mode interaction in rayleigh be nard convection abstract a long wave model for two dimensional thermal convection is examined this model contains both quadratic and cubic nonlinearity unless the problem is symmetrical about the horizontal mid plane in which case only the cubic nonlinearity is present in the asymmetrical case a strong interaction between modes with wavenumbers k and 2k is responsible for wavenumber gaps regions in parameter space where steady states of a given spatial period cease to exist in both the symmetrical and asymmetrical cases a 1 3 resonance generates multiple steady solutions of a given spatial period we calculate steady states and determine their stability for the asymmetrical case we also solve the initial value problem and find stable travelling waves stable modulated travelling waves and stable heteroclinic cycles as predicted by an analysis of the 1 2 resonance

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

spherically symmetric random walks in noninteger dimension a previous article proposed a new kind of random walk on a spherically symmetric lattice in arbitrary noninteger dimension d such a lattice avoids the problems associated with a hypercubic lattice in noninteger dimension this article examines the nature of spherically symmetric random walks in detail a large time asymptotic analysis of these random walks is performed and the results are used to determine the hausdorff dimension of the process exact results are obtained in terms of hurwitz functions incomplete zeta functions for the probability of a walker going from one region of the spherical lattice to another finally it is shown that the probability that the paths of k independent random walkers will intersect vanishes in the continuum limit if d 2k k 1

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

the euler scheme with irregular coefficients weak convergence of the euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of lebesgue measure zero the rate of convergence is presented when coefficients are holder continuous monte carlo simulations are also discussed

on the evolution of periodic plane waves in reaction diffusion systems of l w type lambda omega systems are a class of simple reaction diffusion equations with a limit cycle in the reaction kinetics the author considers the solution of the system given by lambda r lambda 0 r p omega r omega 0 r p on a semi infinite spatial domain with initial data decaying exponentially across the domain numerical evidence is presented showing that this initial condition induces a wave front moving across the domain with periodic plane waves behind the front these periodic waves can move in either direction depending on the parameter values the author uses intuitive criteria to derive an expression for the speed of the advancing front and by reducing the system to ordinary differential equations for similarity solutions the amplitude and speed of the periodic plane waves are determined the speed of these periodic waves varies continuously with the initial data perturbation theory is then used to obtain an analytical approximation to the solutions these anal

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

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

limit cycles in the presence of convection a traveling wave analysis we consider a diffusion model with limit cycle reaction functions in an unbounded domain diffusion spreads the pattern outwards from the source convection adds instability to the reaction diffusion system the result of this instability is a readiness to create a pattern we choose the lambda omega reaction functions for their simple limit cycle we carry out a transformation of the dependent variables into polar form from this we consider the initiation of the pattern to approximate a traveling wave we carry out numerical experiments to test our analysis these confirm the premise of the analysis that the initiation can be modeled by a traveling wave furthermore the analysis produces a good estimate of the numerical results most significantly we confirm that the pattern consists of two different types

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

a class of exactly solvable eigenvalue problems the class of differential equation eigenvalue problems y x x2n 2y x xn ey x n 1 0 1 2 3 on the interval x can be solved in closed form for all the eigenvalues e and the corresponding eigenfunctions y x the eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions the eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as x for odd n the polynomials that are obtained in this way are new and interesting classes of orthogonal polynomials for example when n 1 the eigenfunctions are orthogonal polynomials in x3 multiplying airy functions of x2 the properties of the polynomials for all n are described in detail

numerical integration of stochastic differential equations with nonglobally lipschitz coefficients we propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations sdes with nonglobally lipschitz coefficients following this concept we discard the approximate trajectories which leave a sufficiently large sphere we prove that accuracy of any method of weak order p is estimated by varepsilon o h p where varepsilon can be made arbitrarily small with increasing radius of the sphere the results obtained are supported by numerical experiments

euler maruyama approximations for sdes with non lipschitz coefficients and applications abstract in this paper euler maruyama approximation for sde with non lipschitz coefficients is proved to converge uniformly to the solution in l p space with respect to the time and starting points as an application we also study the existence of solution and large deviation principle for anticipative sde with random initial condition

wavelet projections of the kuramoto sivashinsky equation i heteroclinic cycles and modulated traveling waves for short systems abstract we study fairly low dimensional projections of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a short interval onto bases spanned by periodic wavelets such projections break the translation reflection symmetry o 2 replacing it by a finite dihedral group dk however we show that for the perrier basdevant wavelets used here the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved in particular we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets we use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and fourier modes which preserve full o 2 symmetry and are also optimal in that fourier truncations maximise the energy l2 norm among all finite dimensional models this study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains

spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains invasions in oscillatory systems generate in their wake spatiotemporal oscillations consisting of either periodic wavetrains or irregular oscillations that appear to be spatiotemporal chaos we have shown previously that when a finite domain with zero flux boundary conditions has been fully invaded the spatiotemporal oscillations persist in the irregular case but die out in a systematic way for periodic traveling waves in this paper we consider the effect of environmental inhomogeneities on this persistence we use numerical simulations of several predator prey systems to study the effect of random spatial variation of the kinetic parameters on the die out of regular oscillations and the long time persistence of irregular oscillations we find no effect on the latter but remarkably a moderate spatial variation in parameters leads to the persistence of regular oscillations via the formation of target patterns in order to study this target pattern production analytically we turn to systems numerical simulations confirm analagous behavior in this generic oscillatory system we then repeat this numerical study using piecewise linear spatial variation of parameters rather than random variation which also gives formation of target patterns under certain circum stances which we discuss we study this in detail by deriving an analytical approximation to the targets formed when the parameter 0 varies in a simple piecewise linear manner across the do main using perturbation theory we end by discussing the applications of our results in ecology and chemistry

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

irregular wakes in reaction diffusion waves abstract reaction diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems but chaotic solutions have been very rarely documented we present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems by analysing in detail the bifurcation structure of a particular set of reaction diffusion equations on an infinite one dimensional domain with particular initial conditions we show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations using a combination of analytical and numerical analysis we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front finally we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction diffusion systems

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

existence uniqueness almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the euler maruyama approximation abstract in the first part of this paper the existence uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered under nonlinear growth conditions the results are obtained using the idea from mao and rassias 2005 10 where stochastic differential equation with constant delay are considered however the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea moreover the convergence in probability of the appropriate euler maruyama solution is proved under the same nonlinear growth conditions adding the linear growth condition we show that the almost sure polynomial stability of the euler maruyama solution implies the almost sure polynomial stability of the exact solution this part of the paper represents the extension of the idea from wu et al 2010 17 the main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations

approximation for non smooth functionals of stochastic differential equations with irregular drift abstract this paper aims at developing a systematic study for the weak rate of convergence of the euler maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients we apply our method to obtain the rates of approximation for the expectation of various non smooth functionals of both stochastic differential equations and killed diffusion we also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is holder continuous

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

semi implicit split step numerical methods for a class of nonlinear stochastic differential equations with non lipschitz drift terms abstract in this paper we discuss numerical solutions of a class of nonlinear stochastic differential equations using semi implicit split step methods under some monotonicity conditions on the drift term we study moment estimates and strong convergence properties of the numerical solutions with a focus on stochastic ginzburg landau equations moreover we compare the performance of various numerical methods including the tamed euler truncated euler implicit euler and split step procedures in particular we discuss the empirical rate of convergence and the computational cost of these methods for certain parameter values of the models used

optimal control of a nutrient phytoplankton zooplankton fish system we consider the mathematical formulation analysis and numerical solution of an optimal control problem for a nonlinear nutrient phytoplankton zooplankton fish reaction diffusion system we study the existence of optimal solutions derive an optimality system and determine optimal solutions in the original spatially homogeneous formulation m scheffer oikos 62 1991 pp 271 282 the dynamics of plankton were investigated as a function of parameters for nutrient levels and fish predation rate on zooplankton in our paper the model is spatially extended and the parameter for fish predation treated as a multiplicative control variable the model has implications for the biomanipulation of food webs in eutrophic lakes to help improve water quality in order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi implicit in time finite element method with mass lumping and present the results of numerical experiments in two space dimensions

stability of periodic travelling wave solutions with large spatial periods in reaction diffusion systems abstract we consider a spatially homogeneous system of reaction diffusion equation defined on the interval of the one dimensional spatial variable x it is known that this equation has a one parameter family of periodic travelling wave solutions x ct c if this equation has a spatially homogeneous periodic solution t the spatial period l c of the travelling wave solution satisfies l c c t if c where c is the propagation speed and t is the period of t we prove that in the case c 0 is sufficiently large x ct c is unstable if t is strongly unstable and x ct c is marginally stable if t is strongly stable if the equation is defined on a finite interval 0 l of the variable x with the periodic boundary conditions we can obtain a more precise result regarding the stability of x c t c where c 0 is a speed which satisfies l ml c for some positive integer m we prove that this solution is asymptotically stable in the sense of waveform stability if c 0 is sufficiently large and if t is strongly stable

nonlinear convective stability of travelling fronts near turing and hopf instabilities abstract reaction diffusion equations on the real line that contain a control parameter are investigated of interest are travelling front solutions for which the rest state behind the front undergoes a supercritical turing or hopf bifurcation as the parameter is increased this causes the essential spectrum to cross into the right half plane leading to a linear convective instability in which the emerging pattern is pushed away from the front as it propagates it is shown however that the wave remains nonlinearly stable in an appropriate sense more precisely using the fact that the instability is supercritical it is shown that the amplitude of any pattern that emerges behind the wave saturates at some small parameter dependent level and that the pattern is pushed away from the front interface as a result when considered in an appropriate exponentially weighted space the travelling front remains stable with an exponential in time rate of convergence

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics