in a limited region of the two parameter space of the CGLE. The stable region is dependent on the phase twist imposed on the filament and shrinks in size as the phase twist is increased. It is also shown numerically that the nonlinear evolution of an unstable initial straight filament can lead to a helical filament.

Strongly confined active liquids are subject to unique hydrodynamic interactions due to momentum screening and lubricated friction by the confining walls. Using numerical simulations, we demonstrate that two-dimensional dilute suspensions of fore-aft asymmetric polar swimmers in a Hele-Shaw geometry can exhibit a rich variety of novel phase behaviors depending on particle shape, including coherent polarized density waves with global alignment, persistent counterrotating vortices, density shocks and rarefaction waves. We also explain these phenomena using a linear stability analysis and a nonlinear traffic flow model, both derived from a mean-field kinetic theory.

Coriolis force effects on shear flows are important in geophysical and astrophysical contexts. We here report a study on the linear stability and the transient energy growth of the plane Couette flow with system rotation perpendicular to the shear direction. External rotation causes linear instability. At small rotation rates, the onset of linear instability scales inversely with the rotation rate and the optimal transient growth in the linearly stable region is slightly enhanced, ~Re^2. The corresponding optimal initial perturbations are characterized by roll structures inclined in the streamwise direction and are twisted under external rotation. At large rotation rates, the transient growth is significantly inhibited and hence linear stability analysis is a reliable indicator for instability., Comment: 7 pages, 9 figures

We investigate a control technique for spatially extended systems combining spatial filtering with a previously studied form of time-delay feedback. The scheme is naturally suited to real-time control of optical systems. We apply the control scheme to a model of a transversely extended semiconductor laser in which a desirable, coherent traveling wave state exists, but is a member of a nowhere stable family. Our scheme stabilizes this state, and directs the system towards it from realistic, distant and noisy initial conditions. As confirmed by numerical simulation, a linear stability analysis about the controlled state accurately predicts when the scheme is successful, and illustrates some key features of the control including the individual merit of, and interplay between, the spatial and temporal degrees of freedom in the control., Comment: 9 pages REVTeX including 7 PostScript figures. To appear in Physical Review E

The pattern dynamics of a two-dimensional coupled map lattice system is studied using both analytical and numerical calculations. Two interesting spatiotemporal periodic patterns are solved analytically. Their stability boundaries for small system size are obtained by linear stability analysis. As the system sizes mismatch the spatial periodicity of the patterns, a frozen random chaotic defect cluster and a slow random propagated chaotic defect string appear.

We study the early stage of the phase separation of a binary mixture far from its critical point of demixing. Whenever the mixture of two mutually repulsive species is quenched to a temperature below its critical point of miscibility, the effect of the enthalpic repulsive force prevails upon the entropic tendency to mix, so that the system eventually separates itno two coexisting phases. We have developed a highly nonlinear model, in close analogy with the linear theory of Cahn and Hilliard, where a generalized free energy is defined in terms of two parameters \ensuremath{\psi} and a, the first describing the equilibrium composition of the two phases, ad the second denoting a characteristic length scale that is inversely proportional to the equilibrium surface tension. The linear stability analysis predicts that any perturbation of the initial mixture composition with wave number k smaller than \ensuremath{\surd}2\ensuremath{\psi} /a will grow exponentially in time, with a maximum growth corresponding to ${\mathit{k}}_{\mathrm{max}}$= \ensuremath{\surd}\ensuremath{\psi} /a. A numerical solution of the equation shows that nonlinear effects saturate the exponential growth, and that the concentraiton distribution tends to a steady state, peroidic profile with wavelength \ensuremath{\lambda}=2\ensuremath{\pi}a/ \ensuremath{\surd}\ensuremath{\psi} corresponding to the fastest growing mode of the linear regime. The main result of our theoretical model is that this steady state does not depend on the form of the initial perturbation to the homogeneous composition profile. \textcopyright{} 1996 The American Physical Society.

We study synchronization in a system of phase-only oscillators residing on the sites of a one-dimensional periodic lattice. The oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size-dependent constant. The exponent $\alpha$ of the power law is taken in the range $0 \le \alpha, Comment: 7 pages, 4 figures. v2: new simulation results added, close to the published version

We develop a mathematical model for the drying of porous media in the presence of gravity. The model incorporates effects of corner flow through macroscopic liquid films that form in the cavities of pore walls, mass transfer by diffusion in the dry regions of the medium, external mass transfer over the surface, and the effect of gravity. We consider two different cases: when gravity opposes liquid flow in the corner films and leads to a stable percolation drying front, and when it acts in the opposite direction. In this part, we develop analytical results when the problem can be cast as an equivalent continuum and described as a one-dimensional (1D) problem. This is always the case when gravity acts against drying by opposing corner flow, or when it enhances drying by increasing corner film flow but it is sufficiently small. We obtain results for all relevant variables, including drying rates, extent of the macroscopic film region, and the demarkation of the two different regimes of constant rate period and falling rate period, respectively. The effects of dimensionless variables, such as the bond number, the capillary number, and the Sherwood number for external mass transfer are investigated. When gravity acts to enhance drying, a 1D solution is still possible if an appropriately defined Rayleigh number is above a critical threshold. We derive a linear stability analysis of a model problem under this condition that verifies front stability. Further analysis of this problem, when the Rayleigh number is below critical, requires a pore-network simulator which will be the focus of future work.

We examine the stability of liquids with spherical interfaces evaporating from funnels constructed of different materials. A linear stability analysis predicts stable evaporation for funnels constructed of insulating materials and introduces a stability parameter for funnels constructed of conducting materials. The stability parameter is free of fitting variables since we use the statistical rate theory expression for the evaporation flux. The theoretical predictions are found to be consistent with experimental observations for H(2)O evaporating from a funnel constructed of poly(methyl methacrylate) and for H(2)O and D(2)O evaporating from a funnel constructed of stainless steel.

We study numerically the suppression of the soliton self-frequency shift as well as the compression in the presence of bandwidth-limited optical amplification (BLA). Our results confirm the existence of equilibrium points (stable focuses) for the soliton amplitude and speed identified by the soliton perturbation theory. Analyzing the equilibrium amplitudes as a function of physical parameters, maximum compression factor in the amplification of short pulses is revealed. The results of linear stability analysis allow estimation of the necessary distance for the appearance of equilibrium states from different initial conditions. It has been shown that the shape of the perturbed stationary solution in the presence of intrapulse Raman scattering and BLA can be described by the earlier derived analytic expressions.

The dense radial morphology appears in a number of systems undergoing branched growth. Neither ordered nor fractal, this pattern is characterized by a large number of branches radiating from a central seed and advancing behind a circular envelope. We propose a model for dense radial growth which self-consistently incorporates dissipation in the growth channels. A linear stability analysis of this model delimits conditions under which the dense radial morphology can develop. Predictions of this model are borne out by numerical simlations of evolving resistor bond networks.

The time evolution of a thin conducting liquid film flowing down a porous inclined substrate is investigated when an electric field acts normal to the substrate. It is assumed that the flow through the porous medium is governed by Darcy's law together with Beavers-Joseph condition. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium. A slip condition at the bottom is used to incorporate the effects of the permeability of the substrate. From the set of exact averaged equations derived using integral boundary method for the film thickness and for the flow rate, a nonlinear evolution equation for the film thickness is derived through a long-wave approximation. A linear stability analysis of the base flow is performed and the critical Reynolds number is obtained. The results reveal that the substrate porosity in general destabilizes the liquid film flow and the presence of the electric field enhances this destabilizing effect. A weakly nonlinear stability analysis divulges the existence of supercritical stable and subcritical unstable zones in the wave number/Reynolds number parameter space and the results demonstrate how the neutral curves change as the intensity of the electric filed or the permeability of the porous medium is varied. The numerical solution of the nonlinear evolution equation in a periodic domain reveals that the base flow yields to surface structures that are either time independent waves of permanent form that propagate or time-dependent modes that oscillate slightly in the amplitude. Further, it is observed that the shape and amplitude of long-time waveforms are influenced by the permeability of the porous medium as well as by the applied electric field. The results reveal that the destabilization induced by the electric field in an otherwise stable film over a porous medium is exhibited in the form of traveling waves of finite amplitude. The presence of the porous substrate promotes the oscillatory behavior of the long-time waveform; however, the electric field has a tendency to suppress this oscillatory behavior. � 2010 The American Physical Society.

We study the behavior of a magnetorheological fluid droplet confined to a Hele-Shaw cell in the presence of an applied radial magnetic field. Interfacial pattern formation is investigated by considering the competition among capillary, viscoelastic, and magnetic forces. The contribution of a magnetic field-dependent yield stress is taken into account. Linear stability analysis reveals the stabilizing role played by yield stress. On the other hand, a mode-coupling approach predicts that the resulting fingering structures should become less and less sharp as yield stress effects are increased. By employing a vortex-sheet formalism we have been able to identify a family of exact stationary solutions of the problem, unveiling the development of swollen polygonal patterns. A suggestive magnetically controlled shape transition in which the edges of the patterns change from convex to concave has been also identified.

The present study shows that diffusive connections with two long-time delays can induce the stabilization of a steady state in network oscillators. A linear stability analysis shows that, if the two delay times retain a proportional relation with a certain bias, the stabilization can be achieved independent of the delay times. Furthermore, a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed. The procedure has the following two advantages: one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology. Our analytical results are applied to the well-known double-scroll circuit model on a small-world network.

We propose a mechanism for generating snakelike motion in a micrometer-scale, responsive, synthetic material, which thereby undergoes net movement in a fluid (i.e., swimming). By responsive material, we refer to a material that can expand or contract in response to a chemical concentration change. The concentrations of the chemical species are modeled by simple reaction-diffusion equations with suitably chosen source terms. Using linear stability analysis, we isolate the key properties of the material and reaction rate parameters.

We numerically investigate the stability of one- and two-dimensional gap solitons for very long propagation distances both in self-focusing and in self-defocusing nonlinear photonic media. We demonstrate that the existence of stable solitons in the first gap requires much stronger lattices in a self-focusing than in a self-defocusing medium. Moreover, we present a one-dimensional linear stability analysis of the fundamental solitary mode in the first gap considering a self-focusing photorefractive nonlinearity.

We derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by Vlad and Ross [Phys. Rev. E 66, 061908 (2002)]. Reaction and diffusion processes are separable in a typical reaction-diffusion system, and their combined influence on the evolution of the density of a species is a simple sum. Our derivation shows that this is no longer true for subdiffusive entities undergoing reactions. The strong memory effects in the transport process, i.e., the non-Markovian nature of subdiffusion, results in a nontrivial combination of reactions and spatial dispersal, which we discuss in detail. We carry out a linear stability analysis of the derived reaction-subdiffusion system to understand the effects of memory on pattern formation. We find that the Turing instability persists in the subdiffusive system. However, the memory modifies the Turing threshold and the characteristics of the band of unstable modes close to this threshold.

In this paper, global synchronization in coupled oscillator networks is investigated. We propose an adaptive weighted network and show that such a simple and quite general scheme is able to tip oscillator networks towards collective synchronization. In comparison with the results based on linear stability analysis of unweighted networks, the proposed scheme improves the synchronizability of network dynamics, and is beneficial to analyze the effect of network structure on synchronizability.

Three-dimensional stability of stationary reaction balls is investigated in a system consisting of an autocatalysis accompanied by a slow decay of the autocatalyst. Radially stable stationary spherical structures become unstable to three-dimensional perturbations at small decay rate when the radius of the reaction ball is sufficiently large and the reactant diffuses faster than the autocatalyst. A thorough linear stability analysis and simulations in three spatial dimensions are carried out in the simplest system sustaining stationary reaction balls.

We have investigated the dynamics of the coupled Lorenz oscillators numerically and theoretically. We find the partial amplitude death when the interaction is strong enough. The linear stability analysis of the partial amplitude death is proposed.

This paper concerns the comparison of electroconvective instability in concentration polarization at an ion-selective membrane with previously reported nonequilibrium electro-osmotic instability. Electro-osmotic formulation represents an asymptotic limit case of the electroconvective one. An improved nonequilibrium electro-osmotic slip formula is derived. Linear stability analysis for various nonequilibrium electro-osmotic formulations is carried out, including the analytic studies of the short- and long-wave limits. The obtained results are compared with those for a full electroconvective formulation. It is observed that the shortwave singularity typical for the nonequilibrium electro-osmotic instability is removed in the full electroconvective formulation. The effect of ionic diffusivities ratio on stability is discussed.

A linear stability analysis of a circular crack growing in an elastic medium in two dimensions is presented. Two boundary conditions at the outer boundary are considered, namely, a constant strain and a constant pressure. Size effects are included by assuming a finite distance between the inner and the outer boundaries. If the outer boundary is placed at infinity, the result for the ratio between the instantaneous rates of growth of the perturbation and that of the circular crack is twice that obtained for growth in fields governed by the Laplace equation (diffusion or electrostatic fields) no matter which of the two boundary conditions is imposed. This result is in line with the smaller fractal dimensions obtained in the case of mechanical breakdown.

The nonlinear dynamics of laser beams carrying phase singularity in media with cubic-quintic nonlinearity changing from self-focusing to self-defocusing is examined. A novel kind of stable nonlocalized optical vortices appears in such media as well as localized vortex solitons. Linear stability analysis and numerical simulations show the stability of localized vortices only in the defocusing region.

We report a theoretical investigation of the lateral instability of grafted polymer layers in a poor solvent. Within self-consistent mean-field theory, we carry out a linear stability analysis at the random phase approximation level, for which an explicit incompressibility condition is enforced. Our analysis predicts a stability diagram in which regions of stable and unstable polymer brush profiles are located. Compared with analysis where incompressibility is not taken into account, our results suggest that lateral stability is enhanced.

We study a generalization of globally coupled maps, where the elements are updated with probability $p$. When $p$ is below a threshold $p_c$, the collective motion vanishes and the system is the stationary state in the large size limit. We present the linear stability analysis., 6 pages including 5 figures

A time-headway model is presented to mimic bus behavior on the bus route. The motion of a bus is described in terms of the time headway between its bus and the bus in front. We study the bunching behavior of buses induced by interacting with other buses and passengers. It is shown that the dynamical phase transitions among the inhomogeneous bunching phase, the homogeneous free phase, the coexisting phase, and the homogeneous congested phase occur with varying the initial time headway. We study the effect of not stopping at bus stops on the time-headway profile. It is found that the bunching transition lines are consistent with the neutral stability curves obtained by the linear stability analysis.