Uniform Steady State Research Articles

Taking into account the nonlinear properties of low-temperature plasma in the design of a plasma chemical reactor

An approach to the design of plasma chemical reactors is suggested. The concept of the plasma chemical reactor efficiency is introduced. A correlation between the formation rate of active species and the modification rate of the polymer is established, within the linear approximation, for the stable uniform steady state of plasma. Experiments have demonstrated that, under certain conditions of the interaction between lowpressure plasma and a polymer, the evolution of plasma properties and polymer modification occur in a complicated manner. Models are suggested that take into account the effect of gaseous reaction products on plasma properties as well as the modification rate.

Wave suppression by nonlinear finite-dimensional control

Korteweg–de Vries–Burgers (KdVB) and Kuramoto–Sivashinsky (KS) equations are two nonlinear partial differential equations (PDEs) which can adequately describe motion of waves in a variety of fluid flow processes. We synthesize nonlinear low-dimensional output feedback controllers for the KdVB and KS equations that enhance convergence rate and achieve stabilization to spatially uniform steady states, respectively. The approach used for controller synthesis employs nonlinear Galerkin's method to derive low-dimensional approximations of the PDEs, which are subsequently used for controller synthesis via geometric control methods. The controllers use measurements obtained by point sensors and are implemented through point control actuators. The performance of the proposed controllers is successfully tested through simulations.

Pattern formation in a differential–flow reactor model

A model for a differential–flow reactor is considered, based on cubic autocatalator kinetics in which the substrate is immobilized and the autocatalyst flows through the reactor at a constant rate. Linear stability analysis shows that there is a critical flow rate above which the spatially uniform steady state becomes convectively unstable. Numerical simulations show that this convective instability leads to the formation of a wave packet propagating through the reactor. The nature of this wave packet depends on the flow rate and on the two cases that are identified for the kinetic parameter μ, namely a generic case for general values of μ and when μ is close to the value which gives a Hopf bifurcation in the kinetic system.

A dusty double plasma device

A novel dusty plasma device to create spatially and temporally uniform steady state dusty plasma is described. An ultrasonic vibrator is used to vibrate a dust dispenser which disperses the dust uniformly through a fine mesh. A dusty plasma of large dimension with controllable dust density is produced. Measured dusty plasma parameters are compared with existing theories. Some experimental results related to propagation characteristics of dust-ion-acoustic waves in a dusty plasma column are presented.

Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction

Experimental studies of the formation of Turing patterns in the (chlorine dioxide, iodine, malonic acid) reaction are performed in a spatial open gel disk reactor where all the input species are fed onto one side by a continuous stirred tank reactor. This setup is shown to fit the pool-chemical approximation used in most theoretical approaches. Nonequilibrium phase diagrams are established as a function of concentrations in the input flows. In agreement with theoretical predictions, the location of the transition from uniform steady states to Turing patterns is found to be almost independent of the concentrations of the complexing agent which controls the effective diffusion of activatory species. Extensive analytical and numerical calculations in two and three dimensions are performed on the basis of the Lengyel−Rábai−Epstein kinetic model and its two-variable reduction. This particular experimental configuration is shown to minimize the problems encountered with more commonly used versions of spatial open reactors. In standard conditions, the quantitative agreement with the experiments is excellent in regard to the sketchiness of the model. Finally, we discuss the role of boundary conditions and comment on problems they raise in the use of one-side-fed open spatial reactors.

Linear stability criteria in a reaction-diffusion equation with spatially inhomogeneous delay

Time delays play an important role in many biological and ecological systems. However, they are usually incorporated into mathematical models in a way not explicitly dependent on space, even though the delay may be modelling some environmental aspect of the system. In this paper we study a scalar reaction-diffusion equation with a spatially inhomogeneous delay, taken for simplicity in the form of a step function of the spatial coordinate. We derive the dispersion relation from which analytical results on the stability or instability of the uniform steady states can be determined. We confirm and extended these results by numerical simulations which confirm the possibility of qualitatively different types of behaviour on different parts of the spatial domain.

Flow in multiscale log conductivity fields with truncated power variograms

In a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region (window) being sampled. We did so by demonstrating that such behavior is typical of any random field with a truncated power (semi)variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or Gaussian spatial autocovariance structures. The low‐ and high‐frequency cutoff scales λl and λu are related to the length scales of the sampling window and data support, respectively. We showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes, either at a given locale or between different locales. In this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of Gaussian fields, each associated with an exponential autocovariance. We start by deriving an expression for effective hydraulic conductivity, as a function of the Hurst coefficient H and the cutoff scales in one‐, two‐, and three‐dimensional domains which is qualitatively consistent with experimental data. We then develop leading‐order analytical expressions for two‐ and three‐dimensional autocovariance and cross‐covariance functions of hydraulic head, velocity, and log hydraulic conductivity versus H, λl and λu; examine their behavior; and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance. Our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales. In particular, when λl ≫ λu, the variance of head is infinite in two dimensions and grows in proportion to λl2+2H in three dimensions, while the variance and longitudinal integral scale of velocity grow in proportion to λl2H and λl, respectively, in both cases.

Transport in multiscale log conductivity fields with truncated power variograms

Di Federico and Neuman [this issue] investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic Gaussian fields, each associated with an exponential autocovariance. Here we present leading‐order expressions for the displacement covariance, and dispersion coefficient, of an ensemble of solute particles advected through such a flow field in two or three dimensions. Both quantities are functions of the mean travel distance s, the Hurst coefficient H, and the low‐ and high‐frequency cutoff integral scales λl and λu. The latter two are related to the length scales of the sampling window (region under investigation) and sample volume (data support), respectively. If one considers transport to be affected by a finite domain much larger than the mean travel distance, so that s ≪ λl < ∞, then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s. If one considers transport to be affected by a domain which increases in proportion to s, then λl and s are of similar order and a preasymptotic regime never develops. Instead, transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale λl ∼ s. We show that if, additionally, λu ≪ λl, then the corresponding longitudinal dispersivity grows in proportion to λl1+2H or, equivalently, s1+2H. Both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal Fickian dispersivities with scale. We conclude by investigating the effect of variable separations between cutoff scales on dispersion.

Spatial coherence in an open flow model

A one-dimensional asymmetrically coupled map lattice model is studied extensively by numerical simulations. It is shown that with open boundary conditions, the one-way coupled logistic lattices can exhibit spatially uniform, but temporally chaotic states that are stable in the presence of low-level noise. It is found that the spatially uniform unstable steady state of the system can be stabilized via single point control or pinning at the boundary site. In the spatially coherent and temporally chaotic regimes, a local finite disturbance may generate a propagating localized turbulent patch, which grows in size as it is swept downstream.

Differential-flow-induced instability in a cubic autocatalator system

The formation of spatio-temporal stable patterns is considered for a reaction-diffusion-convection system based upon the cubic autocatalator, A + 2B → 3B, B → C, with the reactant A being replenished by the slow decay of some precursor P via the simple step P → A. The reaction is considered in a differential-flow reactor in the form of a ring. It is assumed that the reactant A is immobilised within the reactor and the autocatalyst B is allowed to flow through the reactor with a constant velocity as well as being able to diffuse. The linear stability of the spatially uniform steady state (a, b) = (µ−1, µ), where a and b are the dimensionless concentrations of the reactant A and autocatalyst B, and µ is a parameter reflecting the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, flow-generated patterns is that the flow parameter φ > φc(µ, λ) where φc(µ,λ) is a (strictly positive) critical value of φ and λ is the dimensionless diffusion coefficient of the species B and also reflects the size of the system. Values of φc at which these bifurcations occur are derived in terms of µ and λ. Further information about the nature of the bifurcating branches (close to their bifurcation points) is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical Hopf bifurcations are possible. The bifurcating branches are then followed numerically by means of a path-following method, with the parameter φ as a bifurcation parameter, for representatives values of µ and λ. It is found that multiple stable patterns can exist and that it is also possible that any of these can lose stability through secondary Hopf bifurcations. This typically gives rise to spatio-temporal quasiperiodic transients through which the system is ultimately attracted to one of the remaining available stable patterns.

Parameter domains for instability of uniform states in systems with many delays

We present a computational method for determining regions in parameter space corresponding to linear instability of a spatially uniform steady state solution of any system of two coupled reaction-diffusion equations containing up to four delay terms. At each point in parameter space the required stability properties of the linearised system are found using mainly the Principle of the Argument. The method is first developed for perturbations of a particular wavenumber, and then generalised to allow arbitrary perturbations. Each delay term in the system may be of either a fixed or a distributed type, and spatio-temporal delays are also allowed.

Pattern selection and the effect of group velocity on interacting oscillatory and stationary instabilities

The effect of mean flows on pattern stability in systems where oscillatory instabilities of the Hopf type interact with stationary ones is investigated. In particular, it is shown that pattern selection may be strongly modified when the absolute instability threshold of the trivial uniform steady state is rejected beyond the stationary instability. The effect of spatially distributed noise including the competition between noise sustained and dynamically sustained structures is also discussed.

Noise-sustained structures in coupled complex Ginzburg-Landau equations for a convectively unstable system.

We investigate a pattern-forming system close to a Hopf bifurcation with broken translational symmetry. In one-dimensional geometries, its evolution is governed by two coupled complex Ginzburg-Landau equations which describe the amplitude of the counterpropagating traveling waves that develop beyond the instability. The convective and absolute instabilities of the possible steady states are analyzed. In the regime of strong cross coupling, where traveling waves are favored by the dynamics, the results of previous analysis are recovered. In the weak cross-coupling regime, where standing waves are favored by the dynamics, traveling waves nevertheless appear, in the absence of noise, between the uniform steady state and the standing-wave patterns. In this regime, standing waves are sustained by spatially distributed external noise for all values of the bifurcation parameter beyond the Hopf bifurcation. Hence, the noise is not only able to sustain spatiotemporal patterns, but also to modify pattern selection processes in regimes of convective instability. In this weak coupling regime we also give a quantitative statistical characterization of the transition between deterministic and noise-sustained standing waves when varying the bifurcation parameter. We show that this transition occurs at a noise-shifted point and it is identified by an apparent divergence of a correlation time and the saturation of a correlation length to a value given by the system size.

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiated in an open isothermal chemical system governed by cubic autocatalytic kinetics are discussed. The system is shown to be capable of sustaining up to three spatially uniform steady states, the (trivial) unreacted state, which is always stable (a node), and two nontrivial states, one of which is always unstable (a saddle point). The third state can change its stability through Hopf bifurcation (both subcritical and supercritical). This allows the possibility of two sorts of travelling wave being established; there are wave profiles which connect the unreacted state ahead to the nontrivial state at the rear, and wave profiles (pulse waves) which have the unreacted state at both the front and rear. The conditions under which a particular wave is initiated are considered by both a discussion of the (ordinary) differential equations governing the travelling waves and by numerical integrations of an initial-value problem. This treatment also reveals the possibility of a stable travelling wave propagating through the system, leaving behind a temporally unstable stationary state. Under these conditions, spatiotemporal chaotic behaviour is seen to develop after the passage of the wave.

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.

Stability analysis of a two-component nonlinear optical system with 2-D feedback

Stationary and oscillatory instabilities of the uniform steady state are found for a Kerr-slice/feedback-mirror model with an added second nonlinear slice. Complex dynamics is predicted to arise from competition between the two instability types.

Instability in a predator-prey system with delay and spatial averaging

We consider a predator-prey reaction-diffusion system containing a term which, in its most general form, represents both spatial and temporal averaging. The eigenvalue equation arising from linearizing the system about a uniform coexistence steady state is studied using the argument principle to obtain a stability criterion, involving a delay measure that is not specific to a particular delay kernel but which works for a wide class of such kernels. In certain particular cases, stability criteria that cannot be improved are obtained using other techniques, and comparisons are made with the general bounds.

Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme

The formation of stable patterns is considered in a reaction-diffusion system based on the cubic autocatalator, A+2B → 3B, B → C, with the reaction taking place within a closed region, the reactant A being replenished by the slow decay of precursor P via the reaction P → A. The linear stability of the spatially uniform Steady state % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaabmGabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyyp% a0ZaaeWaceaacqaH8oqBdaahaaWcbeqaaiabgkHiTiaaigdaaaGcca% GGSaGaeqiVd0gacaGLOaGaayzkaaaaaa!48C3!\[\left( {a,b} \right) = \left( {\mu ^{ - 1} ,\mu } \right)\], where a and b are the dimensionless concentrations of reactant A and autocatalyst B and μ is a parameter representing the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, solutions (patterns) is that the parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadseacqGH8aapcaaIZaGaeyOeI0IaaGOmamaakaaabaGaaGOm% aaWcbeaaaaa!4139!\[D < 3 - 2\sqrt 2 \] where D=D b/Da (D a and D b are the diffusion coefficients of chemical species A and B respectively). The values of μ at which these bifurcations occur are derived in terms of D and λ (a parameter reflecting the size of the system). Further information about the nature of the spatially non-uniform solutions close to their bifurcation points is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical bifurcations are possible. The bifurcation branches are then followed numerically using a path-following method, with μ as the bifurcation parameter, for representative values of D and λ. It is found that the stable patterns can lose stability through supercritical Hopf bifurcations and these stable, temporally periodic, spatially non-uniform solutions are also followed numerically.

Persistent tangles of vortex rings in excitable media

Abstract Vortex filaments in numerical excitable media form ‘stable organizing centers’: compact, spatially symmetric, temporally periodic configurations of linked and knotted rings emitting scroll-shaped wave trains while slowly moving through the medium. All past examples use excitable media in which the 2-dimensional (2d) spiral wave pivots rigidly without meander, and only one local state variable diffuses. Vortex filaments in such special media remain motionless when uncurved and free of ‘twist’. Are these conditions essential for persistence of vortex rings against collapse and reversion to a spatially uniform steady state? This paper shows by example that even if all variables diffuse (equally in this case) or if the 2d vortex ‘meanders’ spontaneously (the usual case), compact organizing centers of topologically distinct kinds can still persist. These ‘persistent organizing centers’ are not ‘stable’: they continually change shape without ever repeating. But their persistence for 70 rotation periods or more under generic conditions in numerical experiments suggests that they should be observable in nature.

The effects of coupling on pattern formation in a simple autocatalytic system

The spatiotemporal structures that can arise in two identical cells, each governed by cubic autocatalator kinetics and coupled via the diffusive interchange of a reactant, are discussed. The coupling gives rise to five spatially uniform steady states, one of which exists in the uncoupled system. By studying the linearized equations, it is found that three of these steady states, including that of the uncoupled system, may give rise to the possibility of bifurcations to spatially nonuniform steady states. In the case of the steady state corresponding to that of the uncoupled system, it is seen that the coupling leads to bifurcations not present in the uncoupled system which give rise to locally stable nonuniform steady states. A weakly nonlinear analysis is developed for both small and large coupling strength a, and for parameter values in a neighbourhood of the bifurcation points on the new steady states. This clarifies the nature of the nonuniform solutions close to bifurcation, which are then followed numerically using a path-following technique. The coupling is found to produce extra nonuniform steady solutions which are stable close to their bifurcation points.