numerical computation of heteroclinic orbits we give a numerical method for the computation of heteroclinic orbits connecting two saddle points in 2 these can be computed to very high period due to an integral phase condition and an adaptive discretization we can also compute entire branches one dimensional continua of such orbits the method can be extended to compute an invariant manifold that connects two fixed points in n as an example we compute branches of traveling wave front solutions to the huxley equation using weighted sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period

soliton like phenomena in one dimensional cross diffusion systems a predator prey pursuit and evasion example abstract we have studied properties of non linear waves in a mathematical model of a predator prey system with pursuit and evasion we demonstrate a new type of propagating wave in this system the mechanism of propagation of these waves essentially depends on the taxis represented by non linear cross diffusion terms in the mathematical formulation we have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms parabolic and linear transition from one form to the other correlates with changes in the shape of the wave profile dependence of the propagation velocity on diffusion in this system differs from the square root dependence typical of reaction diffusion waves we also demonstrate that for systems with negative and positive taxis for example pursuit and evasion there typically exists a large region in the parameter space where the waves demonstrate quasi soliton interaction colliding waves can penetrate through each other and waves can also reflect from impermeable boundaries

comparison principles for strongly coupled reaction diffusion equations in unbounded domains comparison principles for systems of reaction diffusion equations in unbounded domains and coupledvia both reaction and diffusion terms are considered applications are made to the fitzhugh nagumo equations and models of coupled nerve fibres

wave propagation in a fitzhugh nagumo type model with modified excitability we examine a generalized fitzhugh nagumo fhn type model with modified excitability derived from the diffusive morris lecar equations for neuronal activity we obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms we study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms fronts and pulses while the activator wave maintains the usual kink front shape the nonequilibrium ising bloch bifurcation for the wave speed that occurs in the fhn model where the control parameter is the ratio of inhibitor to activator time scales persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter

local and global strong solutions to continuous coagulation fragmentation equations with diffusion we consider the diffusive continuous coagulation fragmentation equations with and without scattering and show that they admit unique strong solutions for a large class of initial values if the latter values are small with respect to a suitable norm we provide sufficient conditions for global in time existence in the absence of fragmentation

existence and uniqueness of traveling wave front of a nonlinear singularly perturbed system of reaction diffusion equations with a heaviside step function abstract the main purpose of this paper is to show the existence and uniqueness of a traveling wave front u w u x v e t w x v e t of the following nonlinear singularly perturbed system of reaction diffusion equations with a heaviside step function u t 2 u x 2 2 h u u w w t 2 w x 2 e u w

spatiotemporal dynamics of continuum neural fields we survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields neural fields model the large scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections they provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and turing like patterns

a parallel algorithm for the sylvester observer equation we present a new algorithm for solving the sylvester observer equation arising in the context of the luenberger observer the algorithm embodies two main computational phases the solution of several independent equation systems and a series of matrix matrix multiplications the algorithm is thus well suited for parallel and high performance computing by reducing the coefficient matrix a to lower hessenberg form one can implement the algorithm efficiently with few floating point operations and little workspace the algorithm has been successfully implemented on a cray c90 a comparison both theoretical and experimental has been made with the well known hessenberg schur algorithm which solves an arbitrary sylvester equation our theoretical analysis and experimental results confirm the superiority of the proposed algorithm both in efficiency and speed over the hessenberg schur algorithm

pointwise bounds for the solutions of the smoluchowski equation with diffusion we prove various decay bounds on solutions fn n 0 of the discrete and continuous smoluchowski equations with diffusion more precisely we establish pointwise upper bounds on nlfn in terms of a suitable average of the moments of the initial data for every positive l as a consequence we can formulate sufficient conditions on the initial data to guarantee the finiteness of l p mathbb r d times 0 t norms of the moments x a x t sum m in mathbb n m a f m x t int 0 infty m a f m x t dm in the case of continuous smoluchowski s equation for every p in 1 infty in previous papers 11 and 5 we proved similar results for all weak solutions to the smoluchowski s equation provided that the diffusion coefficient d n is non increasing as a function of the mass in this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function phi n that is closely related to the total increase of the diffusion coefficient in the interval 0 n

on closed form solutions to the generalized sylvester conjugate matrix equation abstract in this paper several explicit parametric solutions to the generalized sylvester conjugate matrix equation are given the proposed solutions can provide all the degrees of freedom which are represented by a free parameter matrix in addition the approaches adopted in this paper do not require that the coefficient matrices are in any canonical forms

wavefronts in bistable hyperbolic reaction diffusion systems abstract we study the dynamics of wavefronts that arise in bistable systems with correlated diffusion in time for the piecewise linear reaction term the speed of propagation of these fronts is obtained analytically the shape and the width of the front are analyzed by means of two different methods usually employed for parabolic diffusion it is shown that for hyperbolic reaction diffusion equations the wavefront presents a discontinuity in the slope

implementation of a fragmentation coagulation scattering model for the dynamics of stirred liquid liquid dispersions abstract the model for the evolution of a liquid liquid dispersion including the so called volume scattering effect introduced by the authors in a fasano f rosso a new model for the dynamics of dispersions in a batch reactor in h j bungartz r h w hoppe c zenger eds proceedings of the symposium organized by the sonderforschungsbereich 438 on the occasion of karl heinz hoffman s 60th birthday in lectures in applied mathematics springer verlag berlin 2000 pp 123 141 and incorporating multiple breakage see a fasano f rosso dynamics of droplets in an agitated dispersion with multiple breakage i formulation of the model and physical consistency math methods appl sci 28 6 2005 631 659 a fasano f rosso dynamics of droplets in an agitated dispersion with multiple breakage ii uniqueness and global existence math methods appl sci 28 9 2005 1061 1088 is considered from a viewpoint closer to applications the choice of kernels is discussed on the basis of the technical literature not just using very abstract kernels like in most mathematical papers and simulations are performed utilizing data from different sources

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

threshold phenomena for a reaction diffusion system abstract we consider the pure initial value problem for the system of equations t xx w w t e w e 0 the initial data being x 0 w x 0 x 0 here v v h v a where h is the heaviside step function and a 0 1 2 this system is of the fitzhugh nagumo type and has several applications including nerve conduction and distributed chemical biochemical systems it is demonstrated that this system exhibits a threshold phenomenon this is done by considering the curve s t defined by s t sup x v x t a the initial datum x is said to be superthreshold if limt s t it is proven that the initial datum is superthreshold if x a on a sufficiently long interval x is sufficiently smooth and x decays sufficiently fast to zero as x

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

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

wavy fronts and speed bifurcation in excitable systems with cross diffusion a bifurcation of excitation fronts induced by cross diffusion in two component bistable reaction diffusion systems of activator inhibitor type is discovered this bifurcation is similar to the nonequilibrium ising bloch bifurcation a different type of fronts whose spatial profiles are characterized by oscillating tails are associated with this bifurcation these fronts are described using exact analytical solutions of piecewise linear reaction diffusion equations

criteria of spontaneous interconversions of traveling and static arbitrary dimensional dissipative structures abstract the general criteria of spontaneous transformation of traveling dissipative structures dss autowaves into static dss are derived on the basis of translational symmetry of the problem these criteria determine the system parameters in particular the excitation level a am when the velocity of autowaves of arbitrary dimension and shape becomes zero it is demonstrated that the criteria coincide with the instability condition of static dss with respect to the aperiodic increase of a certain fluctuation describing a small translation of the dss along arbitrary axis i e with the condition of spontaneous transformation of static dss into traveling ones it is shown that the velocity of autowaves depends on parameter a near am v a m a 1 2 the models of fitzhugh nagumo type are analyzed and it is shown that the particular conditions obtained earlier follow from the general criteria derived here some scenarios of interconversion of traveling and static dss are discussed

classification of coupled systems with two component nonlinear diffusion equations by the invariant subspace method the invariant subspace method is developed to perform classification of systems with two component nonlinear diffusion equations which was carried out with respect to the invariant subspaces defined by linear ordinary differential equations as a result the corresponding exact solutions generated by invariant subspaces to the resulting systems are obtained in most cases two components of these exact solutions belong to different scalar subspaces behaviour to several exact solutions of the systems is described

robust stability via sign definite decomposition this paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests the problem occurs in the design of a fixed order or fixed structure multivariable feedback controller parametrized by a real design parameter vector for a plant containing a vector of uncertain parameters the characteristic polynomials of such systems often contain coefficients which depend polynomially on and using results on sign definite decomposition a new stability test is developed that gives a sufficient condition for hurwitz stability of the family of closed loop systems that result when and vary over prescribed boxes this test is reminiscent of kharitonov s theorem even though the family of polynomials considered here is certainly not restricted to be interval or even convex moreover the test does reduce to kharitonov s theorem for the special case of interval polynomials using this criterion recursively and modularly sets of controllers that stabilize the family of uncertain plants are determined

traveling waves of calcium in a pancreatic acinar cells model construction and bifurcation analysis abstract we construct and study a model for intracellular calcium wave propagation with particular attention to pancreatic acinar cells the model is based on a model of the inositol trisphosphate ip3 receptor which assumes that calcium modulates the binding affinity of ip3 to the receptor two versions of the model one simpler than the other are studied numerically in both versions solitary waves in the excitable regime arise via homoclinic bifurcations in the traveling wave equations as the background concentration of ip3 is increased the wave speed increases and for some values of the ip3 concentration the initial pulse gives rise to secondary pulses that travel in both directions this can give rise to irregular spatio temporal behavior or to trains of pulses in the simpler model these secondary waves are related to the presence of a t point a heteroclinic cycle and an associated spiral of homoclinic orbits which terminate the branch of homoclinic orbits

speed shift in reaction diffusion equations with cutoff abstract front solutions in a reaction diffusion equations with cutoff are obtained analytically using piecewise linear approximations of the reaction term the piecewise emulation allows us to study analytically the effect of the cutoff for fronts propagating into metastable and unstable states a general relationship between the cutoff threshold and the parameters involved into the reaction term is derived our results are in excellent agreement with that existing in the literature and prove that the piecewise emulation is an useful procedure to deal with complex reaction diffusion equations under cutoff

a new block algorithm for full rank solution of the sylvester observer equation a new block algorithm for computing a full rank solution of the sylvester observer equation arising in state estimation is proposed the major computational kernels of this algorithm are 1 solutions of standard sylvester equations in each case of which one of the matrices is of much smaller order than that of the system matrix and furthermore this small matrix can be chosen arbitrarily and 2 orthogonal reduction of small order matrices there are numerically stable algorithms for performing these tasks including the krylov subspace methods for solving large and sparse sylvester equations the proposed algorithm is also rich in level 3 basic linear algebra subroutine blas 3 computations and is thus suitable for high performance computing furthermore the results on numerical experiments on some benchmark examples show that the algorithm has better accuracy than that of some of the existing block algorithms for this problem

analytic solutions for monotonic and oscillating fronts in a reaction diffusion system under external fields abstract a piecewise linear reaction diffusion system including an external field is considered analytic solutions are obtained for the propagating front the front velocity the perturbed front and the growth rate of perturbations it is found that when the ratio of the time scales null cline parameter and the external field are small enough the moving front has an oscillating behaviour similar to that of a front in an oscillatory medium a comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases in particular we find that for fronts oscillating about a stable state both concentrations must oscillate in two variable models whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic

heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems abstract we apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems using the notion of separation we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space this as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution all the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter e by using singular perturbation techniques especially the slep method a relation between the stability of front or back solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation

biased random walks and propagation failure the critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non markovian biased random walks with reactions from the hamilton jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation if the reaction rate is larger than the jump frequency invasion always succeeds even in the case of maximal bias numerical simulations support our analytical predictions

stability of front solutions in inhomogeneous media abstract we use the fitzhugh nagumo f n model to study front solutions in inhomogeneous media inhomogeneity is modeled by adding a space dependent term z x in the equations changes in z shift the cubic nullcline up and down in the phase plane of the f n model this is used to mimic shifts in the cubic shaped nullcline of a bidomain model of calcium waves in egg cells physica d in press caused by changes in the total calcium concentration conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear we show that spatial inhomogeneities cause pinning and oscillations of front solutions this is best demonstrated when z x is a linear ramp when the slope is large a linear ramp breaks the translational invariance of the stationary front and stabilizes the front the front becomes less stable as the slope decreases at a critical slope the front becomes unstable through a hopf bifurcation beyond which oscillations in the front occur this also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness this bifurcation has been found previously physica d 106 1997 270 phys rev e 55 1997 366 in a local analysis of the f n model near a co dimension two point sometimes called a non equilibrium ising bloch nib bifurcation and with inhomogeneities of small magnitude the analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude we show that this bifurcation need not occur in conjunction with the nib point it is caused exclusively by spatial inhomogeneity a rich variety of wave phenomena including front oscillation front pinning and front reflection are shown to occur in inhomogeneous media

turing pattern formation in the brusselator system with nonlinear diffusion in this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the brusselator system through linear stability analysis of the basic solution we determine the turing and the oscillatory instability boundaries a comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of turing pattern formation we study the process of pattern formation both in one dimensional and two dimensional spatial domains through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns the analysis of the amplitude equations shows the occurrence of a number of different phenomena including stable supercritical and subcritical turing patterns with multiple branches of stable solutions leading to hysteresis moreover we consider traveling patterning waves when the domain size is large the pattern forms sequentially and traveling wave fronts are the precursors to patterning we derive the ginzburg landau equation and describe the traveling front enveloping a pattern which invades the domain we show the emergence of radially symmetric target patterns and through a matching procedure we construct the outer amplitude equation and the inner core solution

the discrete diffusive coagulation fragmentation equations with scattering we consider the discrete coagulation fragmentation equations with diffusion presupposing a maximal cluster size such a feature requires a new interaction mechanism in order to prevent occurrence of too large clusters being produced by coagulation existence of a unique solution for this model is proven and long time behaviour is studied in situations where equilibria are explicitly known

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

evolution of reaction diffusion patterns in infinite and bounded domains abstract we introduce a semi analytical method to study the evolution of spatial structures in reaction diffusion systems it consists in writing an integral equation for the relevant densities from the propagator of the linear part of the evolution operator in order to test the method we perform an exhaustive study of a one dimensional reaction diffusion model associated to an electrical device the ballast resistor we consider the evolution of step and bubble shaped initial density profiles in free space as well as in a semi infinite domain with dirichlet and neumann boundary conditions the piecewise linear form of the reaction term which preserves the basic ingredients of more complex nonlinear models makes it possible to obtain exact wave front solutions in free space and stationary solutions in the bounded domain short and long time behaviour can also be analytically studied whereas the evolution at intermediate times is analyzed by numerical techniques we paid particular attention to the features introduced in the evolution by boundary conditions

influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks abstract the main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts the authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks the mathematical model equation is u t f u r k x y h u y t 1 c x y d y where 0 c 0 and 0 are constants satisfying the condition 0 2 f the function f u represents sodium currents the function k denotes synaptic coupling in a neuronal network and the heaviside step function h is defined by h u 0 for all u 0 h 0 1 2 and h u 1 for all u 0

closed form solutions to sylvester conjugate matrix equations some explicit closed form solutions of homogeneous and nonhomogeneous sylvester conjugate matrix equations are provided in this paper one of the solutions is expressed in terms of controllability matrices and observability matrices the proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix by specifying the solutions of the homogeneous sylvester conjugate equation some new expressions of the solutions of the normal sylvester normal sylvester conjugate and sylvester equations are given this fact reveals that the sylveter conjugate matrix equations are a more general class of some previously investigated matrix equations

stationary wave solutions of a system of reaction diffusion equations derived from the fitzhugh nagumo equations we consider an extension of the fitzhugh nagumo model namely the system u t d 1 u xx f u w qquad w t d 2 w xx varepsilon u gamma w where varepsilon 0 gamma 0 d 1 0 d 2 0 and f u is cubic we allow gamma to be large which implies that there are three constant solutions we show that over an appropriate range of parameters the system has time independent pulse solutions and an infinite number of periodic solutions depending on the particular choice of parameters we show that the pulse solution leads to either the first constant solution or the third constant solution

the influence of fractional diffusion in fisher kpp equations we study the fisher kpp equation where the laplacian is replaced by the generator of a feller semigroup with power decaying kernel an important example being the fractional laplacian in contrast with the case of the standard laplacian where the stable state invades the unstable one at constant speed we prove that with fractional diffusion generated for instance by a stable levy process the front position is exponential in time our results provide a mathematically rigorous justification of numerous heuristics about this model

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

turing instability and pattern induced by cross diffusion in a predator prey system with allee effect to have spatial patterns form small allee effect admits relatively large range of predator mortality rate found cross diffusion is the key mechanism of spatial pattern formation obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur weak allee effect admits supercritical bifurcation while strong one allows subcritical bifurcation in this paper we first propose a mathematical model for a spatial predator prey system with allee effect and then by using the proposed model we investigate the turing instability and the phenomena of pattern formation we show how cross diffusion destabilizes the spatially uniform steady state the method of multiple time scales is employed to derive the amplitude equations which is the cubic stuart landau equation in the supercritical case and the quintic in the subcritical case based on the amplitude equations we obtain the asymptotic solutions of the model close to the onset of instability

evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations abstract the main purposes of this paper are to accomplish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations u t u x x h u u w w t e u w and to establish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation u t u x x h u u under different conditions on the model constants to establish the bifurcation for the system we will study the existence and instability of a standing pulse solution if 0 2 1 the existence and stability of two standing wave fronts if 2 1 and 2 e 1 the existence and instability of two standing wave fronts if 2 1 and 0 2 e 1 the existence and instability of an upside down standing pulse solution if 0 1 2 1 to establish the bifurcation for the scalar equation we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 2 the existence and stability of two standing wave fronts if 2 the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 2 by the way we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation u t u x x h u u w 0 where w 0 2 0 is a positive constant if 0 2 to achieve the main goals we will make complete use of the special structures of the model equations and we will construct evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators it turns out that a complex number 0 is an eigenvalue of the linear differential operator if and only if 0 is a zero of the evans function the stability instability and bifurcations of the nonlinear waves follow from the zeros of the evans functions a very important motivation to study the existence stability instability and bifurcations of the nonlinear waves is to study the existence and stability instability of infinitely many fast slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations the existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience

solitary waves in excitable systems with cross diffusion we consider a fitzhughnagumo system of equations where the traditional diffusion terms are replaced with linear cross diffusion of components this system describes solitary waves that have unusual

front bifurcations in reaction diffusion systems with inhomogeneous parameter distributions abstract fronts between two different stable homogeneous domains are considered in the framework of two component reaction diffusion rd systems with fitzhugh nagumo type dynamics the dominance principle ensuring that for a system with homogeneous parameters the direction of front propagation is uniquely determined by the relative strengths of the two domains fails when the inhibitor s controller s time scale is slowed down sufficiently this leads to a regime of bi directional propagation where for a given set of parameters either of the two domains may win the competition bi directional propagation may be viewed as being organized by a bifurcation of the stationary front obtained for equally strong domains which is destabilized when a critical value of the relative time constant is crossed although generically one would expect a transcritical scenario this bifurcation turns out to be of supercritical pitchfork type for arbitrary nonlinearity i e the bifurcation is independent of an accidental inflection symmetry of the dynamics for inhomogeneous parameter distributions the behavior becomes richer leading to stationary and oscillating pinning as well as to reflections of incoming fronts this situation is analyzed performing a multiple scale perturbation expansion up to third order in the vicinity of the above bifurcation leading to a two dimensional reduced dynamics globally valid in space the two degrees of freedom relevant for the description are the position of the front and a characteristic feature of its shape the latter is reminiscent of the momentum of classical particle physics results are compared with full system numerics

stationary and oscillatory fronts in a two component genetic regulatory network model abstract we investigate a two component gene network model originally used to describe the spatiotemporal patterning of the gene products in early drosophila development by considering a particular mode of interaction between the two gene products denoted proteins a and b we find both stable stationary and time oscillatory fronts can occur in the reaction diffusion system we reduce the system by replacing b with its spatial average shadow system and assume an abrupt on and off switch for the genes in doing so explicit formula are obtained for all steady state solutions and their linear eigenvalues using the diffusion of a d a and the basal production rate r as bifurcation parameters we explore ranges in which a monotone stationary front is stable and show it can lose stability through a hopf bifurcation giving rise to oscillatory fronts we also discuss the existence and stability of steady state and time oscillatory solutions with multiple extrema an intuitive explanation for the occurrence of stable stationary and oscillatory front solutions is provided based on the behavior of a in the absence of b and the opposite regulation between a and b such behavior is also interpreted in terms of the biological parameters in the model including those governing the connection of the gene network

reduced order observers a new algorithm and proof this paper gives a new recursive algorithm to construct a reduced order observer with prescribed spectrum of a given observable system the method is based on the staircase form and implicitly gives a new proof for the existence of such reduced order observers

traveling waves for the fitzhugh nagumo system on an infinite channel abstract we are concerned with the traveling wave solutions for the fitzhugh nagumo system on an infinite channel based on a variational formulation in which a non local term depends on a parameter the speed of a traveling wave can be selected out furthermore to show the existence of a traveling wave solution with such a speed we seek a minimizer subject to a constraint in the way of solving the variational problem we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region

the continuous coagulation fragmentation equations with diffusion existence of global weak solutions to the continuous coagulation fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition our approach relies on weak and strong compactness methods in l 1 in the spirit of the diperna lions theory for the boltzmann equation under the detailed balance condition the large time behaviour is also studied

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

effects of cross diffusion on turing bifurcations in two species reaction transport systems we study the turing bifurcation in general two species reaction transport systems where particle dispersal is governed by diffusion including cross diffusion by performing a linear stability analysis we find conditions for the turing instability and compare the results with the standard turing conditions we apply our results to two model systems the lengyel epstein model and the brusselator and find strong effects of cross diffusion on the turing bifurcation

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

global bifurcation phenomena of traveling wave solutions for some bistable reaction diffusion systems on considere le systeme bistable e u t e 2 u xx f u v v t v xx g u v ou e est petit on etudie les solutions onde de propagation qui tendent vers deux etats stables quand x

coagulation fragmentation processes we study the well posedness of coagulation fragmentation models with diffusion for large systems of particles the continuous and the discrete case are considered simultaneously in the discrete situation we are concerned with a countable system of coupled reaction diffusion equations whereas the continuous case amounts to an uncountable system of such equations these problems can be handled by interpreting them as abstract vector valued parabolic evolution equations where the dependent variables take values in infinite dimensional banach spaces given suitable assumptions we prove existence and uniqueness in the class of volume preserving solutions we also derive sufficient conditions for global existence

exponential decay towards equilibrium for the inhomogeneous aizenman bak model the aizenman bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non degenerate size dependent coefficient both the break up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels we demonstrate that the entropy entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non degenerate diffusion in dimension one we finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial up to our knowledge this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation fragmentation models

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

front propagation and segregation in a reaction diffusion model with cross diffusion abstract a study of front propagation and segregation in a system of reaction diffusion equations with cross diffusion is presented the reaction models predator prey dynamics involving two fields the diffusive part is nonlinear in the sense that the diffusion coefficient instead of being a constant as in the well studied case depends on one of the fields a key element of the model is a cross diffusion term according to which the flux of one of the fields is driven by gradients of the other field the original motivation of the model was the study of the turbulence shear flow interaction in plasmas the model also bears some similarities with models used in the study of spatial segregation of interacting biological species the system has three nontrivial fixed points and a study of traveling fronts solutions joining these states is presented depending on the stability properties of the fixed points the fronts are uniform or have spatial structure in the latter case a cross diffusion driven pattern forming k 0 instability leads to segregation in the wake of the front the segregated state consists of layered structures a ginzburg landau amplitude equation is used to describe the dynamics near marginal stability

implementing linear algebra algorithms for dense matrices on a vector pipeline machine this paper examines common implementations of linear algebra algorithms such as matrix vector multiplication matrix matrix multiplication and the solution of linear equations the different versions are examined for efficiency on a computer architecture which uses vector processing and has pipelined instruction execution by using the advanced architectural features of such machines one can usually achieve maximum performance and tremendous improvements in terms of execution speed can be seen over conventional computers

wavy front dynamics in a three component reaction diffusion system with one activator and two inhibitors an analytic description for traveling waves in a one dimensional reaction diffusion system with one activator and two inhibitors and with equal diffusion constants is developed using a piecewise linear approximation for the nonlinear activator reaction term the case of front waves is examined in more detail the monotonic and oscillating fronts being separately considered the corresponding wave profiles are constructed and the speed equation is obtained and discussed it is found that the fronts in the three component model propagate faster than the fronts in the two component system the front interaction is studied using numerical calculations the results show that at head on collisions two oscillating fronts produce a wavy domain which spreads in space with time

segregation and pursuit waves in activator inhibitor systems we investigate the effects of cross diffusion on propagating waves in an activator inhibitor system the model consists of a piecewise linear approximation of fitzhugh nagumo kinetics and a cross diffusion term for either the activator or the inhibitor we obtain exact analytic solutions for traveling fronts and solitary pulses and discuss the corresponding speed diagrams a detailed comparison with the corresponding rinzel keller model for the usually studied case of self diffusion is performed

front bifurcation in a tristable reaction diffusion system under periodic forcing a piecewise linear tristable reaction diffusion equation under external forcing of periodic type is considered a special feature of the forcing is that the force moves together with the traveling wave front velocity equations are obtained analytically using matching procedures for the front solutions it is noted that there is a restriction in building of null cline for each choice of outer branches of null cline the middle interfacial zone should not exceed some critical value when this zone is larger the front does not exist it is found that in the presence of forcing there exists a set of front solutions with different phases matching point coordinates the periodic forcing produces a change in the velocity versus phase diagram for a specific choice of wave number there is a bubble formation which corresponds to additional solutions when the velocity bifurcates to form three fronts

an exactly solvable model of subdiffusion reaction front propagation the propagation of a reaction front is considered in the framework of the seki lindenberg reaction subdiffusion model which is appropriate in the case of a diffusion limited reaction typically this kind of problem is solved using approximate approaches or numerical methods we apply a model piecewise linear reaction function which allows for exact analytical solutions for a front between two stable homogeneous states a unique value of the front velocity is found in the case of fronts between a stable and an unstable state a family of traveling wave solutions is revealed the conditions for the existence of a special pushed front are considered the influence of advection on the front velocity is also analyzed

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space