analysis of complex and chaotic patterns near a codimension 2 turing hopf point in a reaction diffusion model we study a reaction diffusion system of activator inhibitor type and characterize the resulting complex and chaotic spatio temporal patterns by their lyapunov spectrum and by a karhunen loeve decomposition into empirical orthogonal eigenmodes different periodic patterns corresponding to localized hopf and turing modes and mixed modes including subharmonic spatio temporal spiking are found near a codimension 2 bifurcation point the asymptotic patterns are preceded by transient spatio temporal chaos before the system abruptly locks into a periodic state in space and time the karhunen loeve decomposition is shown to be a powerful tool for extracting detailed quantitative information of complex space time data

development of the kansa method for solving seepage problems using a new algorithm for the shape parameter optimization abstract in this research the kansa or multiquadric method mq has been developed for solving the seepage problems in 2d and 3d arbitrary domains this research is the first application of this method for seepage analysis in both confined and unconfined porous media the domain decomposition approach has been employed for applying mq method easily in inhomogeneous and irregular complex geometries and decreasing the computational costs for determining the optimum shape parameter that affects strongly the accuracy of mq and other rfb methods a new scheme that decreases drastically the computational time is introduced the efficiency of the proposed algorithm has been examined under various radial basis functions variations of number of interpolating points and points distribution through a numerical example with analytical solution eventually three examples including different boundary conditions are presented comparing results of the examples with other numerical methods indicates that the present approach has high capability and accuracy in solving seepage problems

global hopf bifurcation of two parameter flows the behavior of center indices as introduced by j mallet paret j yorke is analyzed for two parameter flows the integer sum of center indices along a one dimensional curve in parameter space is called the h index a nonzero h index implies global hopf bifurcation the index h is not a homotopy invariant this fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero which we call b points to each b point we assign an integer b index such that the h index relates to the b indices by a formula such as occurs in the calculus of residues

spatiotemporal chaos stimulated by transverse hopf instabilities in an optical bilayer system we study two dimensional spatiotemporal dynamics in an optical system of two identical nonlinear films irradiated from both sides by equal plane waves and show that a wide variety of chaotic behaviors can be obtained near the subcritical pitchfork bifurcation point the regimes arising in the vicinity of asymmetrical steady state depend on stability of the symmetrical steady state at the same driving field on interactions of hopf and turing instabilities occurring at the asymmetrical branch of solutions and on a transverse wave number of the hopf instability band center the wave number is controlled by a phase shift of the field passed between two films and the relative order of the turing and hopf bifurcations is controlled by a diffusion of charge carrier in a semiconductor media chaotic oscillating patterns are formed mainly by transverse hopf modes rising due to a time delay in the system

solving time dependent engineering problems with multiquadrics abstract this paper deals with the solution of time dependent problems the multiquadric radial basis function method is formulated with a new approach for transient problems one and two dimensional problems are considered the forward difference and the crank nicolson time marching schemes for parabolic cases are considered the central difference integration method of the newmark family is considered for hyperbolic problems the method proves its accuracy in four numerical examples

self replicating spots in reaction diffusion systems in this article we discuss the self replicating spot a particlelike phenomenon that occurs in reaction diffusion rd systems 1 the spots consist of localized regions in which the concentrations of the reactants differ from the surrounding concentration fields they grow reaching a critical size at which time they divide in two the two resulting spots again grow and divide this process which is visually similar to cell division continues indefinitely the long time behavior depends on the precise values of the external control parameters but typically consists of a chaotic soup in which many spots compete for resources as illustrated in fig 1 those spots that find adequate resources continue to grow and divide those that are unable to find adequate resources decay into the background the spots observed in 1 were found during an attempt to model labyrinthine patterns in the ferrocyanide iodate sulfate fis reaction 2 since then replicating spot patterns have been observed both numerically and experimentally in the fis reaction 3 4 there are obvious differences in the gaspar showalter model of the fis reaction 5 and of the models discussed by us and others this fact suggests that replication is a generic feature characterizing a broad class of reaction diffusion systems in 6 we presented some arguments in support of this proposition these arguments included both a heuristic description of the process of replication and demonstrations of analytic features common to several related model rd systems it turns out that replication is more general than our analysis accounts for nevertheless we think it worthwhile to spell out the details of the theory presented in 6 we remark here that various aspects of the replication phenomenon have been discussed by other authors 7 8 kerner and osipov have a large body of work on largeamplitude dissipative structures including an analysis of the static division of one dimensional pulses as the system size is changed gurevich and mints have a body of work on replication of thermal hot spots in composite superconductors in the article by petrov scott and showalter replication

a characterization of all elliptic algebro geometric solutions of the akns hierarchy here 9 x p x wl w3 denotes the elliptic weierstrass function with fundamental periods 2wl and 2w3 im w3 wl 0 in the special case where wl is real and w3 is purely imaginary the potential q x in 1 1 is real valued and ince s striking result 51 in modern spectral theoretic terminology yields that the spectrum of the unique self adjoint operator associated with the differential expression l2 d2 dx 2 q x in l2 r exhibits finitely many bands and gaps respectively that is

slowly modulated two pulse solutions in the gray scott model i asymptotic construction and stability two pulse solutions play a central role in the phenomena of self replicating pulses in the one dimensional 1 d gray scott model in the present work part i of two parts we carry out an existence study for solutions consisting of two symmetric pulses moving apart from each other with slowly varying velocities this corresponds to a mildly strong pulse interaction problem in which the inhibitor concentration varies on long spatial length scales critical maximum wave speeds are identified and odes are derived for the wave speed and for the separation distance between the pulses in addition the formal linear stability of these two pulse solutions is determined good agreement is found between these theoretical predictions and the results from numerical simulations the main methods used in this paper are analytical singular perturbation theory for the existence demonstration and the nonlocal eigenvalue problem nlep method developed in our earlier work for the stability analysis the analysis of

numerical solution for biharmonic equation using multilevel radial basis functions and domain decomposition methods biharmonic equation has significant applications in physics and engineering but is difficult to solve due to the existing fourth order derivatives one of the domain type mashless methods is obtained by simply applying the radial basis functions rbfs as a direct collocation which has shown to be effective in solving complicated physical problems with irregular domains in this paper we utilize overlapping domain decomposition and multilevel rbf methods for solving biharmonic equation numerical results indicate that these two methods circumvent the ill conditioning problem resulted from using the radial basis function as a global interpolant

spatiotemporal dynamics near a codimension two point spatiotemporal dynamics resulting from the interaction of two instabilities breaking respectively spatial and temporal symmetries are studied in the framework of the amplitude equation formalism the corresponding bifurcation scenarios feature steady hopf bistability with corresponding localized structures but also different types of mixed states some of these mixed modes result from self induced subharmonic instabilities of the pure steady and hopf modes the bifurcation schemes are then used to organize the results of numerical simulations of a one dimensional reaction diffusion model these dynamics are relevant to experimental chemical systems featuring a codimension two turing hopf point but also to any experimental setup where homogeneous temporal oscillations and spatial patterns are obtained for nearby values of parameters s1063 651x 96 03707 5

existence and stability of multiple spot solutions for the gray scott model in r2 abstract in this paper we rigorously prove the existence and stability of multiple spot patterns for the gray scott system in a two dimensional domain which are far from spatial homogeneity the green s function and its derivatives together with two nonlocal eigenvalue problems both play a major role in the analysis we establish a threshold behavior for stability if a certain inequality for the parameters holds then we get stability otherwise we get instability of multiple spot solutions the exact asymptotics of the critical thresholds are obtained

strange attractors in coupled reaction diffusion cells abstract a numerical study of two identical reaction cells with diffusion coupling has shown that the structure of motion in the system in principle agrees with results reported for variety of dynamic systems when the characteristic parameter is varied alternating sub intervals of stable periodic p k and stable aperiodic a l solutions appear the sub intervals are connected by intervals where tangent bifurcations and infinite sequences of subharmonic bifurcations occur feigenbaum relation holds for the studied sequence of subharmonic bifurcations aperiodic chaotic states are characterized by a complete set of one dimensional lyapunov exponents by power spectra and by corresponding poincare maps the spectra of lyapunov exponents are of the type 0 and show that the topological dimension of the chaotic attractor is two the power spectra are of two different types a the spectra with sharp peaks located above the broad band noise showing statistical phase coherence of the attractor b the flat spectra showing only broad band noise corresponding to phase incoherent attractor the phase coherence is present close after every point of accumulation phase incoherence arises when the strange attractors contain unstable periodic orbits with different topology the relations of bifurcated stable and unstable periodic solutions computed by means of continuation techniques to the structure of the strange attractor is discussed and poincare maps are used to illustrate the dependence of the structure of the attractors on the value of the characteristic parameter

mesa type patterns in the one dimensional brusselator and their stability the brusselator is a generic reaction diffusion model for a tri molecular chemical reaction we consider the case when the input and output reactions are slow in this limit we show the existence of k periodic spatially bi stable structures mesas and study their stability using singular perturbation techniques we find a threshold for the stability of k mesas this threshold occurs in the regime where the exponentially small tails of the localized structures start to interact by comparing our results with turing analysis we show that in the generic case a turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state we also study a breather type instability of a mesa which occurs due to a hopf bifurcation full numerical simulations are shown to confirm the analytical results in 1952 turing proposed that the formation of spatial patterns during morphogenesis could be explained in terms of the instability of a homogeneous steady state solution of a reaction diffusion network describing the evolution of a set of morphogens 31 turing himself illustrated his ideas on two chemical models turing s original work is primarily concerned with the stability analysis of the homogeneous steady state solution of the rate equations for the interacting morphogens 22 the main point of biological interest however is whether stable spatial structures may be generated beyond the instability i e whether the rate equations admit stable and positive inhomogeneous solutions exhibiting the most characteristic features of morphogenetic patterns this point has been taken up seriously in the early seventies more systematic numerical studies of turing s model have been performed showing irregular spatial structures 4 this led to serious reservations about the relevance of turing s theory in developmental biology particularly its ability to generate regular patterns but these criticisms originate from a somewhat unfortunate choice of turing s example and they do not touch the essential points of turing s theory more specifically the obvious requirement that the rate equations must admit positive and bounded solutions is not satisfied in turing s example 11

dynamics of two compartment gray scott equations abstract in this work the existence of a global attractor is proved for the solution semiflow of the coupled two compartment gray scott equations with the homogeneous neumann boundary condition on a bounded domain of space dimension n 3 the grouping estimation method combined with a new decomposition approach is introduced to overcome the difficulties in proving the absorbing property and the asymptotic compactness of this four component reaction diffusion systems with cubic autocatalytic nonlinearity and linear coupling the finite dimensionality of the global attractor is also proved

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

a multiquadric quasi interpolation with linear reproducing and preserving monotonicity in this paper we develop a multiquadric mq quasi interpolation which has the properties of linear reproducing and preserving monotonicity moreover we give its approximation error by theoretic analysis and illustrate the effect by means of two examples one of the examples is to approach the linear combination of two sine functions with different frequencies another is to approximate a function with discontinuity from the results of the examples we believe that the present mq quasi interpolation is feasible

stationary patterns caused by cross diffusion for a three species prey predator model we study a system arising from a three component prey predator model with prey dependent and ratio dependent functional responses where cross diffusion is included in such a way that predator chases the prey and the prey runs away from the predator we prove that the cross diffusion can generate the stationary patterns nonconstant positive steady states

applying multiquadric quasi interpolation to solve burgers equation abstract in this paper we develop a kind of univariate multiquadric mq quasi interpolation and use it to solve burgers equation with viscosity at first we construct the mq quasi interpolation which possesses the properties of linear reproducing and preserving monotonicity next we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable then we verify our method for two examples with distinguishing initial value condition one example is tested for three reynolds number that is r 10 r 100 and r 10 000 from the numerical experiments we see that the presented method in this paper is valid although the accuracy of the technique is not higher than hon and mao s one another example is used to examine the travelling of the shock we can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate mq quasi interpolation the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

a theory for one dimensional asynchronous chemical waves we present a theory for an experimentally observed phenomenon of one dimensional asynchronous waves it has been proposed that two oppositely moving travelling wave states can localize an oscillatory core structure a class of even and odd parity localized solution of the same system has been explicitly shown it has been argued that a combination of these local and global states in an extended system can produce asynchronous waves we also produce numerical results in support of our analytic predictions

synchronization and desynchronization in a network of locally coupled wilson cowan oscillators a network of wilson cowan wc oscillators is constructed and its emergent properties of synchronization and desynchronization are investigated by both computer simulation and formal analysis the network is a 2d matrix where each oscillator is coupled only to its neighbors we show analytically that a chain of locally coupled oscillators the piecewise linear approximation to the wc oscillator synchronizes and we present a technique to rapidly entrain finite numbers of oscillators the coupling strengths change on a fast time scale based on a hebbian rule a global separator is introduced which receives input from and sends feedback to each oscillator in the matrix the global separator is used to desynchronize different oscillator groups unlike many other models the properties of this network emerge from local connections that preserve spatial relationships among components and are critical for encoding gestalt principles of feature grouping the ability to synchronize and desynchronize oscillator groups within this network offers a promising approach for pattern segmentation and figure ground segregation based on oscillatory correlation

dynamical effects of integrative time delay coupling we study coupled dynamical systems wherein the influence of one system on the other is cumulative coupling signals are integrated over a time interval t a major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space for coupled limit cycle oscillators the landau stuart model we obtain an analytic estimate for the boundary of this region while for coupled chaotic lorenz oscillators numerical results are presented for given t we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously

self replication of mesa patterns in reaction diffusion systems abstract certain two component reaction diffusion systems on a finite interval are known to possess mesa box like steady state patterns in the singularly perturbed limit of small diffusivity for one of the two solution components as the diffusivity d of the second component is decreased below some critical value d c with d c o 1 the existence of a steady state mesa pattern is lost triggering the onset of a mesa self replication event that ultimately leads to the creation of additional mesas the initiation of this phenomena is studied in detail for a particular scaling limit of the brusselator model near the existence threshold d c of a single steady state mesa it is shown that an internal layer forms in the centre of the mesa the structure of the solution within this internal layer is shown to be governed by a certain core problem comprised of a single nonautonomous second order ode by analysing this core problem using rigorous and formal asymptotic methods and by using the singular limit eigenvalue problem slep method to asymptotically calculate small eigenvalues an analytical verification of the conditions of nishiura and ueyama y nishiura d ueyama a skeleton structure of self replicating dynamics physica d 130 1 1999 73 104 believed to be responsible for self replication is given these conditions include 1 the existence of a saddle node threshold at which the steady state mesa pattern disappears 2 the dimple shaped eigenfunction at the threshold believed to be responsible for the initiation of the replication process and 3 the stability of the mesa pattern above the existence threshold finally we show that the core problem is universal in the sense that it pertains to a class of reaction diffusion systems including the gierer meinhardt model with saturation where mesa self replication also occurs

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions the development of numerical methods for the solution of partial differential equations pdes with nonlocal boundary conditions is important since such type of problems arise as mathematical models of various real world processes we use radial basis function rbf collocation technique for the solution of a multidimensional linear elliptic equation with classical dirichlet boundary condition and nonlocal integral conditions rbf based meshless methods are easily implemented and efficient especially for multidimensional problems formulated on complexly shaped domains in this paper properties of the method are investigated by studying two and three dimensional test examples with manufactured solutions we analyze the influence of the rbf shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix

traveling spike autosolitons in the gray scott model abstract we developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons self sustained solitary waves in the gray scott model of an autocatalytic reaction we found that when the inhibitor diffusion is sufficiently slow the ultrafast traveling spike autosolitons are realized in a wide range of parameters when the diffusion of the inhibitor is sufficiently fast the slower traveling spike autosolitons with the diffusion precursor are realized we asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the gray scott model we also showed that in certain parameter regions the traveling spike autosolitons coexist with static

kansa rbf algorithms for elliptic problems in axisymmetric domains we employ a kansa radial basis function method for the numerical solution of elliptic boundary value problems in three dimensional axisymmetric domains we consider problems governed by the poisson equation the inhomogeneous biharmonic equation and the inhomogeneous cauchy navier equations appropriate discretizations lead to linear systems the coefficient matrices of which possess block circulant structures these systems are then solved efficiently by means of matrix decomposition algorithms and fast fourier transforms methods for choosing appropriate values of the shape parameter are also proposed the effectiveness of the proposed algorithms is demonstrated by considering several numerical examples

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

trust region methods for nonlinear elliptic equations with radial basis functions we consider the numerical solution of nonlinear elliptic boundary value problems with kansas method we derive analytic formulas for the jacobian and hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust region algorithm this ansatz is tested on semilinear quasilinear and fully nonlinear elliptic pdes including plateaus problem heleshaw flow and the mongeampre equation with excellent results the new approach distinctly outperforms previous ones based on linearization or finite difference jacobians

an asymptotic study of the inductive pattern formation mechanism in drosophila egg development the complexity and nonlinear dynamics of patterning networks in development make modeling an important approach for the evaluation of the experimentally derived pattern formation mechanisms as a rule mechanistic models of patterning networks have large number of uncertain parameters model analysis requires extensive computational searches of the parameter space analytical techniques can circumvent these difficulties and offer important insights into the networks functional capabilities here we present an asymptotic analysis of the multiple steady states and transitions between them in a mechanistic model of patterning events specifying the formation of a pair organ in drosophila oogenesis the model describes the interaction between the spatially nonuniform inductive signal and a network of spatially distributed feedback loops our approach dramatically reduces the complexity of the problem and provides an explicit analytical method for the construction and parametric analysis of the patterned states responsible for signaling the analysis reveals a skeleton structure for the patterning capability of the considered regulatory module and demonstrates how a single regulatory network can be used to generate a variety of developmental patterns

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

a skeleton structure of self replicating dynamics abstract self replicating patterns srp have been observed in several chemical reaction models such as the gray scott gs model as well as in physical experiments watching these experiments computational and physical is like watching the more familiar coarsening processes but in reverse the number of unit localized patterns increases until they fill the domain completely self replicating dynamics then can be regarded as a transient process from a localized trigger to a stable stationery or oscillating turing pattern since it is a transient process it is very difficult to give a suitable definition to characterize srp it cannot be described in terms of well studied structures such as the attractor or a singular saddle orbit for a dynamical system in this paper we present a new point of view to describe the transient dynamics of srp over a finite interval of time we focus our attention on the basic mechanism causing srp from a global bifurcational point of view and take our clues from two model systems including the gs model a careful analysis of the anatomy of the global bifurcation diagram suggests that the dynamics of srp is related to a hierarchical structure of limit points of folding bifurcation branches in parameter regions where the branches have ceased to exist thus the skeleton structure mentioned in the title refers to the remains of bifurcation branches the aftereffects of which are manifest in the dynamics of srp one of the natural and important problems is about the existence of an organizing center from which the whole hierarchical structure of limit points emerges in our setting the numerics suggests a strong candidate for that i e bogdanov takens turing btt singularity together with the presence of a stable critical point and so this indicates a universality of the above structure in the class of equations sharing this characteristic

algorithms defined by nash iteration some implementations via multilevel collocation and smoothing abstract we describe the general algorithms of nash iteration in numerical analysis we make a particular choice of algorithm involving multilevel collocation and smoothing our test case is that of a linear differential equation although the theory allows for the approximate solution of nonlinear differential equations we describe the general situation completely and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps we had earlier shown how these ideas apply to scattered data approximation but in this work we are interested in the application of the ideas to the numerical solution of differential equations we make use of approximate smoothers involving the solution of evolution equations with calibrated time steps

quartic gaussian and inverse quartic gaussian radial basis functions the importance of a nonnegative fourier transform we catalogue the numerical properties of approximations using two novel types of radial basis functions f r the qg species is a basis of exponentials of quartic argument f x f r b f x a h j 1 na jexp a h 4 x x j 4 where the x j are the rbf centers and also the interpolation points we show that quartic gaussian rbfs fail at many discrete values of the shape parameter a we show through a detailed analysis that these singularities are directly related to zeros of q k the fourier transform of exp x 4 if we reverse the roles and take q x as the rbf all difficulties disappear because these iqg rbfs have a fourier transform which is nonnegative for all real k we explain that although the quartic gaussian exp x 4 is positive definite in the physics dynamical systems sense of being zero free and nonnegative it lacks the crucial property of being positive definition in the rbf analysis sense

reaction diffusion patterns in confined chemical systems we review on the effects of the feed mode on pattern selection observed in chemical systems operated in open spatial reactors in two side fed reactors strong parameter ramps naturally confine patterns in a stratum the reactor thickness acts both as a genuine bifurcation parameter and on the pattern dimensionality depending on that thickness standard 2d hexagon and stripe turing patterns or more complex 3d planforms are observed in thin one side fed reactors patterning process must escape the imposed fixed boundary conditions either by devices introducing mixed boundary conditions or by an intrinsic phenomenon dubbed spatial bistability we show that in most cases for a comprehensive understanding of experimental observations the full 3d aspects have to be taken into account

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

devising efficient numerical methods for oscillating patterns in reaction diffusion systems in this paper we consider the numerical approximation of a reaction diffusion system 2d in space whose solutions are patterns oscillating in time or both in time and space we present a stability analysis for a linear test heat equation in terms of the diffusion d and of the reaction timescales given by the real and imaginary parts and of the eigenvalues of j p e the jacobian of the reaction part at the equilibrium point p e focusing on the case 0 0 we obtain stability regions in the plane where h d h t h t h t time stepsize lumped diffusion scale depending also from the space stepsize h and from the spectral properties of the discrete laplace operator arising from the semi discretization in space in space we apply the extended central difference formulas ecdfs of order p 2 4 6 in time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators the explicit euler and adi methods the symplectic euler and a partitioned runge kutta method that are symplectic in the absence of diffusion hence by estimating for each method we derive stepsize restrictions h t f m e t h d p in terms of the stability curve f m e t depending on diffusion and reaction timescales and from the approximation order in space for the same schemes we provide also a dispersion error analysis we present numerical simulations for the test heat equation and for the lotka volterra pde system with solutions oscillating only in time for the presence of a centre type dynamics in these cases the implicit symplectic schemes provide the best choice we solve also the schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the turing hopf instability in this case all schemes attain closed orbits in the phase space but the explicit adi method is the best choice from the computational point of view

analytical approach to localized structures in a simple reaction diffusion system we study from an analytical point of view a simple reaction diffusion model which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction diffusion model capturing all the essential ingredients of these breathing particlelike solutions

global dynamics and robustness of reversible autocatalytic reaction diffusion systems abstract global asymptotic dynamics of a representative cubic autocatalytic reaction diffusion system the reversible selkov equations are investigated this system features two pairs of oppositely signed nonlinear terms so that the asymptotic dissipative condition is not satisfied which causes substantial difficulties in an attempt to attest that the longtime dynamics are asymptotically dissipative an l 2 to h 1 global attractor of finite fractal dimension is shown to exist for the semiflow of the weak solutions of the reversible selkov equations with the dirichlet boundary condition on a bounded domain of dimension n 3 a new method of rescaling and grouping estimation is used to prove the absorbing property and the asymptotical compactness importantly the upper semicontinuity robustness in the h 1 product space of the global attractors for the family of solution semiflows with respect to the reverse reaction rate as it tends to zero is proved through a new approach of transformative decomposition to overcome the barrier of the perturbed singularity between the reversible and non reversible systems by showing the uniform dissipativity and the uniformly bounded evolution of the union of global attractors under the bundle of reversible and non reversible semiflows

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

forced patterns near a turing hopf bifurcation we study time periodic forcing of spatially extended patterns near a turing hopf bifurcation point a symmetry based normal form analysis yields several predictions including that i weak forcing near the intrinsic hopf frequency enhances or suppresses the turing amplitude by an amount that scales quadratically with the forcing strength and ii the strongest effect is seen for forcing that is detuned from the hopf frequency to apply our results to specific models we perform a perturbation analysis on general two component reaction diffusion systems which reveals whether the forcing suppresses or enhances the spatial pattern for the suppressing case our results are consistent with features of previous experiments on the chlorine dioxide iodine malonic acid chemical reaction however we also find examples of the enhancing case which has not yet been observed in experiment numerical simulations verify the predicted dependence on the forcing parameters

synchronization of random linear maps abstract we study synchronization of random one dimensional linear maps for which the lyapunov exponent can be calculated exactly certain aspects of the dynamics of these maps are explained using their relation with a random walk we confirm that the lyapunov exponent changes sign at the complete synchronization transition we also consider partial synchronization of nonidentical systems it turns out that the way partial synchronization manifests depends on the type of differences in lyapunov exponent or in contraction points between the systems the crossover from partial to complete synchronization is also examined

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

space fractional advection dispersion equations by the kansa method the paper makes the first attempt at applying the kansa method a radial basis function meshless collocation method to the space fractional advection dispersion equations which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer order derivative however because of non local property of integro differential operator of space fractional derivative numerical solution of these novel models is very challenging and little has been reported in literature it is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non local property thus the global methods appear to have certain advantages in numerical simulation of these non local models because of their high accuracy and smaller size resultant matrix equation compared with the finite difference method popular in the solution of fractional equations the kansa method is a recent meshless global technique and is promising for high dimensional irregular domain problems in this study the resultant matrix of the kansa method is accurately calculated by the gauss jacobi quadrature rule numerical results show that the kansa method is highly accurate and computationally efficient for space fractional advection dispersion problems

six strategies for defeating the runge phenomenon in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation because it is meshless there is no canonical grid to act as a starting point for function adaptive grid modification uniform grids are therefore common with rbfs like polynomial interpolation which is the limit of rbf interpolation when the rbf functions are very wide the flat limit rbf interpolants are vulnerable to the runge phenomenon whether the grid is uniform or irregular the n point interpolant may diverge as n on the interval spanned by the interpolation points even for a function f x that is analytic everywhere on and near this interval in this work we discuss six strategies for defeating the runge phenomenon specializing to a uniform grid on a finite interval and one particular species of rbfs in one space dimension f x exp a 2 h 2 x 2 where h is the grid spacing three strategies fail but three are successful one good strategy is to vary a with the number of interpolation points n as n 1 4 unfortunately this gives both a subgeometric rate of convergence for the approximation error falling as exp qn for some constant q and a matrix condition number that grows as exp pn for some p 0 in order to explain why fixed a independent of n is not a convergent strategy we digress to discuss error saturation and show experimentally that the saturation error on the finite interval is about 0 6exp 0 47 a 2 f if a user chosen error tolerance d is acceptable then the optimum choice is a o p t i m u m d 1 2log d 0 06 the second good strategy is rbf extension which uses rbf functions with centers on a interval slightly larger than the target interval to approximate f x a third strategy is to split the interval into two boundary layers and a large middle interval and apply separate rbf approximations on each the slow decrease of a strategy is much cheaper but rbf extension is much more resistant to ill conditioning and therefore can achieve much lower errors the three interval method is the least accurate but is robust and inexpensive

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

cross correlation of instantaneous phase increments in pressure flow fluctuations applications to cerebral autoregulation we investigate the relationship between the blood flow velocities bfv in the middle cerebral arteries and beat to beat blood pressure bp recorded from a finger in healthy and post stroke subjects during the quasisteady state after perturbation for four different physiologic conditions supine rest head up tilt hyperventilation and co2 rebreathing in upright position to evaluate whether instantaneous bp changes in the steady state are coupled with instantaneous changes in the bfv we compare dynamical patterns in the instantaneous phases of these signals obtained from the hilbert transform as a function of time we find that in post stroke subjects the instantaneous phase increments of bp and bfv exhibit well pronounced patterns that remain stable in time for all four physiologic conditions while in healthy subjects these patterns are different less pronounced and more variable we propose an approach based on the cross correlation of the instantaneous phase increments to quantify the coupling between bp and bfv signals we find that the maximum correlation strength is different for the two groups and for the different conditions for healthy subjects the amplitude of the cross correlation between the instantaneous phase increments of bp and bfv is small and attenuates within 3 5 heartbeats in contrast for post stroke subjects this amplitude is significantly larger and cross correlations persist up to 20 heartbeats further we show that the instantaneous phase increments of bp and bfv are cross correlated even within a single heartbeat cycle we compare the results of our approach with three complementary methods direct bp bfv cross correlation transfer function analysis and phase synchronization analysis our findings provide insight into the mechanism of cerebral vascular control in healthy subjects suggesting that this control mechanism may involve rapid adjustments within a heartbeat of the cerebral vessels so that bfv remains steady in response to changes in peripheral bp

homoclinic bifurcations at the onset of pulse self replication we establish a series of properties of symmetric n pulse homoclinic solutions of the reduced gray scott system u uv 2 v v uv 2 which play a pivotal role in questions concerning the existence and self replication of pulse solutions of the full gray scott model specifically we establish the existence and study properties of solution branches in the plane that represent multi pulse homoclinic orbits where and are the central values of u x and v x respectively we prove bounds for these solution branches study their behavior as and establish a series of geometric properties of these branches which are valid throughout the plane we also establish qualitative properties of multi pulse solutions and study how they bifurcate i e how they change along the solution branches

the existence and stability of spike equilibria in the one dimensional gray scott model the pulse splitting regime abstract the existence stability and pulse splitting behavior of spike patterns in the one dimensional gray scott model on a finite domain is analyzed in the semi strong spike interaction regime this regime is characterized by a localization of one of the components of the reaction near certain spike locations while the other component exhibits a more global spatial variation across the domain the method of matched asymptotic expansions is then used to construct k spike equilibria in terms of a certain core problem this core problem is studied numerically and asymptotically for each integer k 1 it is shown that there are two branches of k spike equilibria that meet at a saddle node bifurcation value for small values of the diffusivity d of the second component these saddle node bifurcation points occur at approximately the same value a combination of asymptotic and numerical methods is used to analyze the stability of these branches of k spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile in this way the key bifurcation and spectral conditions of ei et al s ei y nishiura k ueda 2 n splitting or edge splitting a manner of splitting in dissipative systems jpn j ind appl math 18 2001 181 205 believed to be essential for pulse splitting behavior in a reaction diffusion system are verified by having verified these conditions a simple analytical criterion for the occurrence of pulse splitting is then formulated and confirmed with full numerical simulations of the gray scott model this criterion verifies a conjecture based on numerics and topological arguments reported in a doelman r a gardner t j kaper stability analysis of singular patterns in the 1d gray scott model a matched asymptotics approach physica d 122 1 4 1998 1 36 the analytical results are compared with previously obtained results for pulse splitting behavior

total and partial amplitude death in networks of diffusively coupled oscillators networks of weakly nonlinear oscillators are considered with diffusive and time delayed coupling averaging theory is used to determine parameter ranges for which the network experiences amplitude death whereby oscillations are quenched and the equilibrium solution has a large domain of attraction the amplitude death is shown to be a common phenomenon which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions in addition when the network consists of dissimilar oscillators there exist parameter values for which only parts of the network are suppressed sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities coupling coefficients and connection delays

spatio temporal chaos for the gray scott model abstract a new geometrical criterion for the transition to spatio temporal chaos stc arising in the gray scott model is presented this is based on the inter relationship of global bifurcating branches of ordered patterns with respect to supply and removal rates contained in the model this viewpoint not only gives us a new criterion for the onset of stc but also clarifies how the orbit itinerates among several ordered patterns in infinite dimensional space moreover the geometrical characterization gives us a universal viewpoint about the onset and termination of stc there are at least two different mechanisms that cause re injection dynamics and drive the stc one is a generalized heteroclinic cycle consisting of self replication and self destruction processes and the other involves annihilation of colliding waves instead of self destruction

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

slowly modulated two pulse solutions in the gray scott model ii geometric theory bifurcations and splitting dynamics with great sadness we note the passing away of our mentor and colleague wiktor abstract in this second paper we develop a geometrical method to systematically study the singular perturbed problem associated to slowly modulated two pulse solutions it enables one to see that the characteristics of these solutions are strongly determined by the flow on a slow manifold and hence also to identify the saddle node bifurcations and bifurcations to classical traveling waves in which the solutions constructed in part i are created and annihilated moreover we determine the geometric origin of the critical maximum wave speeds discovered in part i in this paper we also study the central role of the slowly varying inhibitor component of the two pulse solutions in the pulse splitting bifurcations finally the validity of the quasi stationary approximation is established here and we relate the results of both parts of this work to the literature on self replication

second order implicit explicit scheme for the gray scott model a second order scheme for the gray scott gs model used to describe the pattern formation is studied the linear part of the gs equation for the time derivative and the viscous terms is discretized implicitly while the other or nonlinear part of the gs equation explicitly galerkin finite element approximation methods are presented and analyzed as well as methods for solving the resulting system of equations the optimal l 2 norm error estimates are derived numerical experiments are presented

stability analysis of singular patterns in the 1d gray scott model a matched asymptotics approach abstract in this work we analyze the linear stability of singular homoclinic stationary solutions and spatially periodic stationary solutions in the one dimensional gray scott model this stability analysis has several implications for understanding the recently discovered phenomena of self replicating pulses for each solution constructed in a doelman et al nonlinearity 10 1997 523 563 we analytically find a large open region in the space of the two scaled parameters in which it is stable specifically for each value of the scaled inhibitor feed rate there exists an interval whose length and location depend on the solution type of values of the activator autocatalyst decay rate for which the solution is stable the upper boundary of each interval corresponds to a subcritical hopf bifurcation point and the lower boundary is explicitly determined by finding the parameter value where the solution disappears i e below which it no longer exists as a solution of the steady state system explicit asymptotic formulae show that the one pulse homoclinic solution gains stability first as the second parameter is decreased and then successively the spatially periodic solutions with decreasing period become stable moreover the stability intervals for different solutions overlap these stability results are derived via the reduction of a fourth order slow fast eigenvalue problem to a second order nonlocal eigenvalue problem nlep explicit determination of these stability intervals plays a central role in understanding pulse self replication numerical simulations confirm that the spatially periodic stationary solutions are attractors in the pulse splitting regime and moreover whenever for a given solution the value of the activator decay rate was taken to lie in the regime below that solution s stability interval initial data close to that solution were observed to evolve toward a different spatially periodic stationary solution one whose stability interval inclucded the parameter value the main analytical technique used is that of matched asymptotic expansions

axisymmetric ring solutions of the 2d gray scott model and their destabilization into spots this article concerns annular ring solutions of the gray scott model in the monostable regime annular rings are far fromequilibrium patterns supported on annuli inside of which the activator is concentrated the diffusive flux of inhibitor over long length scales toward such an annulus feeds the production of activator there and the interaction is semi strong numerical and experimental observations show that annular rings often split into spots and the main result presented in this article is a method to predict the number of spots that an annular ring unstable to angular disturbances will split into this method is an extension to 2d circular geometries of the nonlocal eigenvalue problem nlep method developed for pulse solutions of the 1d gray scott problem in which the full eigenvalue problem a pair of second order nonautonomous coupled equations is recast as a single second order equation with a nonlocal term we also continue the results for the monostable regime into the bistable regime of the gray scott model where target patterns exist and their rings are observed to destabilize into rings of spots as may be shown using a classical turing ginzburg landau analysis thus for these 2d circular geometries the nlep method is to the instability of annular rings in the monostable regime what the turing analysis is to the instability of target patterns in the bistable regime near criticality 2004 elsevier b v all rights reserved

a meshfree method for solving multidimensional linear fredholm integral equations on the hypercube domains the main purpose of this article is to describe a numerical scheme for solving multidimensional linear fredholm integral equations of the second kind on the hypercube domains the method is based on interpolation by radial basis functions rbfs to approximate the solution of the linear fredholm integral equations error analysis is presented for this method finally several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method

exact homoclinic and heteroclinic solutions of the gray scott model for autocatalysis in this paper we obtain explicit nontrivial stationary patterns in the one dimensional gray scott model for cubic autocatalysis involved in the reaction are two chemicals a and b whose diffusion coefficients are denoted by da and db respectively the chemical a is fed into the system at a rate kf reacts with the catalyst b at a rate k1 and the catalyst decays at a rate k2 if these parameters obey the relation kf da k2 db then for appropriate values of the rate constants we present explicit expressions for two families of pulses and one kink we also show that if is only satisfied approximately these families of pulses are preserved and there exists a smooth branch of kinks leading from the explicit one obtained when is satisfied we determine the local behavior of this branch near the explicit kink

the dynamics and pinning of a spike for a reaction diffusion system the motion of a one spike solution to a simplified form of the gierer meinhardt activator inhibitor model is studied in both a one dimensional and a two dimensional domain the pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator referred to as precursor gradients is studied in detail in the one dimensional case we derive a differential equation for the trajectory of the spike in the limit e 0 where e is the activator diffusivity a similar differential equation is derived for the two dimensional problem in the limit for which e 1 and d 1 where d is the inhibitor diffusivity a numerical finite element method is presented to track the motion of the spike for the full problem in both one and two dimensions finally the numerical results for the spike motion are compared with corresponding asymptotic results for various examples

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

differential flow induced transition of hopf instability to turing instability and pattern formation abstract an external electric field may affect mass transport in a reaction diffusion system we show that a symmetry breaking instability leading to formation of spatial structures may result in when a constant external electric field normal to the reaction plane causes a diffusion driven stable state in absence of diffusion the steady state is homogeneous and unstable due to hopf bifurcation to become unstable we trace its origin in an interesting transition of hopf instability to reaction diffusion instability triggered by a differential flow in presence of electric field we carry out a numerical simulation in three dimensions on a chemical system chlorite iodide malonic acid the pattern is generically different from the electric field deformed turing pattern

pattern formation in a simple chemical system with general orders of autocatalysis and decay i stability analysis in this preliminary paper we introduce a mathematical model based on an autocatalytic reaction scheme with termination and general reaction orders rigorous stability analysis and numerical solutions are presented which illustrate that the problem can for certain parameter values admit a stable stationary pattern in the concentration of the autocatalyst and reactant this pattern forms at the rear of a permanent form travelling wave or pulse solution

global bifurcations at the onset of pulse self replication in this work we carried out an extensive numerical exploration of the delicate global dynamics of pulse self replication and analyzed the stability of singular homoclinic stationary solutions and their bifurcations in the one dimensional gray scott model this stability analysis has several implications for understanding the recently discovered phenomena of self replicating pulses the solutions of the ordinary differential equation are organized around a codimension 2 global bifurcation from which two or n branches of homoclinic orbits originate corresponding to solitary pulse solutions in the partial differential equation a careful analysis of the bifurcation scenarios in the global bifurcation diagram suggests that the dynamics of the self replicating system are related to a hierarchy structure of folding bifurcation branches in parameter space the numerical simulation suggests that the bogdanov takens points together with the presence of critical points emanating from a particular codimension 2 homoclinic orbit play a central role for the global bifurcation of periodic orbits the homoclinic solutions and the complex chaotic dynamics the numerical simulation also reveals the existence of a modulating two pulse or multipulse which accompanies the procedure of pulse self replication in reaction diffusion systems

solving partial differential equation by using multiquadric quasi interpolation abstract in this paper we use a kind of univariate multiquadric mq quasi interpolation to solve partial differential equation pde we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement our numerical experiment includes two examples one is solving viscid burgers equation for initial trapezoidal conditions another is simulating the interaction of two waves travelling in opposite direction from the numerical experiment we can see that the present scheme is valid