a cellular automaton model of excitable media iii fitting the belousov zhabotinskiic reaction abstract we obtain a set of parameter values for our cellular automaton model that gives a reasonably good description of the belousov zhabotinskii reaction as determined by comparison with experimental results and with calculations on the oregonator a differential equation model of the chemical system

traveling wave solutions in a diffusive system with two preys and one predator abstract this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator by constructing upper and lower solutions the existence of nontrivial traveling wave solutions is established the asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles applying the theory of asymptotic spreading the nonexistence of traveling wave solutions is proved

classification of steady solutions of the full kinematic model abstract the geometric type of kinematic model governs any moving plane curve which propagates in the normal direction in the medium there have appeared some results in the literature via this model however these results are all based upon some simplification on the model or prior phenomenological assumptions on the solutions in this paper we use really full kinematic model to classify steady solutions i e curvature is independent of time with positive or negative curvature various type of solutions have been obtained such as rotating and translating waves especially for the rotating spiral waves spiral waves arise in many biological chemical and physiological systems the kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane in fact using our results theorem 3 8 we can answer the following questions is there any steadily rotating wave for a given weakly excitable medium if yes what kind of information we can know about these rotating waves e g the tip s curvature the tip s tangential velocity and the rotating frequency comparing our results with previous ones in the literature there are some differences between them the only evolving plane curves with positive or negative curvature must have monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

traveling wave solutions for gause type predator prey systems with density dependence a heteroclinic orbit in r4 abstract this paper investigates the existence of traveling wave solutions for a class of diffusive gause type predator prey systems with linear density dependence for predator using the shooting method we establish the existence of a non negative traveling wave solution connecting a boundary equilibrium point to the coexistence equilibrium point with the help of topological method of wazewski theorem and lasalle s invariance principle

a cellular automaton model of excitable media ii curvature dispersion rotating waves and meandering waves abstract a cellular automaton model of excitable media recently introduced by the authors is examined in more detail here we show that i the model is consistent with a linear dependence of wave speed on wave front curvature in two dimensional space ii the model exhibits a stepwise increasing relation between wave speed and temporal period the dispersion relation whose precise shape can be controlled by parameter settings iii the model generates rigidly rotating spiral waves whose frequency speed and core size are consistent with theoretical expectations and iv spiral waves rotate non rigidly showing short term flowerlet meandering which seems to arise by bifurcation from the rigidly rotating solution and long term irregular wandering over large distances these observations are consistent with many known features of wave propagation in two dimensional continuous excitable media governed by reaction diffusion equations suggesting that our cellular automaton model is not only a fast but also a reliable method for investigating the behavior of excitable media

wavefront solutions of degenerate quasilinear reaction diffusion systems with mixed quasi monotonicity abstract this paper is concerned with the existence of wavefront solutions to a system of degenerate quasilinear reaction diffusion equations of mixed quasi monotone properties in the form u i t d i u i u i f i u x t 0 for i 1 n the important features of this system are that some of the diffusion coefficients d i u i are density dependent and may vanish at certain value of u i and that each function f i is quasi monotone increasing for some components of u u 1 u n and decreasing for other components of u such systems model reaction diffusion processes with density driven diffusion mechanism under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions a predator prey model with nonlinear diffusion is used as an illustration of application the presence of wavefront solutions flowing toward the coexistence states is established by constructing appropriate upper and lower solutions

spiral selection as a free boundary problem abstract we present a new formulation of the spiral selection problem for the belousov zhabotinsky reaction in particular we focus on deriving an exact integro differential shape equation and discuss the possible behavior of solutions to this equation we also present some new results on the asymptotic far from core structure of spirals in the fife scaling regime

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

a geometric approach in the study of traveling waves for some classes of non monotone reaction diffusion systems abstract in this paper we further extend a recently developed method to investigate the existence of traveling waves solutions and their minimum wave speed for non monotone reaction diffusion systems our approach consists of two steps first we develop a geometrical shooting argument with the aid of the theorem of homotopy invariance on the fundamental group to obtain the positive semi traveling wave solutions for a large class of reaction diffusion systems including the models of predator prey interaction for both predator independent dependent functional responses the models of combustion belousov zhabotinskii reaction si type of disease transmission and the model of biological flow reactor in chemostat next we apply the results obtained from the first step to some models such as the beddinton deangelis model and the model of biolocal flow reactor to show the convergence of these semi traveling wave solutions to an interior equilibrium point by the construction of a lyapunov type function or the convergence of semi traveling waves to another boundary equilibrium point by the further analysis of the asymptotical behavior of semi traveling wave solutions

on global leader following consensus of identical linear dynamic systems subject to actuator saturation abstract this paper studies the leader following consensus problem for a group of agents with identical linear systems subject to control input saturation we focus on two classes of linear systems neutrally stable systems and double integrator systems for neurally stable systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with proper choices of relative potential functions global consensus can also be achieved over a switching communication topology for double integrator systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with the help of a simple saturation function in the local feedback laws global consensus can also be achieved over a switching communication topology simulation results illustrate the theoretical results

on stability and uniqueness of stationary one dimensional patterns in the belousov zhabotinsky reaction in this paper a numerical solution technique is presented for obtaining time independent spatial patterns of chemical concentration in the belousov zhabotinsky reaction the oregonator model of the reaction is assumed and the concentrations of each of the three principal intermediate species are expressed as fourier series in the spatial variable and the coefficients are then obtained by a straightforward galerkin technique results of extensive numerical calculation are discussed and it is demonstrated that in many instances the spatial pattern obtained is not unique in the sense that at least two different patterns could be produced at the same values of the chemical rate constants and diffusion coefficients the problem of whether the steady patterns are stable in time to small perturbations is discussed in detail a criterion for stability of patterns of infinitesimal amplitude is established and it is shown that when all the diffusion coefficients are equal there can be no stable small amplitude solutions however in the non linear case large amplitude quasi stable patterns have been discovered and these are discussed

a predator prey model with ivlev s functional response abstract the uniqueness of limit cycles is proved for a two dimensional predator prey system with a functional response of ivlev type the system of planar autonomous ode s is transformed to a lienard system to which a modified theorem of zhang is applied

effect of harvesting delay and diffusion in a generalist predator prey model in compared with specialist predators which feed almost exclusively on a specific species of prey generalist predators feed on many types of species consequently their dynamics is not coupled to the dynamics of a specific prey population and the generalist predators has itself growth function which be extended a well known logistic growth term we develop a generalist predator prey model with diffusion and study the effect of harvesting and delay under neumann conditions the stability of the equilibria is firstly investigated and the existence of traveling wave solutions is then established by constructing a pair of upper lower solutions and using the cross iteration method and schauder s fixed point theorem

spirals in excitable media ii meandering transition in the diffusive free boundary limit abstract in this paper we present a numerical stability calculation for steadily rotating spirals in an excitable medium while experiments as well as numerical simulations of two field reaction diffusion models have shown the existence of a hopf bifurcation from steady rotations to a meandering state all the analytical approaches so far have failed to predict this transition this mismatch between analysis and simulations raises the question whether meandering critically depends on the finite diffusivity of the interface separating between the excited and the refractory phases our calculations show that this is not the case the meandering transition takes place even in the limit of an infinitely sharp interface the boundaries of the meandering transition as function of the model parameters are traced we discuss possible explanations for the failure of previous analytical approaches

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

invasion traveling wave solutions of a predator prey system abstract this paper is concerned with the traveling wave solutions of a predator prey system in which intra specific conflictions are involved by using the comparison principle and the asymptotic speed of spreading the existence nonexistence and minimal wave speed of non negative traveling wave solutions are established such a traveling wave solution can model the spatial temporal process where the predator invades the territory of the prey and they eventually coexist

asymptotic spreading in a lotka volterra predator prey system abstract this paper is concerned with the asymptotic spreading of a lotka volterra predator prey system with self limit effect in which the intrinsic growth rates of both species are positive the spreading speeds of both species are estimated which implies the role of nonlinearities in asymptotic spreading biologically our results confirm that the prey may contribute to the asymptotic spreading of the predator

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

curvature dependence of a model for calcium wave propagation abstract the propagation of a two dimensional wave front in an excitable medium is dependent on the curvature of the front current theories of excitable reaction diffusion models predict that when reaction is much faster than diffusion the normal wave speed n is approximately related to the curvature of the wave front the plane wave speed c and the diffusion coefficient of the propagator variable d by the eikonal equation n c d we show that a simple model for intracellular calcium ca2 wave propagation does not obey the eikonal equation and postulate an alternative curvature equation that is dependent on the parameter values used in the model this new curvature relation is confirmed by numerical simulations we raise the possibility that different models for ca2 wave propagation will have qualitatively different spiral wave patterns providing a new way of distinguishing between proposed models the theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of ca2

minimal wave speed for a class of non cooperative diffusion reaction system abstract in this paper we consider a class of non cooperative diffusion reaction systems which include prey predator models and disease transmission models the concept of weak traveling wave solutions is proposed the necessary and sufficient conditions for the existence of such solutions are obtained by the schauder s fixed point theorem and persistence theory the introduction of persistence theory is very technical and crucial the lasalle s invariance principle is applied to show that traveling wave solutions connect two equilibria the nonexistence of traveling wave solutions is proved by introducing a negative one sided laplace transform the results are applied to a prey predator model and a disease transmission model with specific interaction functions we find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition which is a new phenomenon

two parameter bifurcation in a predator prey system of ivlev type abstract our concern is to consider an ecological system with ivlev s functional response 1 e axof predator to prey in addition toa this system contains parametersrandd which represent the intrinsic rate of increase for the prey population and the death rate for the predator population respectively a necessary and sufficient condition for the uniqueness of limit cycles of the predator prey system is presented this result gives the bifurcation curve in the a d plane

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

second order leader following consensus of nonlinear multi agent systems via pinning control abstract without assuming that the interaction diagraph is strongly connected or contains a directed spanning tree this paper studies the second order leader following consensus problem of nonlinear multi agent systems with general network topologies based on graph theory matrix theory and lasalle s invariance principle a pinning control algorithm is proposed to achieve leader following consensus in a network of agents with nonlinear second order dynamics furthermore a pinning consensus protocol is developed for coupled double integrators with a constant reference velocity in particular this paper addresses what kind of agents and how many agents should be pinned and establishes some sufficient conditions to guarantee that all agents asymptotically follow the virtual leader numerical simulations are given to verify the theoretical analysis

traveling wave solutions for diffusive predator prey type systems with nonlinear density dependence abstract traveling wave solution for a class of diffusive predator prey system with nonlinear density dependence is considered using methods of topological shooting we show the existence of a non negative traveling wave solution connecting a boundary equilibrium to the co existence steady state with the help of a wazewski like set together with lyapunov function constructed elaborately this means that the traveling wave solution established by huang 2012 can be preserved in the presence of the nonlinear density dependence for the predator and the results of huang 2012 are generalized

analysis of consensus protocols with bounded measurement errors this paper analyzes two classes of consensus algorithms in the presence of bounded measurement errors the considered protocols adopt an updating rule based either on constant or vanishing weights under the assumption of bounded error the consensus problem is cast in a set membership framework and the agreement of the team is studied by analyzing the evolution of the feasible state set bounds on the asymptotic difference between the states of the agents are explicitly derived in terms of the bounds on the measurement noise and the values of the weight matrix

consensus conditions of multi agent systems with time varying topologies and stochastic communication noises this paper investigates the average consensus problem of first order discrete time multi agent networks in uncertain communication environments each agent can only use its own and neighbors information to design its control input to attenuate the communication noises a distributed stochastic approximation type protocol is used by using probability limit theory and algebraic graph theory consensus conditions for this kind of protocols are obtained a for the case of fixed topologies a necessary and sufficient condition for mean square average consensus is given which is also sufficient for almost sure consensus b for the case of time varying topologies sufficient conditions for mean square average consensus and almost sure consensus are given respectively especially if the network switches between jointly containing spanning tree instantaneously balanced graphs then the designed protocol can guarantee that each individual state converges both almost surely and in mean square to a common random variable whose expectation is right the average of the initial states of the whole system and whose variance describes the static maximum mean square error between each individual state and the average of the initial states of the whole system

minimal wave speed for a class of non cooperative reaction diffusion systems of three equations abstract in this paper we study the traveling wave solutions and minimal wave speed for a class of non cooperative reaction diffusion systems consisting of three equations based on the eigenvalues a pair of upper lower solutions connecting only the invasion free equilibrium are constructed and the schauder s fixed point theorem is applied to show the existence of traveling semi fronts for an auxiliary system then the existence of traveling semi fronts of original system is obtained by limit arguments the traveling semi fronts are proved to connect another equilibrium if natural birth and death rates are not considered and to be persistent if these rates are incorporated then non existence of bounded traveling semi fronts is obtained by two sided laplace transform then the above results are applied to some disease transmission models and a predator prey model

finite time observers for multi agent systems without velocity measurements and with input saturations abstract this paper is devoted to the distributed finite time observers for multi agent systems where the control inputs are required to be bounded and the velocities are assumed to be not available for feedback an effective framework through defining a class of coordinated saturation functions is introduced under which both a first order finite time observer and a high order finite time observer are constructed by applying the homogeneous theory for stability analysis it is proven that all the states of the followers can converge to that of the leader in finite time under our proposed observers with mild modifications of our control strategies the foregoing results are then extended to the distributed finite time containment control problem where the states of the followers converge to the convex hull spanned by the multiple dynamic leaders numerical simulations are presented to demonstrate the efficiency of our methods

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

a geometrical approach to wave type solutions of excitable reaction diffusion systems we formulate the eikonal equation approximation for travelling waves in excitable reaction diffusion systems which have been proposed as models for a large number of biomedical situations this formulation is particularly suited in a natural way to numerical solution by finite difference methods we show how this solution is independent of the parametric variable used for expressing the eikonal equation and how a reduction of dimensionality implies a major saving over the time taken to solve the original reaction diffusion system neumann boundary conditions on reactants in the original system translate into a geometric constraint on the wave boundary itself we show how this leads to geometrically stable stationary wave boundaries in appropriately shaped non convex domains this analytical prediction is verified by numerical solution of the eikonal equation on a domain which supports geometrically stable stationary wave boundary configurations we show how the concepts of geometrical stability and wave front stability relate to a problem where a bi stable reaction diffusion system has a stable stationary wave front configuration

traveling waves for a generalized holling tanner predator prey model abstract we study traveling wave solutions for holling tanner type predator prey models where the predator equation has a singularity at zero prey population the traveling wave solutions here connect the prey only equilibrium 1 0 with the unique constant coexistence equilibrium u v first we give a sharp existence result on weak traveling wave solutions for a rather general class of predator prey systems with minimal speed explicitly determined such a weak traveling wave u v connects 1 0 at but needs not connect u v at next we modify the holling tanner model to remove its singularity and apply the general result to obtain a weak traveling wave solution for the modified model and show that the prey component in this weak traveling wave solution has a positive lower bound and thus is a weak traveling wave solution of the original model these results for weak traveling wave solutions hold under rather general conditions then we use two methods a squeeze method and a lyapunov function method to prove that under additional conditions the weak traveling wave solutions are actually traveling wave solutions namely they converge to the coexistence equilibrium as

existence of traveling wave solutions for gause type models of predator prey systems this paper deals with the existence of three types of traveling waves for a general predator prey systems of gause type traveling wave train solution point to point and point to periodic traveling wave solutions applying the methods of wazewski theorem lasalle s invariance principle and hopf bifurcation theorem we obtain the existence results also the minimal wave speed for biological invasion is obtained furthermore some applications are given to illustrate our results

hyperbolic reaction diffusion equations and chemical oscillations in the brusselator abstract hyperbolic reaction diffusion equations are discussed with the example of the brusselator in this paper their behavior is compared with that of the parabolic reaction diffusion equations conventionally used in connection with oscillating chemical reactions and chemical waves it is shown by using the example of the brusselator that an approximate dispersion relation computed from the hyperbolic reaction diffusion equations has a qualitatively correct wave number dependence whereas the parabolic equations do not yield a dispersion relation when a similar method is used it is also shown that chaotic solutions can arise in the case of hyperbolic reaction diffusion equations linear stability characteristics can be quite different in the two cases of evolution equations

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

shaping wave patterns in reaction diffusion systems we present a method to control the two dimensional shape of traveling wave solutions to reaction diffusion systems such as interfaces and excitation pulses control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation while the control enforces a desired wave shape perpendicular to the local propagation direction the wave profile along the propagation direction itself remains almost unaffected provided that the one dimensional wave profile of all state variables and its propagation velocity can be measured experimentally and the diffusion coefficients of the reacting species are given the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown

global consensus for discrete time multi agent systems with input saturation constraints in this paper we consider the global consensus problem for discrete time multi agent systems with input saturation constraints under fixed undirected topologies we first give necessary conditions for achieving global consensus via a distributed protocol based on relative state measurements of the agent itself and its neighboring agents we then focus on two special cases where the agent model is either neutrally stable or a double integrator for the neutrally stable case any linear protocol of a particular form which solves the consensus problem for the case without input saturation constraints also solves the global consensus problem for the case with input saturation constraints for the double integrator case we show that a subset of linear protocols which solve the consensus problem for the case without saturation constraints also solve the global consensus problem for the case with input saturation constraints the results are illustrated by numerical simulations

effect of diffusion on patterns in excitable belousov zhabotinskii systems travelling wave patterns occur frequently in chemically reacting systems these include planar fronts target patterns and spiral structures we review the dispersion relation for planar waves including the effects of diffusion in the slow field for a simplified piecewise linear model with a focus on the scaling behavior with respect to the relative rate of the fast and slow reactions we discuss the implications of these results for spirals showing the origins of fife scaling and deriving a boundary integral formulation of the spiral equations we discuss the generalization to more realistic models in particular the popular oregonator model for belousov zhabotinskii reactions

symmetric spirals in media with relaxation kinetics and two diffusing species abstract we find an approximate solution for the shape and rotation frequency of spirals in excitable media with anti symmetric kinetics and diffusion in both species using singular perturbation theory we approximate the governing diffusion reaction system with a free boundary problem and then map this free boundary problem onto a fixed domain problem the resulting fixed domain problem is viewed as a nonlinear eigenvalue problem which can be solved iteratively using a numerical shooting algorithm solutions of the approximate free boundary problem are compared with numerical solutions of the full system of diffusion reaction equations and shown to be in good agreement

a free boundary problem arising in some reacting diffusing system we prove the well posedness for a one dimensional free boundary problem arising from some reaction diffusion system the interfacial point hits a boundary point in finite time or remains inside for all time in the large diffusion limit the system is reduced to ordinary differential equations of finite dimension

dynamics of wave fronts in excitable media abstract the dynamics of wave fronts in two dimensional excitable media is analytically studied by means of the asymptotic perturbation method for wave fronts in reaction diffusion systems we derive a space derivative burgers equation which governs the shape of boundary layers and their propagation application of lie group analysis of differential equations yields group invariant solutions of this equation and their classification some classes of the group invariant solutions determine the remarkable patterns expanding circle waves rotating spiral waves initial value problems for the burgers equation are also studied and a spiral wave is shown to be asymptotically realized under some proper boundary conditions

travelling waves and static structures in a two dimensional exactly solvable reaction diffusion system exact travelling wave solutions of a fisher type reaction diffusion equation in two spatial dimensions are obtained the waves have nontrivial geometries and possess velocities smaller and larger than that of a plane wave the dispersion relationship and velocity curvature dependence generated by these solutions are characterized static structures with and without circular symmetry are constructed

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

brief paper robust consensus of multi agent systems with diverse input delays and asymmetric interconnection perturbations the consensus problem of second order multi agent systems with diverse input delays is investigated based on the frequency domain analysis decentralized consensus conditions are obtained for the multi agent system with symmetric coupling weights then the robustness of the symmetric system with asymmetric perturbation is studied a bound of the largest singular value of the perturbation matrix is obtained as the robust consensus condition simulation examples illustrate the design procedure of consensus protocols and validate the correctness of the results

finite time convergent gradient flows with applications to network consensus this paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function extending recent results on nonsmooth stability analysis we characterize their asymptotic convergence properties and identify conditions that guarantee finite time convergence we discuss the application of the results to consensus problems in multi agent systems and show how the proposed nonsmooth gradient flows achieve consensus in finite time

the existence of spiral waves in an oscillatory reaction diffusion system rotating waves are proven to exist on the unit disk for an oscillatory reaction diffusion equation with neuman boundary conditions the method of proof relies on a two parameter shooting argument for the ensemble frequency and the radial derivative of the magnitude numerical solutions indicate that the waves are stable if the diffusion is sufficiently small it is also shown that these solutions cease to exist for large diffusion the origin of the rotating waves is discussed key words reaction diffusion equations spiral waves oscillatory media

dynamical behavior in linearly coupled oregonators abstract tyson ann ny acad sci 316 1979 279 conjectured that the stable homogeneous positive steady state may coexist with stable echo wave meaning anti phase wave in linearly coupled oregonators and thus gave a conjecture on the bifurcation diagram of this system in this paper we rigorously prove stability of the in phase wave and existence of the anti phase wave our proof procedure actually gives a general method or line to deal with the analogous problem for instance to prove stability of the in phase wave following our line one may decompose the corresponding variational equations a four dimensional system into two independent planar systems also for instance existence of the anti phase wave can be concluded as existence and uniqueness of limit cycle of the associate oscillator in addition according to parameter regimes of existence of the anti phase wave and the stable homogeneous positive steady state we give their coexistence regime and specify it and in particular give the regime of the coupled coefficient the specified results show that the theoretical results are in good accord with tyson s numerical results

existence of travelling wave solutions of predator prey systems via the connection index we consider a reaction diffusion system which arises as a model of predator prey interactions in mathematical ecology the system admits four constant equilibria two of which are stable the main result is an existence theorem for travelling wave solutions which connect the two stable equilibria most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit the new aspect of the present discussion lies in devising some novel methods for locating the connecting solution in phase space which do not require that the equations be singularly perturbed in certain cases the main result holds for an arbitrary choice of diffusion coefficients

the core of the spiral the problem of finding the unique frequency and shape including the core of spirals in excitable media is studied a leading order approximation of the problem is formulated as a nonlinear eigenvalue problem involving only ordinary differential equations odes shooting and comparison arguments are used to show why solutions of the leading order approximate problem exist and some of their quantitative features are explored numerically because of inaccuracies of the results in certain parameter ranges an improved formulation of the problem is proposed when solved numerically as in the special case of symmetric spirals solutions of the improved formulation differ significantly from solutions of the leading order problem but agree closely with the numerical solution of the full partial differential equation pde system

diffusion and wave propagation in cellular automaton models of excitable media in this paper the authors 1 examine general masks as discrete approximations to the diffusion equation showing how to calculate the diffusion coefficient from the elements of the mask 2 combine the mask with a thresholding operation to simulate the propagation of waves in excitable media showing that for well chosen masks the waves obey a linear speed curvature relation with slope given by the predicted diffusion coefficient and 3 assess the utility of different masks in terms of computational efficiency and adherence to a linear speed curvature relation

the effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex ginzburg landau equation abstract in this paper we consider an oscillatory medium whose dynamics are modeled by the complex ginzburg landau equation in particular we focus on n armed spiral wave solutions of the complex ginzburg landau equation in a disk of radius d with homogeneous neumann boundary conditions it is well known that such solutions exist for small enough values of the twist parameter q and large enough values of d we investigate the effect of boundaries on the rotational frequency of the spirals which is an unknown of the problem uniquely determined by the parameters d and q we show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak we use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q

a heuristic description of spiral wave instability in discrete media abstract a set of simple low dimensional differential equations representing a chain of coupled oscillators is suggested as a model for explaining the instability of a rigidly rotating spiral wave that has been observed in spatially discrete and continuous models bifurcation to an invariant circle on a torus appears to be the mechanism responsible for this instability in the discrete case and is thus in agreement with barkley s assertion of an instability due to a hopf bifurcation

existence of traveling wave solutions for diffusive predator prey type systems abstract in this work we investigate the existence of traveling wave solutions for a class of diffusive predator prey type systems whose each nonlinear term can be separated as a product of suitable smooth functions satisfying some monotonic conditions the profile equations for the above system can be reduced as a four dimensional ode system and the traveling wave solutions which connect two different equilibria or the small amplitude traveling wave train solutions are equivalent to the heteroclinic orbits or small amplitude periodic solutions of the reduced system applying the methods of wazewski theorem lasalle s invariance principle and hopf bifurcation theory we obtain the existence results our results can apply to various kinds of ecological models

the eikonal equation stability of reaction diffusion waves on a sphere abstract the eikonal approach to excitable reaction diffusion systems provides an analytical hold on the phenomenon of wave propagation on curved surfaces and the evolution of toroidal scroll structures in the neighbourhood of organising filaments the extent to which the eikonal equation contains information on the stability of these wave solutions is important from a theoretical and experimental point of view we initiated this study in 1987 by demonstrating the stability of an expanding spherical wave recent experiments on wave propagation on a sphere encouraged us further to investigate stability of waves on surfaces we analyse in this paper the geometrical stability of symmetric counter rotating spiral waves propagating on the unit sphere we demonstrate that these solutions are stable under small perturbations normal to the wave fron lying on the unit sphere

periodic traveling wave train and point to periodic traveling wave for a diffusive predator prey system with ivlev type functional response abstract a diffusive predator prey system with ivlev type scheme is investigated in this article the existences of a small amplitude periodic traveling wave train p and the traveling wave solution connecting the boundary equilibrium e u 1 0 to the periodic traveling wave p are obtained the existence of this point to periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain and thus plays a mild role in the evolution of predator prey communities the techniques used here are the hopf bifurcation theorem the improved shooting method combining with the geometric singular perturbation method

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

alternative stable rotors in an excitable medium abstract a given excitable medium continuous uniform isotropic simply connected two or three dimensional may support alternative types of vortex like solution radiating from a rotor these have distinct periods wavelengths behaviors near interfaces and in rotationally symmetric vortex rings different contraction or expansion rates and drift rates initial conditions select among the discrete alternatives rotors of different types may coexist within one medium until the interface between their domains encroaches upon the rotor of longer period examples are given using the fitzhugh nagumo model they are interpreted in terms of the measured dispersion curve for one dimensional propagation and a hypothetical curvature relation with one undetermined parameter