singularly perturbed ordinary differential equations abstract we show the existence of periodic solutions for a couple of ordinary differential equations depending on a parameter e when e presents a small parameter for the majorest derivative of the first equation the key idea is to connect a coincidence degree theory with a method from a previous paper by the author j differential equations 89 1991 203 223 to avoid the use of a hard implicit function theorem of nash moser

a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

nonlinear evolution equations and the euler flow abstract a new theorem on abstract nonlinear equations of evolution is proved as an application the existence uniqueness regularity and continuous dependence on the data are proved for the solution of the euler equation for incompressible fluids in a bounded domain in r m

finite element error estimates for 3d exterior incompressible flow with nonzero velocity at infinity we consider a stationary incompressible navier stokes flow in a 3d exterior domain with nonzero velocity at infinity in order to approximate this flow we use the stabilized p1 p1 finite element method proposed by rebollo numer math 79 283 319 1998 following an approach by guirguis and gunzburger model math anal numer 21 445 464 1987 we apply this method to the navier stokes system with oseen term in a truncated exterior domain under a pointwise boundary condition on the artificial boundary this leads to a discrete problem whose solution approximates the exterior flow as is shown by error estimates

stochastic hopf bifurcation the effect of colored noise on the bifurcation interval we consider a general nonlinear dynamical system undergoing a supercritical hopf bifurcation and driven by a gaussian colored noise using normal form theory and an approximate effective fokker planck equation valid for weak intensity and small correlation time of the noise we obtain an explicit expression for the stationary probability distribution of the full general system close to the bifurcation and analyze the changes in the shape of the distribution to discuss the effect of the noise on the bifurcation interval textcopyright 1996 the american physical society

on the approximation solvability of a class of strongly nonlinear elliptic problems on unbounded domains a class of strongly nonlinear boundary value problems posed on unbounded regions is considered a nonlocal coupling of the linearized far field exterior to an auxiliary boundary allows for approximations to be defined on domains of finite extent constructive existence results for bounded domains are then extended by employing an exhausting sequence of approximating domains in particular well posedness is seen to be equivalent to unique approximation solvability with the rate of convergence dependent upon the radius of the auxiliary boundary application to a model of proteins immersed in an electrolyte solution is made

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

error analysis and preconditioning for an enhanced dtn fe algorithm for exterior scattering problems in this paper we present an error analysis for a high order accurate combined dirichlet to neumann dtn map finite element fe algorithm for solving two dimensional exterior scattering problems we advocate the use of an exact dtn or steklov poincare map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region the advantage of using an exact dtn map is that it provides a transparent condition which does not reflect scattered waves unphysically our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle to compute the dtn map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the dtn map we also present some preliminary work concerning the preconditioning of the resulting system of linear equations including numerical experiments

a comparison of nrbcs for pufem in 2d helmholtz problems at high wave numbers in this work exact and approximate non reflecting boundary conditions nrbcs are implemented with the partition of unity finite element method pufem to solve short wave scattering problems governed by the helmholtz equation in two dimensions by short wave problems we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer various nrbcs are implemented and a comparison of their performance is carried out based on the accuracy of the results ease of implementation and computational cost the aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

finite element methods applied to nearly one way wave propagation abstract a numerical method is developed for approximately solving stationary wave propagation problems for which most of the energy propagates in an angle band of less than 90 about a fixed direction this property often occurs in realistic physical models the numerical method is based on factoring out the principal propagation direction in the solution and then applying the finite element method the method is applied to wave propagation in a duct or waveguide and to the exterior scattering problem it is shown both theoretically and by means of numerical examples that this method requires considerably less grid points and computational cost than standard discretization methods

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

effects of a periodic drive and correlated noise on birhythmic van der pol systems abstract this paper considers the dynamics of a van der pol birhythmic oscillator submitted both to colored noise and harmonic excitation applying the quasi harmonic assumption to the corresponding langevin equation we derive an approximated fokker planck equation that is compared with the results of computer simulations we thus derive both the effects of the correlation time and the harmonic excitation on the parameter space where birhythmicity appears in this region we find that the multi limit cycle van der pol oscillator reduces to an asymmetric bistable system where the sinusoidal drive intensity plays the role of asymmetric parameter and noise can lead to stochastic bifurcations consisting in a qualitative change of the stationary amplitude distribution under both influence of noise and harmonic excitation the dynamics can be well characterized through the concepts of pseudo potential that regulates the low noise arrhenius like behavior

effects of non gaussian noise near supercritical hopf bifurcation abstract we have studied the effects of non gaussian colored noise in a chemical oscillation system the well known brusselator model in the parameter region close to the supercritical hopf bifurcation with the variation of the parameter q which quantifies the deviation from gaussian character the signal to noise ratio of noise induced oscillation exhibits a bell shaped change indicating the presence of resonant activity the cooperative effects of q and the correlation time on the performance of noise induced oscillation are also investigated interestingly resonance like behavior can be induced by either q or when the other parameter is properly fixed stochastic normal form theory is used to analyze these nontrivial effects and the simulation results are well reproduced this work provides us comprehensive understanding of how non gaussian noise influences the dynamics in chemical oscillation systems

stochastic bifurcation for a tumor immune system with symmetric levy noise abstract in this paper we investigate stochastic bifurcation for a tumor immune system in the presence of a symmetric non gaussian levy noise stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index the effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation we find that stochastic dynamics induced by gaussian and non gaussian levy noises are quite different

a path integral method for coarse graining noise in stochastic differential equations with multiple time scales abstract we present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales the objective of this method is to derive from the original system a new stochastic differential equation describing the system s evolution on slow time scales for this purpose we start from the corresponding path integral representation of the stochastic system and apply a multi scale expansion to the associated path integral kernel of the corresponding lagrangian as a concrete example we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion managed fiber optic communications moreover we show that for this particular example the new path integration method yields the same result at leading order as an asymptotic expansion of the associated fokker planck equation

absorbing boundary conditions difference operators and stability in this paper we present a review of some of the methods currently used for solving the absorbing boundary problem for the two dimensional scalar wave equation we show the relationship between the methods of lindman and clayton and engquist through this relationship we can derive discretizations of any rational approximation to the one way wave equation we prove that for all the cases considered here which can be solved in a manner similar to lindman s approach the bo unds imposed on the courant number for stability at the boundary are no more severe than the bound 1 fi required for stability of the interior scheme these bounds are however necessary but not sufficient we also compare the methods reviewed numerically it is demonstrated that lindman s scheme is no better than a sixth order approximation of halpern and trefethen for low order approximations higdon s one dimensional equations are satisfactory but as the order increases the two dimensional form of the equations as derived by halpern and trefethen is preferable 0 1992 academic pws

nonreflecting boundary conditions in elastodynamics for finite element methods based upon off surface boundary integral equations in this work an off surface boundary integral osbi method is presented as a mesh termination scheme for solving large or infinite domain problems of elastodynamics the boundary integral equation is discretized using finite element shape functions and the neumann boundary condition term is solved for in terms of the dirichlet boundary condition term this expression is then substituted into associated finite element formulation for the interior problem comparisons are made using the new osbi technique the dtn method of givoli and keller and several other local nonreflecting boundary conditions the proposed boundary condition is shown to be accurate is well suited for use with finite element methods and is competitive with the dtn method

a semiclassical initial value approximation for the trace of green s function a semiclassical initial value approximation for the trace of green s function is derived in contrast to the well known formula of gutzwiller applicability of the present expression does not require knowledge of the system s periodic orbits but constructs the trace from classical trajectories originating from all points on a poincare surface a given trajectory provides a contribution to the trace each time it returns to the surface with a weight based in part on the inner product on this surface of coherent states associated with the initial and returning points the treatment is generalized to obtain a version of the initial value formula that is useful for systems having discrete symmetries the initial value trace expression is shown to be semiclassically valid for chaotic systems by a stationary phase treatment that demonstrates its reduction to gutzwiller s formula in the classical limit numerical calculations of energy eigenvalues verify the applicability of the approximation not only to chaotic systems but to integrable systems and systems with mixed phase space the approximation presented here has numerical advantages over methods for determining the trace based on initial value treatments of the time dependent propagator especially for systems with homogeneous potential energy functions

a finite element method for 3d exterior oseen flows error estimates stationary oseen flows in a 3d exterior domain are discretized by applying a p1 p1 stabilized finite element method to the oseen system in a truncated exterior domain with an implicit pointwise artificial boundary condition on the truncating boundary error estimates are proved for this discretization the paper extends an approach which was introduced by guirguis and gunzburger rairo model math anal numer 21 1987 pp 445 464 for exterior stokes flows the stabilized p1 p1 method in question was introduced by rebollo numer math 79 1998 pp 283 319

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

construction of local boundary conditions for an eigenvalue problem using micro local analysis application to optical waveguide problems we present a general procedure based on the pseudo differential calculus for deriving artificial boundary conditions for an eigenvalue problem that characterizes the propagation of guided modes in optical waveguides this new approach allows the construction of local conditions that a are independent of the frequency regime b preserve the sparsity pattern of the finite element discretization and c are applicable to arbitrarily shaped convex artificial boundaries the last feature has the potential for reducing the size of the computational domain numerical results are presented to highlight the potential of conditions of order 1 2 and 1 for improving significantly the computational efficiency of finite element methods for the solution of optical waveguide problems

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations

stochastic modelling replacing fast degrees of freedom by noise in systems with timescale separation where the fast degrees of freedom exhibit chaotic motion the latter are replaced by suitable stochastic processes a projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones the resulting equations can be approximated in a controlled way by fokker planck equations or equivalently by stochastic differential equations for slow degrees of freedom we discuss some model situations and explore the accuracy of the approximations by numerical simulations

general tooth boundary conditions for equation free modeling we are developing a framework for multiscale computation which enables models at a microscopic level of description for example lattice boltzmann monte carlo or molecular dynamics simulators to perform modeling tasks at macroscopic length scales of interest the plan is to use the microscopic rules restricted to small patches of the domain the teeth using interpolation to bridge the gaps here we explore general boundary conditions coupling the widely separated teeth of the microscopic simulation that achieve high order accuracy over the macroscale we present the simplest case when the microscopic simulator is the quintessential example of a pde we argue that classic high order interpolation of the macroscopic field provides the correct forcing in whatever boundary condition is required by the microsimulator such interpolation leads to tooth boundary conditions which achieve arbitrarily high order consistency the high order consistency is demonstrated on a class of linear pdes in two ways first through the eigenvalues of the scheme for selected numerical problems and second using the dynamical systems approach of holistic discretization on a general class of linear pdes analytic modeling shows that for a wide class of microscopic systems the subgrid fields and the effective macroscopic model are largely independent of the tooth size and the particular tooth boundary conditions when applied to patches of microscopic simulations these tooth boundary conditions promise efficient macroscale simulation we expect the same approach will also accurately couple patch simulations in higher spatial dimensions

singularly perturbed variational problems abstract the existence of periodic solutions for some singularly perturbed variational equations is shown the main idea is to connect the galerkin approximation method with a procedure from a previous paper by the author j differential equations 89 1991 203 223 to avoid the use of a hard implicit function theorem of nash moser

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

phase descriptions of a multidimensional ornstein uhlenbeck process stochastic oscillators play a prominent role in different fields of science their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case one notion of phase that we put forward previously the asymptotic phase of a stochastic oscillator is based on the eigenfunction expansion of its probability density more specifically it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least negative eigenvalue formally besides the backward phase one can also define the forward phase as the complex argument of the eigenfunction of the forward kolomogorov operator corresponding to the least negative eigenvalue until now the intuition about these phase descriptions has been limited here we study these definitions for a process that is analytically tractable the two dimensional ornstein uhlenbeck process with complex eigenvalues for this process i we give explicit expressions for the two phases ii we demonstrate that the isochrons are always the spokes of a wheel but that iii the spacing of these isochrons their angular density is different for backward and forward phases iv we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field whereas the forward phase also depends on the noise matrix and v we demonstrate that the mean progression of the backward phase in time is always uniform whereas this is not true for the forward phase except in the rotationally symmetric case we illustrate our analytical results for a number of qualitatively different cases

on convergence of higher order schemes for the projective integration method for stiff ordinary differential equations we present a convergence proof for higher order implementations of the projective integration method pi for a class of deterministic multi scale systems in which fast variables quickly settle on a slow manifold the error is shown to contain contributions associated with the length of the microsolver the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro and macrosolvers respectively we also provide stability conditions for the pi methods under which the fast variables will not diverge from the slow manifold we corroborate our results by numerical simulations

a lagrange multiplier method for the interface equations from electromagnetic applications in the vector scalar potential formulation for three dimensional electromagnetic field applications one arises at a system of interface equations in r 3 coupling the two potentials in this paper a lagrange multiplier method for solving the coupled equations is presented and it is shown that the resulting lagrange multiplier operator defined on the interface is not only symmetric positive definite but also spectrally equivalent to the identity operator with respect to an appropriate inner product hence the lagrange multiplier equation can be solved efficiently using iterative solution schemes such as the conjugate gradient iterations the exterior problem is approximated by truncating the equation on a bounded domain finite element methods are chosen for the approximation of interior and truncated exterior problems as well as for the lagrange multiplier problem error estimates are proved for the finite element approximations which guarantee convergence as the mesh is refined an alternative operat

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

high frequency numerical analytic approximations for the separable acoustic wave equation abstract an algorithm which employs a combination of asymptotic and numerical methods to solve a sturm liouville problem arising from a separable acoustic wave equation is presented error criteria for switching between methods are derived implementation requires a decomposition of the index of refraction profile into separable subprofiles computing times and accuracies are compared with conventional methods

determination of the complex acoustic scattering matrix of a right angled duct a method for determining the complete higher order scattering matrix of an acoustic discontinuity is developed the method is demonstrated for a right angled waveguide bend and the magnitude and phase of the reflection and transmission coefficients are extracted precisely the procedure is straightforward and based on the solutions to the helmholtz equation by the finite element method fem the consistency of the scattering coefficients found by this method is verified by their properties of symmetry and their accuracy is established by the conservation of energy the reliability of the new technique is further proved by means of an arbitrary sound source and by comparing the direct fem response to the reflection matrix calculation some features of the scattering matrix as a function of frequency are surprising such as the steps and reversion of the phase evolution or the complete loss of transmission of the incoming wave the methodology detailed in this paper can be extended to other multiport junctions such as t junctions or size discontinuities in ducts

analytical and numerical investigation of the performance of the bgt2 condition for low frequency acoustic scattering problems a mathematical and numerical analysis is performed to assess the performance of the second order bayliss gunzburger turkel bgt2 condition when applied to solving low frequency acoustic scattering problems in the case of elongated scatterers this investigation suggests that bgt2 retains an acceptable level of accuracy for relatively low wavenumber a damping effect is incorporated to the bgt2 condition in order to extend the range of satisfactory performance this damping procedure consists in adding only a constant imaginary part to the wavenumber the numerical results indicate that the modified version of bgt2 extends the range of satisfactory performance by improving the level of accuracy by up to two orders of magnitude guidelines on the appropriate choice of the damping coefficient are provided

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

a third order boundary condition for the exterior stokes problem in three dimensions approximation de l operateur de stokes sur un domaine exterieur a trois dimensions par un probleme tronque sur un sous domaine fini introduction d une condition aux limites artificielle d ordre 3 comportement du probleme tronque et de sa discretisation dans un espace a elements finis estimations d erreurs

stochastic bifurcations in a bistable duffing van der pol oscillator with colored noise this paper aims to investigate gaussian colored noise induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable duffing van der pol oscillator by using the stochastic averaging method theoretically one can obtain the stationary probability density function of amplitude for the duffing van der pol oscillator and can reveal interesting dynamics under the influence of gaussian colored noise stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution which indicates that system parameters noise intensity and noise correlation time respectively can be treated as bifurcation parameters they also imply that the effects of multiplicative noise are rather different from that of additive noise the results of direct numerical simulation verify the effectiveness of the theoretical analysis moreover the largest lyapunov exponent calculations indicate that p and d bifurcations have no direct connection

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

high order numerical simulation of the transmission and scattering of waves using the method of difference potentials the method of difference potentials generalizes the method of calderon s operators from pdes to arbitrary difference equations and systems it offers several key advantages such as the capability of handling boundaries interfaces that are not aligned with the discretization grid variable coefficients and nonstandard boundary conditions in doing so the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid previously we have applied the method of difference potentials to solving several variable coefficient interior helmholtz problems with fourth and sixth order accuracy we have employed compact finite difference schemes as a core discretization methodology those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself numerical experiments corroborate the high order accuracy of our method for variable coefficients regular grids and non conforming boundaries in the current paper we extend the previously developed methodology to exterior problems we present a complete theoretical analysis of the algorithm as well as the results of a series of numerical simulations specifically we study the scattering of time harmonic waves about smooth shapes subject to various boundary conditions we also solve the transmission scattering problems in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside in all the cases our methodology guarantees high order accuracy for regular grids and non conforming boundaries and interfaces

a combined method for solving elliptic problems on unbounded domains abstract a noncomforming combined method of the ritz galerkin and finite element methods is applied in solving unbounded domain problems of the equation u u 0 the ritz galerkin method is used in an exterior subdomain and the traditional finite element method is used in a bounded subdomain errors of numerical solutions have been derived in the sobolev norms and a significant coupling relation n 1 o ln h as h 0 is found where n 1 is the total number of asymptotic particula ritz galerkin method and h is the maximal boundary length of triangular elements used in the finite element method numerical experiments have been carried out for a model problem on unbounded domains by using the nonconforming combination and the coupling strategy 1 showing that only two or three terms i e n 1 2 or 3 of particular solutions near the infinity are in general cases required for good approximate solutions in engineering application

stochastic bifurcations and coherencelike resonance in a self sustained bistable noisy oscillator we investigate the influence of additive gaussian white noise on two different bistable self sustained oscillators duffing van der pol oscillator with hard excitation and a model of a synthetic genetic oscillator in the deterministic case both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state we find that under the influence of noise their dynamics can be well characterized through the concept of stochastic bifurcation consisting in a qualitative change of the stationary amplitude distribution for the duffing van der pol oscillator analytical results obtained for a quasiharmonic approach are compared with the result of direct computer simulations in particular we show that the dynamics is different for isochronous and anisochronous systems moreover we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line this effect is similar to coherence resonance however in the case of anisochronous systems this effect breaks down and a new phenomenon anisochronous based stochastic bifurcation occurs

geometric singular perturbation theory for stochastic differential equations abstract we consider slow fast systems of differential equations in which both the slow and fast variables are perturbed by noise when the deterministic system admits a uniformly asymptotically stable slow manifold we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold which we construct explicitly depending on the dynamics of the reduced system the results cover time spans which can be exponentially long in the noise intensity squared that is up to kramers time we obtain exponentially small upper and lower bounds on the probability of exceptional paths if the slow manifold contains bifurcation points we show similar concentration properties for the fast variables corresponding to non bifurcating modes we also give conditions under which the system can be approximated by a lower dimensional one in which the fast variables contain only bifurcating modes

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

sensitivity of a hopf bifurcation to multiplicative colored noise the location of a hopf bifurcation in parameter space may be postponed or advanced by multiplicative colored noise depending on the interplay between the time scale of the noise and the rotation period of the phase variable

slow manifold and averaging for slow fast stochastic differential system abstract we consider stochastic dynamical systems with multiple time scales an intermediate reduced model is obtained and explored for a slow fast system when the fast mode is driven by white noise first and for later comparison one approximation to the reduced stochastic system on the exponentially attracting stochastic slow manifold is derived to errors of order o second because the noise only drives the fast modes averaging derives an autonomous deterministic system with errors of order o then a martingale argument accounts for fluctuations about the averaged system to form an intermediate reduced model with errors of order o the autonomous deterministic system now driven by white noise this intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold these results not only connect averaging with the stochastic slow manifold they also provide a martingale method for improving averaged models of stochastic systems

finite difference analysis of borehole flows involving domain contractions around three dimensional anomalies a finite difference analysis in cylindrical coordinates of borehole flows governed by poisson s equation yields an infinite cylindrical electrical grid as a model the customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks when the medium around the borehole is either uniform or cylindrically layered except for a three dimensional anomalous region near the borehole infinite electrical network theory allows one to represent the medium outside any cylinder s containing the anomaly and centered on the borehole by an infinite continued fraction of circulant laurent operators the continued fraction in turn can be represented by scalar conductances connected to the grid s nodes on s by choosing s just large enough to contain the anomaly one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on s an operational calculus for the circulant laurent operators and the fast fourier transform expedites the computations and moreover allows a rapidly computed determination of the solution outside s

the role of additive and multiplicative noise in filtering complex dynamical systems covariance inflation is an ad hoc treatment that is widely used in practical real time data assimilation algorithms to mitigate covariance underestimation owing to model errors nonlinearity or an

a fast finite difference method for propagation predictions over irregular inhomogeneous terrain propagation of electromagnetic waves over irregular inhomogeneous terrain is solved by a finite difference scheme the method is fast and requires considerably less memory than the integral equation methods the method requires a storage space of order o n and an execution time of order o n sup 2 fields generated by a te sub 2 line source are represented in an integral form in terms of the field over a flat constant impedance plane the incident field and the field scattered by the terrain irregularities and inhomogeneities accurate expressions are provided for the incident field and the green s function whose evaluation is otherwise accomplished by the rather time consuming sommerfeld s integrals measured equation of invariance is used to terminate the computational domain the sparse matrix generated by the method is inverted by the ricatti transform numerical results are presented for the ground wave as well as for the sky wave comparison is made for known geometries to establish the validity and limitations of the method

non lorentzian spectral lineshapes near a hopf bifurcation the effects of additive white noise upon the dynamics of a system described by its hopf normal form are investigated with particular reference to the well known model of a detuned single mode laser the power spectrum corresponding to the laser amplitude is determined by finite difference solution of a partial differential equation and analytical formulas are determined in the asymptotic limits of large parameters the effect of the amplitude phase coupling parameter in generating non lorentzian lineshapes is highlighted and the regions of parameter space where accurate first eigenvalue approximations of the fokker planck equation exist are indicated

co existing responses and stochastic resonance in post buckled structures a combined numerical and experimental study abstract much of what is known about co existing responses in nonlinear systems under both deterministic and random dynamic loading is limited to phenomenological investigations of discrete systems most commonly the duffing equation from these results alone it is difficult to extrapolate the behavior of the distributed nonlinear systems more commonly seen in real structures such as buckled beams and curved panels this is because beyond the simple increase in dimension real systems bring with them imperfections and more complex forms of energy dissipation the possibility of co existing responses particularly in the case of simultaneous safe and unsafe solutions e g snap through and non snap through poses potential problems for engineers as it multiplies the workload since one must be very careful to ensure that a particular simulation or experiment has captured the most critical response even in the case where random forces dominate the overall loading of a system a situation in which one might anticipate an equally random response a very small harmonic component can have an influence beyond its proportion this effect known as stochastic resonance is quite counterintuitive to the analyst more familiar with linear systems where the principles of superposition and scalar multiplication of solutions make this impossible in this paper the effect of the damping and noise level on the number of co existing responses in nonlinear systems is investigated stochastic resonance is also demonstrated first with a double well duffing oscillator and then it is shown to exist experimentally we believe for the first time on a macroscopic structure that being an imperfect buckled beam

an analysis of quadrature errors in second kind boundary integral methods this paper considers the effect of quadrature errors on galerkin methods for approximating the solution of second kind fredholm integral equations with weakly singular kernels defined on a smooth closed surface in three dimensions a quadrature strategy for galerkin methods that use piecewise poly nomials both continuous and discontinuous on a quadrilateral based mesh is proposed the error resulting from this further discretization is proved to be no larger than the original discretization error and moreover the amount of work needed to compute the resulting matrix for the corresponding linear system is shown to be proportional to the number of matrix entries

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

positive lyapunov exponents in the kramers oscillator the maximum lyapunov exponent is computed numerically for the double well oscillator in a heat bath positive exponents are found in a wide range of friction coefficients in the low damping regime

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

axial green s function method for multi dimensional elliptic boundary value problems axial green s function method is proposed to solve multi dimensional second order elliptic partial differential equations with the geometrical complexity of solution domains instead of directly dealing with 2d or 3d green s functions the proposed method systematically decomposes a multi dimensional elliptic differential operator into id ones consequently the method only requires id green s function associated with each coordinate axis theoretical formulation and simple axial discretization result in the system of equations of which the solutions represent the numerical solution of the original multi dimensional elliptic problem which is otherwise intractable to obtain by the traditional multi dimensional green s function technique the method employs analytical form of id green s function and integral schemes so that it can potentially be more robust than both the finite difference method and the boundary element method in addition the simple discretization procedure which uses only axially orthogonal straight lines and boundary data demonstrates its convenience over the finite element method of comparable accuracy numerical examples verify these advantages by demonstrating the second order convergence in various discretized models of complex geometries

convergence of the t matrix approach in scattering theory ii the t matrix numerical scheme is widely used in practice convergence of this scheme was not proved a proof of convergence is given in this paper

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

three dimensional approximate local dtn boundary conditions for prolate spheroid boundaries we propose a new class of approximate local dtn boundary conditions to be applied on prolate spheroidal shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles these conditions are a exact for the first modes b easy to implement and to parallelize c compatible with the local structure of the computational finite element scheme and d applicable to exterior ellipsoidal shaped boundaries that are more suitable in terms of cost effectiveness for surrounding elongated scatterers we investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions we also compare their performance to the second order absorbing boundary condition bgt2 designed by bayliss gunzburger and turkel when expressed in prolate spheroid coordinates the analysis reveals that in the low frequency regime the new second order dtn condition dtn2 retains a good level of accuracy regardless of the slenderness of the boundary in addition the dtn2 boundary condition outperforms the bgt2 condition such superiority is clearly noticeable for large eccentricity values

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

normal forms for stochastic differential equations we address the following problem from the intersection of dynamical systems and stochastic analysis two sde dx t j 0 m f j x t dw t j and dx t j 0 m g j x t dw t j in d with smooth coefficients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random diffeomorphism coordinate transformation h with h 0 0 and dh 0 id which conjugates the corresponding local flows

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme

application of bogomolny s transfer operator to a circular harmonic oscillator plus potential bogomolny s transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two dimensional integrable system the system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential the classical trajectories are used to construct the transfer matrix and an expression is derived for the eigenvalues of this matrix as a function of the energy these eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system which turn out to be exactly the same as the results obtained by solving the schrodinger equation some insight into this unexpected agreement is provided by considering an exact transfer operator we show that when this operator is expanded in powers of planck s constant the leading term in the expansion is bogomolny s transfer operator

power spectrum of a noisy system close to a heteroclinic orbit we consider a two dimensional dynamical system that possesses a heteroclinic orbit connecting four saddle points this system is not able to show self sustained oscillations on its own if endowed with white gaussian noise it displays stochastic oscillations the frequency and quality factor of which are controlled by the noise intensity this stochastic oscillation of a nonlinear system with noise is conveniently characterized by the power spectrum of suitable observables in this paper we explore different analytical and semianalytical ways to compute such power spectra besides a number of explicit expressions for the power spectrum we find scaling relations for the frequency spectral width and quality factor of the stochastic heteroclinic oscillator in the limit of weak noise in particular the quality factor shows a slow logarithmic increase with decreasing noise of the form q sim ln 1 d 2 our results are compared to numerical simulations of the respective langevin equations