computing the 1 norm of the impulse response of linear time invariant systems lower and upper bounds to the 1 norm of the impulse response of linear time invariant systems are derived it is shown that the bounds can be adjusted so that the difference between them is arbitrarily small an algorithm is described that uses the bounds to compute the 1 norm of the impulse response to a pre specified tolerance details are given of how to compute the bounds for state space systems finally two numerical examples of using the algorithm are given

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

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

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

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

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

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

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

comparison of peak and rms gains for discrete time systems abstract a convolution system can have a frequency response which is small for all frequencies yet still greatly amplify the peaks of signals passing though it for finite dimensional systems however we establish the simple bound h 1 2 n 1 h h h where h 1 is the peak gain of the system h h is the maximum frequency response of the system and n its dimension the same result for continuous time systems is due to gohberg and doyle and it is mentioned in 6 the bound implies that h optimal controllers which minimize the maximum of some disturbance to error transfer function cannot have very large peak gains from the disturbance to error

on performance criteria two theorems generalize previous work lack of performance m p is defined as the least upper bound of the absolute error as time t ranges over the real numbers r and the input ranges over a space fm this space contains all r r such that t 0 for t o and such that sup d m p if and only if the transfer function has a zero at the origin and all its poles have negative real parts some of the implications of the theorems in control theory and design are discussed

a nonequilibrium statistical model of spectrally truncated burgers hopf dynamics exact spectral truncations of the unforced inviscid burgers hopf equation are hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior for this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies in the present paper the burgers hopf dynamics truncated to n fourier modes is treated by a new statistical model reduction technique and a closed system of evolution equations for the mean values of the m lowest modes is derived for m n in the reduced model the m mode macrostates are associated with trial probability densities on the phase space of the n mode microstates and a cost functional is introduced to quantify the lack of fit of paths of these densities to the liouville equation the best fit macrodynamics is obtained by minimizing the cost functional over paths and the equations governing the closure are then derived from hamilton jacobi theory the resulting reduced equations have a fractional diffusion and modified nonlinear interactions and the explicit form of both are determined up to a single closure parameter the accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles and the predicted behavior is found to be well represented by the reduced equations 2014 wiley periodicals inc

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

kinematical theory of spiral waves in excitable media comparison with numerical simulations abstract the kinematical theory of spiral waves is compared with the data of numerical simulations using complete partial differential equations of a reaction diffusion model we find good agreement for temporal periods of spiral waves which rotate around minimal holes and in the free medium no adjustable parameters were used in the calculations

the principle of matching practical conditions for systems with inputs restricted in magnitude and rate of change the main objective of matching a principle of design is to ensure that the set of possible inputs to a system is a subset of the set of tolerable inputs those which are judged to produce an acceptable response from the system this note presents practical conditions some necessary and sufficient and others only sufficient for a matched design when the set of possible inputs is characterized by restrictions on the magnitude and the rate of change of the input and the set of tolerable inputs is characterized by magnitude bounds on the response of the system

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

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

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

non minimal factorization approach to the l gain of discrete time linear systems a method is presented to compute the gain of discrete time linear mimo systems which is based on a newly derived sequence of factorizations for g z by resorting to a chain scattering representation of the system the proposed approach offers a criterion enabling one to cope with the static output feedback 1 control design by solving a finite set of linear inequalities numerical examples are reported as well as an application to the synthesis of a performing decentralized 1 controller

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

an optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics a general method for deriving closed reduced models of hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation given a vector of resolved variables selected to describe the macroscopic state of the system a family of quasi equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities specifically a cost function is constructed to quantify the lack of fit to the microscopic dynamics of any feasible path of densities from the statistical model it is an ensemble averaged weighted squared norm of the residual that results from submitting the path of densities to the liouville equation the path that minimizes the time integral of the cost function determines the best fit evolution of the mean resolved vector the closed reduced equations satisfied by the optimal path are derived by hamilton jacobi theory when expressed in terms of the macroscopic variables these equations have the generic structure of governing equations for nonequilibrium thermodynamics in particular the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes the adjustable closure parameters in the best fit reduced equations depend explicitly on the arbitrary weights that enter into the lack of fit cost function two particular model reductions are outlined to illustrate the general method in each example the set of weights in the optimization principle contracts into a single effective closure parameter

a path integral formalism for non equilibrium hamiltonian statistical systems a path integral formalism for non equilibrium systems is proposed based on a manifold of quasi equilibrium densities a generalized boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full liouvillean evolution the likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction the lagrangian here is of modified generalized onsager machlup form for large times and long slow timescales the thermodynamics is of ottinger form the proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field it is however entirely of a wiener form and so practical to compute finally it is shown that providing certain reasonable conditions are met then there exists a unique steady state consistency distribution

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

hamiltonian structure and statistically relevant conserved quantities for the truncated burgers hopf equation in dynamical systems with intrinsic chaos many degrees of freedom and many conserved quantities a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes the galerkin truncation of the burgers hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments here it is established that the truncated burgers hopf model is a hamiltonian system with hamiltonian given by the integral of the third power this additional conserved quantity beyond the energy has been ignored in previous statistical mechanics studies of this equation thus the question arises of the statistical significance of the hamiltonian beyond that of the energy first an appropriate statistical theory is developed that includes both the energy and hamiltonian then a convergent monte carlo algorithm is developed for computing equilibrium statistical distributions the probability distribution of the hamiltonian on a microcanonical energy surface is studied through the monte carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the hamiltonian empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy the predictions of the theory for relevant and irrelevant values for the hamiltonian are confirmed through systematic numerical simulations for statistically relevant values of the hamiltonian these simulations show a surprising spectral tilt rather than equipartition of energy this spectral tilt is predicted and confirmed independently by monte carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt on the other hand the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the hamiltonian with excellent accuracy the results established here for the burgers hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications several interesting mathematical problems suggested by this study are mentioned in the final section 2002 wiley periodicals inc

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

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

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

excitation wave propagation within narrow pathways geometric configurations facilitating unidirectional block and reentry abstract the propagation of excitation waves in narrow pathways of 2d excitable media through regions of impermeable media and decreased excitability media is investigated as a model of reentrant arrhythmias in the heart arrhythmia occurrences are common in the presence of infarct scars where regions of normal tissue are interspersed with inexcitable ones propagation through narrow pathways with three idealized geometric configurations are considered propagation through narrow paths with either parallel borders tapered borders or a combination of these the focus of this study is on the conditions for unidirectional block and reentry appearance it is shown that in narrow paths with only parallel borders unidirectional block is impossible regardless of the properties of the border media in this case only bidirectional propagation or block is possible depending on the pathway width unidirectional block can occur however when the narrow path has a tapered shape waves propagating from the wide end of the pathway die out at the narrow end while waves propagating in the opposite direction are able to pass through an approximate relationship between the geometry of the pathway and wave front curvature is obtained this provides a simple approach to estimate the critical curvature value in computer simulations of various excitable media or in experiments with real media a new computer simulation approach is proposed to closely approximate the dependence of stationary speed of propagation on wave front curvature in computer simulations of a two dimensional grid of 128 128 membrane segments using the modified fitzhugh nagumo equations the arrangement of two or more pathways in parallel permits reentry if at least one pathway has a configuration causing unidirectional block and the medium at the site of unidirectional block has sufficient time to recover from the previous excitation the latter is facilitated by anisotropic conduction properties of the media especially in the case with impermeable borders thus there exist specific geometric configurations of two dimensional narrow pathways which allow reentry even when the media properties are uniform

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

rotating spiral waves in λ ω systems on circular domains abstract systems are reaction diffusion systems whose reaction kinetics admit a stable limit cycle it is known that systems can possess various types of solutions among them spiral waves are the most fascinating pattern however the effects of the diffusivity the sizes of the domains and the reaction kinetics on spiral waves are largely unknown in this paper we investigate how these quantities affect the properties of m armed spiral waves in a generalized class of system on a circular disk with no flux boundary condition first we derive a criterion for the existence of m armed spiral waves specifically we show that m armed spiral waves do not exist for d 0 r 2 j m 2 while for d 0 0 r 2 j m 2 there exists an m armed spiral wave if the twist parameter q is small here d is the diffusivity for the system r is the radius of the circular disk 0 is the value of the function a at a 0 and j m is the first positive zero of the first derivative of the bessel function of the first kind of order m we also show that the critical diffusivity d 0 r 2 j m 2 is a bifurcation point next we use the numerical simulation to show that for small twist parameter the rotational frequency increases with increasing domain size while for large twist parameter the dependence of the rotational frequency on the domain size is not monotonic moreover small circular domains may change the properties of spiral waves drastically these numerical results are in contrast to those in excitable media finally the stability of spiral waves is investigated numerically

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

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

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

on computing the worst case peak gain of linear systems based on the bounds due to doyle and boyd we present simple upper and lower bounds for the l1 norm of the tail of the impulse response of finite dimensional discrete time linear time invariant systems using these bounds we may in turn compute the l gain of these systems to any desired accuracy by combining these bounds with results due to khammash and pearson we derive upper and lower bounds for the worst case l gain of discrete time systems with diagonal perturbations

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

on computing the worst case norm of linear systems subject to inputs with magnitude bound and rate limit in this paper we propose a practical and effective approach to compute the worst case norm of finite dimensional convolution systems system inputs are modelled to have bounded magnitude and rate limit the computation of the worst case norm is formulated as a fixed terminal time optimal control problem applying pontryagin s maximum principle with the generalized karush kuhn tucker theorem we obtain necessary conditions which are subsequently exploited to characterize the worst case input furthermore we develop a novel algorithm called successive pang interval search spis to construct the worst case input for general finite dimensional convolution systems the algorithm is guaranteed to converge and give an accurate solution within a prescribed error bound to verify the accuracy of the algorithm we derive bounds on computational errors including the truncation error and the discretization error then the bounds on the errors yielded by our algorithm are compared with those of a comparative discr

a criterion of approximation for the method of inequalities a criterion of approximation is derived for replacing a plant transfer function which is either non rational or is rational but large or is not accurately known by a convenient rational approximant for the purposes of designing a control system the criterion provides simple sufficient conditions on the accuracy of approximation to ensure that designs carried out with the approximant are valid

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

a performance criterion a natural measure of performance is defined for control systems the input to the system is known to the extent that it belongs to a known function space lack of performance is then defined as the least upper bound of the absolute error that occurs over all time and all inputs the main problems of computing a value of the controller parameter that satisfies prescribed performance are considered and useful expressions are derived for such computations in particular a simple formula is derived for calculating performance the paper discusses the implications of the new measure of performance in control theory and design

theory of the spiral core in excitable media abstract a rigorous asymptotic spiral solution to an excitable reaction diffusion system is found by separating space into two scaling regions an outer region having the same spatial scaling as the overall spiral structure which exhibits a singularity at the spiral tip and a core region around the spiral tip where space is scaled so as to resolve the tip singularity the stability of the spiral structure is investigated for both the outer region which is found to possess no intrinsic instability and the core region which is found to be unstable both the zero and small diffusion systems are found to exhibit qualitatively similar behaviour the implication of these results for both experiment and simulation in particular within the context of the observed meandering instability are addressed

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

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

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

bifurcation and stability analysis of rotating chemical spirals in circular domains boundary induced meandering and stabilization recent experimental and model studies have revealed that the domain size may strongly influence the dynamics of rotating spirals in two dimensional pattern forming chemical reactions hartmann et al phys rev lett 76 1384 1996 report a frequency increase of spirals in circular domains with diameters substantially smaller than the spiral wavelength in a large domain for the catalytic no co reaction on a microstructured platinum surface accompanying simulations with a simple reaction diffusion system reproduced the behavior here we supplement these studies by a numerical bifurcation and stability analysis of rotating spirals in a simple activator inhibitor model the problem is solved in a corotating frame of reference no flux conditions are imposed at the boundary of the circular domain at large domain sizes eigenvalues and eigenvectors very close to those corresponding to infinite medium translational invariance are observed upon decrease of domain size we observe a simultaneous change in the rotation frequency and a deviation of these eigenvalues from being neutrally stable zero real part the latter phenomenon indicates that the translation symmetry of the spiral solution is appreciably broken due to the interaction with the now nearby wall various dynamical regimes are found first the spiral simply tries to avoid the boundary and its tip moves towards the center of the circular domain corresponding to a negative real part of the translational eigenvalues this effect is noticeable at a domain radius of r r c r 1 the spiral subsequently exhibits an oscillatory instability the tip trajectory displays a meandering motion which may be characterized as boundary induced spiral meandering a systematic study of the spiral rotation as a function of a control parameter and the domain size reveals that the meandering instability in large domains becomes suppressed and the spiral rotation becomes rigid at a critical radius r c r 0 boundary induced meandering arises below a second critical radius r c r 2 r c r 0

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

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

input spaces and output performance the performance of an output is defined as the least upper bound of the absolute value of that output as time ranges over the real numbers and the input ranges over an appropriate function space the paper extends previous results by introducing useful input spaces a simple relation is found between performance the step response and a characteristic of the input space useful necessary and sufficient conditions for finite performance are derived the new results are extended to multivariable systems the appropriateness of various conventional performance indices is assessed by considering the appropriateness of their corresponding input spaces

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

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

dynamics of spiral waves on unbounded domains using center manifold reductions abstract an equivariant center manifold reduction near relative equilibria ofg equivariant semiflows on banach spaces is presented in contrast to previous results the lie groupginduces a strongly continuous group action on the underlying function space in fact gmay act discontinuously the results are applied to bifurcations of stable patterns arising in reaction diffusion systems on the plane or in three space modeling chemical systems such as catalysis on platinum surfaces and belousov zhabotinsky reactions these systems are equivariant under the euclidean symmetry group hopf bifurcations from rigidly rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated