localized patterns for the quintic complex swift hohenberg equation abstract we show using numerical simulations that a variety of localized patterns arise in a model equation the quintic swift hohenberg equation with complex coefficients we demonstrate that various sizes of localized standing wave patterns are possible when the imaginary part of the complex coefficient is small localized traveling waves as well as localized standing waves with a fixed size are observed when the imaginary part is rather large we also present stable localized patterns in two spatial dimensions and study their interaction

the eikonal equation some results applicable to computer vision in this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape particular emphasis is given to eikonal equations two results are presented first we prove that a special type of eikonal equation has only one convex and positive c2 solution in some neighborhood of a singular point using this result we show that a restricted form of this equation has exactly two solutions these results have application in scanning electron microscopy

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

noise reduction in surface reconstruction from a given gradient field we present a gradient space technique for noise reduction in surfaces reconstructed from a noisy gradient field we first analyze the error sources in the recovered gradient field of a surface using a three image photometric stereo method based on this analysis we propose an additive noise model to describe the errors in the surface gradient estimates we then use a vector space formulation and construct a multiscale orthonormal expansion for gradient fields using the sparse representation properties of this expansion we develop techniques for reducing the gradient field noise by coefficient selection with thresholding the simulation results indicate that the proposed technique provides significant improvement on the noise levels of both the estimated gradient fields and the reconstructed surfaces under heavy noise levels furthermore the experiments using noisy photometric stereo image triplets of real range data suggest that the additive model remains viable after the nonlinear photometric stereo operation to provide accurate noise removal

reflection and attachment of spirals at obstacles for the fitzhugh nagumo and beeler reuter models in this paper the fitzhugh nagumo fhn equations and a modified fhn mfhn are considered for the modified version the recovery variable v has three different time scales by considering different parameters in the local dynamics of the mfhn equations it is observed that the phenomenon of reflection and annihilation at an impermeable boundary is observed just as in the beeler reuter model the interaction of spirals obtained with the fhn mfhn and beeler reuter model and an obstacle is also considered the phenomenon of reflection of the spiral wave at a boundary changes when the boundary becomes an obstacle four properties for attachment of a spiral wave to an obstacle are presented in this work

discrete groups and holomorphic functions we determine conditions under which the quotient of a kahler hadamard manifold by a torsion free discrete group of isometries admits a big set of nontrivial holomorphic functions we also generalize a theorem of g a margulis on centerless of cocompact discrete groups of automorphisms of a caratheodory hyperbolic manifold by a different approach based on the caratheodory pseudodistance

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

bifurcation characteristics and spatial patterns in an integro differential equation abstract spatial patterns are studied in reaction diffusion systems where chemical reactions are coupled with diffusion an integro differential equation is obtained when the inhibitor has a much larger reaction rate than the activator the periodic spatial pattern near the bifurcation point c is described with a time dependent ginzburg landau equation derived using a perturbative method which can be applied to the situation where the amplitude of the patterned solution develops gradually and also where it appears discontinuously at c furthermore a new type of potential is proposed to explain comprehensively the mechanisms of the bifurcation and the localized spatial pattern appearing very far from equilibrium since the potential is constructed from two internal variables the bifurcation can be understood from the analogy with the first order phase transition in equilibrium systems and at the same time the appearance of patterns is shown to be analogous with the phase separation the theoretical results are compared with numerical calculations

drift bifurcation of dissipative solitons due to a change of shape experiment and theory abstract the subject of this paper is self organized solitary current filaments being observed experimentally in a planar dc gas discharge system with high ohmic barrier we report on the dynamical behavior of these objects which we refer to as dissipative solitons an application of stochastic data analysis methods enables the separation of intrinsic dynamics from stochastic contributions using the externally applied voltage as control parameter we observe a drift bifurcation that is accompanied by a change of shape of the bifurcating dissipative soliton in order to interpret the experimental finding a theoretical model in form of a three component reaction diffusion system is considered allowing for a qualitative understanding of the obtained results we discuss an analytical expression for the velocity of the dissipative soliton as a function of an appropriate control parameter and demonstrate that the change of the shape can induce the drift bifurcation this analytical finding is supported by numerical simulations of the model equations

patterns and localized structures in a hybrid non equilibrium ising model we study a hybrid non equilibrium pattern formation model combining short scale ising model with a continuous slow or long scale inhibitor the computation combines alternates monte carlo algorithm with updating of the inhibitor field it is very fast and allows us to study the influence of various factors such as scale ratios coupling strength bias temperature level of noise anisotropy etc on the pattern formation and behavior of emerging non equilibrium structures

existence and uniqueness in photometric stereo we analyze the problem of recovering the shape of a smooth lambertian surface from two images obtained by consecutive illumination of the surface by distant point light sources from two different directions we discuss the issues of existence and uniqueness for the corresponding system of two first order nonlinear partial differential equations we show that generically this system has up to a constant a unique solution we also examine exceptional cases for which there is no such uniqueness

analytic solutions for nonlinear waves in coupled reacting systems abstract we analyze a system of reacting elements harmonically coupled to nearest neighbors in the continuum limit an analytic solution is found for traveling waves the procedure is used to find oscillatory as well as solitary waves a comparison is made between exact solutions and solutions of the piecewise linearized system showing how the linearization affects the amplitude and frequency of the solutions

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

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

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

regular patterns in dichotomically driven activator inhibitor dynamics we investigate turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and fitzhugh nagumo kinetics the fluctuations vary in space and or time depending on the realization of the dichotomous switching the system is at a given time for spatial disorder at a given position in one of two possible excitable dynamical regimes each of the two excitable dynamics for itself does not support pattern formation with proper dichotomous fluctuations however the homogeneous steady state is destabilized via a turing instability we investigate the influence of different switching rates different correlation length of the spatial disorder on pattern formation we find three distinct mechanisms for slow switching existing boundaries become unstable for high rates the system exhibits effective bistability which allows for a turing instability for medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold first they produce local inhomogeneities which then grow again caused by fluctuations until the whole space is covered utilizing a nonlinear map approach we show bistability of a period one and a period two orbit being associated with the steady homogeneous and the turing pattern state respectively finally for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in turing patterns

singular filaments organize chemical waves in three dimensions iv wave taxonomy abstract this is the last of four papers on the anatomy of three dimensional organizing centers in excitable media these organizing centers are made of scroll ring waves with twists and knots in them diversely linked together each such aggregate of linked rings is characterized by a handful of topological quantum numbers which partition the family of organizing centers into discrete classes or atoms we derive the rules which govern nuclear structure in this periodic table as well as the selection rules for transmutation

on complete integrals and uniqueness in shape from shading in this paper we analyze the problem of representing solutions of a first order partial differential equation in terms of complete integrals we show that some of the existing results referring to the representability problem are incomplete additionally we analyze in the context of complete integrals the uniqueness problem for the shape recovery of a smooth lambertian surface from an image obtained by illuminating this surface by an overhead distant point light source specifically we revisit the uniqueness results already existing in the shape from shading literature that concern eikonal equations corresponding to the images of a lambertian hemisphere and a lambertian plane we show that the latter results are incomplete and indicate how to fill the gaps in the corresponding proofs

oscillations of fronts and front pairs in two and three component reaction diffusion systems abstract we investigate multi co ponent reaction diffusion systems on a finite one dimensional domain the first component is assumed both to react on a short time scale and to have a small diffusion length leading to patterns with sharp fronts to analyze stability we apply the slep method by nishiura and fujii this method is superior to standard perturbation methods we illustrate this using a two component system in the three component system in a situation where both the influence of the first component on the other components and the interaction of the latter is only weak we show that there is a unique hopf destabilization of stationary fronts when changing time constants as an example we treat a system with global coupling used e g for semi conductor devices for filaments front pairs we find two types of hopf destabilizations breathing and swinging the global coupling controls which type occurs this corresponds to recent experimental results and is confirmed via numerical calculations

two dimensional spectrophotometry of spiral wave propagation in the belousov zhabotinskii reaction i experiments and digital data representation abstract the spiral shaped wave of chemical activity propagating in a 1 mm layer of an excitable solution of the belousov zhabotinskii reaction is analysed quantitatively for a 5 min time interval in a 4 5 4 5 mm area the spatial distribution of the reaction catalyst and indicator ferroin is measured by light absorption at 490 nm with a video and computer based two dimensional spectrophotometer digital images with 10 m spatial and 256 digital units intensity resolution are acquired at a frequency of 20 per minute initially the spiral has a pitch of 1 18 mm a revolution time of 17 2 s and the wavefronts propagate outwards with a velocity of 69 m s after 5 min these quantities have changed by up to 5 three dimensional graphic procedures and image analysis reveal that the wave profiles are symmetric close to the spiral center and become highly asymmetric during their outward displacement the core of the spiral has a radius of 0 35 mm its center is a singular site diameter

some remarks on the shape from shading problem in computer vision the shape from shading problem in computer vision see horn 11 deals with the reconstruction of the shape of an object from a two dimensional image we consider an object illuminated by a point source the reflected light being recorded on a photographic plate at a fixed position the intensity of the reflected light is a function of the position of the source the position of the plate and the normal to the surface of the object if the surface is given as u u x jj this leads see horn i to a first order partial differential equation of the form

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

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

singular filaments organize chemical waves in three dimensions iii knotted waves abstract this is the third of a series of papers on the anatomy of three dimensional organizing centers in excitable media we here ask whether all self consistent waves in excitable media are topologically equivalent to the experimentally verified scroll ring whose axis lies in a plane as a test case we examine a scroll ring whose axis contains a knot it proves to be incompatible with the requirements of physical chemistry unless simultaneously twisted by an amount equal to the writhing number of its axis which is zero for planar closed curves appropriate initial conditions are suggested for experimentally creating a wave whose source is a scroll ring knotted and twisted in this way

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

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

instabilities of front patterns in reaction diffusion systems abstract recent experiments in chemical reaction diffusion systems with externally imposed concentration gradients may provide access to a host of spatio temporal pattern formation phenomena these systems tend to form steep reaction fronts in response to the external gradient we use singular perturbation techniques normal form calculations and numerical simulations to investigate the existence and the stability of such sustained fronts in one dimensional systems the theoretical predictions are found in quantitative agreement with direct simulations of the hopf bifurcation from steady to periodically oscillating front structures observed in the couette flow reactor also conditions are found under which oscillations of the spatial structure become chaotic in two dimensional systems we address the issue of realizing an experimental situation hitherto unattained a one dimensional chain of coupled oscillators at the onset of the hopf destabilization of the front structure we point out the intimate relationship between the frequency of oscillation of the homogeneous front pattern and the characteristic wavelength of the turing pattern that can develop along the front 2 d 1 2 we comment on subsequent bifurcations that may result from the nonlinear interaction between hopf and turing instabilities as the precursors to spatio temporal chaos in conclusion we emphasize the possibility of probing the transition to mediated defect turbulence in thin film gel reactors

effects of transport memory and nonlinear damping in a generalized fisher s equation memory effects in transport require for their incorporation into reaction diffusion investigations a generalization of traditional equations the well known fisher s equation which combines diffusion with a logistic nonlinearity is generalized to include memory effects and traveling wave solutions of the equation are found comparison is made with alternative generalization procedures

spherically symmetric internal layers for activator inhibitor systems i existence by a lyapunov schmidt reduction abstract reaction diffusion systems of activator inhibitor type are studied on an n dimensional ball with the homogeneous neumann boundary conditions under the condition that the activator diffuses slowly reacts rapidly and the inhibitor diffuses rapidly reacts moderately we show that the system admits a family of spherically symmetric internal transition layer equilibria the method of proof consists of rigorous asymptotic expansions and a lyapunov schmidt reduction

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

an electrothermal instability the influence of albedo boundary conditions on the stationary states of an exactly solvable model abstract in order to study the effect of partially reflective albedo boundary conditions on pattern formation and stability in reaction diffusion systems we have analyzed an exactly soluble model of an electrothermal instability the ballast resistor the present results allow a continuous interpolation between the results corresponding to neumann and dirichlet boundary conditions and they allow qualitative changes in the patterns to be followed

self organized pulse generator abstract numerical simulations of a reaction diffusion system of the bonhoeffer van der pol type will be presented the system admits a coesistence of a stable stationary solution and a stable limit cycle which leads to the generation of a self organized pulse generator pattern dynamics of the system are explored in one and two dimensional spaces

growth of domains under global constraints abstract we develop growth rates for domains in bistable systems far from equilibrium the governing equation is a bistable reaction diffusion equation including a nonlocal inhibiting term the latter one will be responsible for the existence of stable localized domains the growth of the size of different domains shows quite surprising similarities with coalescence phenomena of droplets in van der waals gases eventually we discuss the possibility of breathing of the domains induced by a time delayed nonlocal interaction

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

pattern selection in a general model of convection diffusion and catalytic reaction abstract we study the process of pattern selection in a large class of heterogeneous models of catalytic reactor by analyzing the behavior of a simple condensed model which obeys reflection and inversion symmetries and which captures the main features of a detailed model the three variable model incorporates a very fast and long ranged variable which may describe the fluid phase in a mixed plug flow or an axial dispersion reactor or the pore phase in a catalytic particle and two variables a medium ranged activator and a localized inhibitor that describe the solid phase patterns are classified according to their symmetry properties patterns may emerge already with simple bistable kinetics in a plug flow reactor i e a single integrodifferential equation pattern selection is determined by the phase planes spanned by the reactor by the ratio of front residence time to the period of oscillations and by the ratio of convection to fluid phase diffusions terms patterns include stationary or oscillatory fronts or pulses antiphase oscillations which appear at the transition from homogeneous solution unidirectional pulses that are the dominant patterns when mixing is weak and stationary or oscillatory waves that emerge with strong fluid mixing the simple form of the model allows us to suggest a general classification of emerging patterns and to identify the corresponding conditions in term of realistic models of reactors

nonequilibrium potential approach local and global stability of stationary patterns in an activator inhibitor system with fast inhibition abstract we study the formation and global stability of stationary patterns in a finite one dimensional reaction diffusion model of the activator inhibitor type the analysis proceeds through the study of the nonequilibrium potential or lyapunov functional for this system considering the fast inhibitor case and in order to obtain analytical results the adoption of a piecewise linear version of the model we have studied the changes in relative stability among the different patterns as the ratio between the diffusion coefficients is varied and have discussed the meaning of the different contributions to the nonequilibrium potential

wave front interaction in steadily rotating spirals abstract we develop a self consistent model for excitable media the so called wave front interaction model wfm in which two waves the front and the back interact with a relaxational field c 2 we study numerically the solutions corresponding to spirals rotating steadily with an angular velocity with a tip rotating around an unexcited region of radius r 0 it is found that these two quantities are determined uniquely as a function of e the ratio between fast reaction time and refractory time and the excitability corresponding shapes for spirals are determined and in particular the shape and the size of the core a semi analytical solution for the spiral shape is found in the large r 0 limit

spirals in excitable media the free boundary limit with diffusion abstract we solve numerically for the steady state spiral in the thin interface limit including the effects of diffusion of the slow field the calculation is performed using a generalization of the hybrid scheme of keener in this method the diffusion equation is solved on a suitable mapped lattice while the eikonal equation relating the field on the interface to the interfacial velocity and curvature is solved independently we present results for the selected frequency and tip radius as a function of the various parameters we note that a stability analysis based on these results may be performed

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

dynamics of dissipative structures in reaction diffusion equations the authors investigate the dynamics of dissipative structures in a reaction diffusion system they propose an analytical theory for the behavior of dissipative structures and show that a dissipative structure ds that is stable in a one dimensional homogeneous medium can be induced to drift by slow variation of the diffusion coefficients or by curvature of the ds in higher dimensions in one spatial dimension this motion can be in the direction of increasing or decreasing diffusion coefficient depending on properties of the ds which can be determined analytically in two and three dimensions this drift is proportional to the sum of the curvature of the ds and the gradient of the diffusion coefficient of the medium the analysis of this motion uses standard ideas from perturbation theory to find an equation of motion for the location of the ds numerical simulations in one and two dimensions show good quantitative agreement with the theoretical results

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

the dynamics of three dimensional scroll waves in excitable media abstract we investigate the motion of the central filament of a scroll wave in excitable medium we assume that two dimensional spiral solutions are known and use these solutions to produce slowly varying scroll waves in three dimensional space using standard ideas from perturbation theory multiscale techniques we find equations governing the motion of the central filament of the scroll and the twist of the scroll wave these equations are specialized to the case of a circular filament with uniformly distributed twist to determine the collapse or expansion and vertical drift of the spiral filament and to a helical filament with drift the predictions of these equations are in good qualitative agreement with the numerical experiments

knotted scroll wave filaments in excitable media abstract it is shown that certain general systems of diffusion reaction equations can have invariant knotted scroll wave solutions the theory is based on the assumption that the system of equations supports permanently rotating spiral solutions and from this assumptions equations governing the motion of the scroll wave filament can be derived then bifurcatilon theory is used to show that the equations of motion have torus knots as invariant solutions analytical methods are used to determine qualitative and quantitative features of these knotted solutions

robin type boundary conditions in transition from reaction diffusion equations in 3d domains to equations in 2d domains abstract we consider a singular limit of diffusion equations in 3d domains of thickness converging to zero in the 2d limit the resulting reaction diffusion equation has a source term resulting from the robin type boundary conditions imposed on boundaries of the original 3d domain the proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar cylindrical or spherical layers between two membranes as an example we refer to the problem of activation of b lymphocytes which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell for this example assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators

effects on global coupling of an interface problem in an activator inhibitor system in this paper we shall explore the dynamics of interfaces in the problem with global coupling in order to investigate the existence of time periodic solutions as the global coupling intensity and the bifurcation parameter vary

piecewise linear emulation of propagating fronts as a method for determining their speeds we use piecewise linear terms to emulate the polynomial nonlinear terms in a typical reaction diffusion equation replacing it thus with a set of simple linear inhomogeneous differential equations the resulting analytic solution constitutes an excellent approximation to the exact propagating front as is explicitly shown in the case of cubic and quintic nonlinearities yielding also the correct value for the physically selected speed of the observable front such a piecewise linear emulation can be used for any nonlinearity giving therefore a very reliable and accurate method for determining the selected speed of fronts invading unstable states especially pushed fronts

dispersion of traveling waves in the belousov zhabotinskii reaction abstract we use perturbation and numerical methods to calculate the dispersion relation for periodic wave train solutions of diffusion reaction equations modeling the belousov zhabotinskii reaction these results are compared with experimental data for the belousov zhabotinskii reagent and shown to be in good agreement

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

coupled three state oscillators abstract we investigate globally coupled stochastic three state oscillators which we consider as general models of stochastic excitable systems we compare two situations in the first case the transitions between the three states of each unit 1 2 3 1 are determined by poissonian waiting time distributions in the second case only transition 1 2 is poissonian whereas the others are deterministic with a fixed delay when coupled the second system shows coherent oscillations whereas the first remains in a stable stationary state we show that the coherent oscillations are due to a hopf bifurcation in the dynamics of the occupation probabilities of the discrete states and discuss the bifurcation diagram