stability of neuronal pulses composed of concatenated unstable kinks we demonstrate that a traveling pulse solution emerging from the concatenation of two unstable kinks can be stable by means of stability analysis and numerical simulations we show the stability of neuronal pulses action potentials with increasing refractory periods which decompose into two radiationally unstable kinks in the limit these action potentials are solutions of an ultrarefractory version of the fitzhugh nagumo system

pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

analytical and simulation results for the stochastic spatial fitzhugh nagumo model neuron for the fitzhugh nagumo system with space time white noise we use numerical methods to consider the generation of action potentials and the reliability of transmission in the presence of noise the accuracy of simulated solutions is verified by comparison with known exact analytical results noise of small amplitude may prevent transmission directly whereas larger amplitude noise may also interfere by producing secondary nonlocal responses the probability of transmission as a function of noise amplitude is found for both uniform noise and noise restricted to a patch for certain parameter ranges the recovery variable may be neglected to give a single component nonlinear diffusion with space time white noise in this case analytical results are obtained for small perturbations and noise which agree well with simulation results for the voltage variable expressions are given for the mean covariance and variance and their steady state forms the spectral density of the voltage is also obtained numerical examples are given of the difference between the properties of nonlinear and linear cables and the validity of the expressions obtained for the statistical properties is investigated as a function of noise amplitude for given parameters analytical results are in good agreement with simulation until a certain critical noise amplitude is reached which can be estimated the role of trigger zones in increasing the reliability of transmission is discussed

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

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

a space time discontinuous galerkin method for the elastic wave equation abstract in this work we present a new high order space time discretization method based on a discontinuous galerkin paradigm for the second order visco elastodynamics equation after introducing the method we show that the resulting space time discontinuous galerkin formulation is well posed stable and retains optimal rate of convergence with respect to the discretization parameters namely the mesh size and the polynomial approximation degree a set of two and three dimensional numerical experiments confirms the theoretical bounds

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

combined dg cg time stepping for wave equations the continuous and discontinuous galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order this eliminates the doubling of the number of variables that results when a second order problem is written as a first order system stability convergence and accuracy of these schemes is established in the context of the wave equation it is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy and that this problem can be circumvented using interpolants matched with the time stepping scheme

stabilization of pulse waves through inhibition in a feedforward neural network we consider a firing rate model of a neural network of excitatory and inhibitory populations with an excitatory feedforward connectivity we analyze traveling wave solutions and determine the conditions for their existence and stability our study demonstrates the role of inhibition in stable pulse propagation in a purely excitatory network pulse waves are unstable because of the existence of stable front wave and back wave with different velocities pulse waves can propagate stably in the network where excitation is appropriately balanced by inhibition analytical results on the wave speeds and the shape of waves are obtained

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

an extension of the decomposition method for solving nonlinear equations and its convergence in this paper we propose an extension of the decomposition method for solving nonlinear equations after a summary of adomian s decomposition method adm a new kind of decomposition strategy for nonlinear functions in nonlinear equations is proposed and a generalized decomposition method gdm for solving nonlinear equations is presented under a rather mild condition the convergence of the gdm is proved also some concrete examples are studied to illustrate with numerical results how powerful the gdm is finally some general remarks conclude this study

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous l 2 dirichlet boundary data this paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with l 2 boundary terms and smooth right hand side for the case of l 2 boundary data the rates of convergence in negative norms are derived in the case of smooth forcing term and zero boundary data optimal rates of convergence in positive norms are provided

pulse bifurcations and instabilities in an excitable medium computations in finite ring domains we investigate the instabilities and bifurcations of traveling pulses in a model excitable medium in particular we discuss three different scenarios involving either the loss of stability or disappearance of stable pulses in numerical simulations beyond the instabilities we observe replication of pulses backfiring resulting in complex periodic or spatiotemporally chaotic dynamics as well as modulated traveling pulses we approximate the linear stability of traveling pulses through computations in a finite albeit large domain with periodic boundary conditions the critical eigenmodes at the onset of the instabilities are related to the resulting spatiotemporal dynamics and act upon the back of the pulses the first scenario has been analyzed earlier m g zimmermann et al physica d 110 92 1997 for high excitability low excitation threshold it involves the collision of a stable pulse branch with an unstable pulse branch in a so called t point in the framework of traveling wave ordinary differential equations pulses correspond to homoclinic orbits and the t point to a double heteroclinic loop we investigate this transition for a pulse in a domain with finite length and periodic boundary conditions numerical evidence of the proximity of the infinite domain t point in this setup appears in the form of two saddle node bifurcations alternatively for intermediate excitation threshold an entire cascade of saddle nodes causing a spiraling of the pulse branch appears near the parameter values corresponding to the infinite domain t point backfiring appears at the first saddle node bifurcation which limits the existence region of stable pulses the third case found in the model for large excitation threshold is an oscillatory instability giving rise to breathing traveling pulses that periodically vary in width and speed

continuous finite elements in space and time for the heat equation on analyse une nouvelle methode variationnelle pour l approximation de l equation de la chaleur a l aide des elements finis continus dans l espace et le temps on presente les estimations d erreurs et les resultats d experiences numeriques

a discontinuous galerkin method for higher order ordinary differential equations in this paper we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal o tp 1 convergence rate in the l2 norm we further show that the p degree discontinuous solution of differential equation of order m and its first m 1 derivatives are o t2p 2 m superconvergent at the end of each step we also establish that the p degree discontinuous solution is o tp 2 superconvergent at the roots of p 1 m degree jacobi polynomial on each step finally we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates

bifurcation structure of dissipative solitons in this paper we analyze in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity we compare our results with the general picture described in p d woods a r champneys physica d 129 1999 147 p coullet c riera c tresser phys rev lett 84 2000 3069 and find that the stable and unstable manifolds of homogeneous and patterned solutions present a much higher level of complexity than predicted including the existence of additional localized solutions and fronts this extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low amplitude periodic solutions and to the fact that these periodic solutions together with the high amplitude ones constitute a one parameter family generating a closed line on the symmetry plane