numerical computation of heteroclinic orbits we give a numerical method for the computation of heteroclinic orbits connecting two saddle points in 2 these can be computed to very high period due to an integral phase condition and an adaptive discretization we can also compute entire branches one dimensional continua of such orbits the method can be extended to compute an invariant manifold that connects two fixed points in n as an example we compute branches of traveling wave front solutions to the huxley equation using weighted sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period

computation of homoclinic orbits in partial differential equations an application to cylindrical shell buckling this paper concerns numerical computation of localized solutions in partial differential equations pdes on unbounded domains the application is to the von karman donnell equations a coupled system of elliptic equations describing the equilibrium of an axially compressed cylindrical shell earlier work suggests that axially localized solutions are the physically preferred buckling modes hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the axial variable and periodic circumferentially the numerical method is based on a galerkin spectral decomposition circumferentially to pose ordinary differential equations odes in the unbounded coordinate methods for location and parameter continuation of homoclinic solutions of odes are then adapted making special use of the symmetry and reversibility properties of solutions observed experimentally thus a formally well posed continuation problem is reduced to a rotational subgroup circumferentially and posed over a truncation of the half interval axially the method for location of solutions makes use of asymptotic approximations where available more generally an adaptation of the successive continuation shooting method is used in the lowest possible number of circumferential modes followed by additional homotopies to add more modes by continuation in the strength of nonlinear mode interaction terms the method is illustrated step by step to produce a variety of homoclinic solutions of the equations and compute their bifurcation diagrams as the loading parameter varies good agreement with experimental data is found all computations are performed using auto the techniques illustrated here for the von karman donnell equations are applicable to a wider class of pdes

spectral stability of small amplitude viscous shock waves in several space dimensions a planar viscous shock profile of a hyperbolic parabolic system of conservation laws is a steady solution in a moving coordinate frame the asymptotic stability of viscous profiles and the related vanishing viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions it is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile the evans function method provides a geometric dynamical systems framework to study the eigenvalue problem in this approach eigenvalues correspond to zeros of an essentially analytic function mathcal e rho lambda rho omega which detects nontrivial intersections of the so called stable and unstable spaces that is spaces of solutions that decay on one or the other side of the shock wave respectively in a series of pioneering papers kevin zumbrun and collaborators have established in various contexts that spectral stability that is the non vanishing of mathcal e rho lambda rho omega and the non vanishing of the lopatinski kreiss majda function imply nonlinear stability of viscous shock profiles in several space dimensions in this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions this is done by exploiting the slow fast nature of the small amplitude limit which was used in a previous paper by the authors to prove spectral stability of small amplitude shock waves in one space dimension geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior three qualitatively different regimes are distinguished that relate the small strength epsilon of the shock wave to appropriate ranges of values of the spectral parameters various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality

computation of homoclinic solutions to periodic orbits in a reduced water wave problem abstract this paper concerns homoclinic solutions to periodic orbits in a fourth order hamiltonian system arising from a reduction of the classical water wave problem in the presence of surface tension these solutions correspond to travelling solitary waves which converge to non decaying ripples at infinity an analytical result of amick and toland 1992 showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit is extended numerically also a related result by amick and mcleod 1991 showing the non existence of homoclinic solutions to zero is motivated geometrically a general boundary value method is constructed for continuation of homoclinic orbits to periodic orbits in hamiltonian and reversible systems numerical results are presented using the path following software auto showing that the amick toland solutions persist well away from the singular limit and for large amplitude periodic orbits special account is taken of the phase shift between the two periodic solutions in the asymptotic limits furthermore new multi modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis which is verified a posteriori by explicit computation continuation of these new solutions reveals limit points with respect to the singular parameter

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

analysis of a singularly perturbed traveling wave problem a singularly perturbed traveling wave problem derived from the drift diffusion model describing electron transport in two valley semiconductor materials is analysed it is shown that the reduced problem defined on a submanifold of the phase space has homoclinic orbits for certain parameters by using methods from invariant manifold theory and methods from homoclinic continuation theory it is proved that the homoclinic orbits of the reduced problem persist in a global center manifold near the manifold of the reduced problem

solitary waves in semiconductors with finite geometry and the gunn effect an asymptotic analysis of the gunn effect oscillations of the current through a semiconductor under constant voltage bias is given the electric field inside the one dimensional semiconductor is described by a parabolic equation with small diffusivity plus zero field boundary conditions and a constant voltage integral constraint nonuniform stationary solutions are constructed and their stability analyzed under both current and voltage bias the oscillations are known to be produced by repeated generation at one boundary of a solitary wave that moves towards and is annihilated at the other boundary far from the boundaries the solitary wave is constructed and its stability analyzed the effect of the boundaries is qualitatively discussed and several open problems pointed out

the application of bifurcation method to a higher order kdv equation abstract bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the higher order kdv equation u t au n u x u xxx 0 where n 1 and a r numbers of solitary waves are given for each parameter condition under some parameter conditions explicit solitary wave solutions are obtained specially some new solitary wave solutions are found for kdv or mkdv equation

heteroclinic bifurcation and singularly perturbed boundary value problems abstract we study a singularly perturbed boundary value problem in r m n x f x y e e y g x y e b 1 x 0 y 0 e 0 b 2 x 0 y 0 e 0 given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions similar results concerning the existence of periodic solutions relaxation oscillations are also presented several ideas from dynamical systems theory are employed e g exponential dichotomies fredholm alternatives and heteroclinic bifurcations

the numerical computation of homoclinic orbits for maps transversal homoclinic orbits of maps are known to generate shift dynamics on a set with cantor like structure in this paper a numerical method is developed for computation of the corresponding homoclinic orbits they are approximated by finite orbit segments subject to asymptotic boundary conditions we provide a detailed error analysis including a shadowing type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment this approach is applied to several examples in some of them parameters appear and closed loops of homoclinic orbits are found by a path following algorithm

the onset and end of the gunn effect in extrinsic semiconductors a hopf bifurcation analysis of the spontaneous current oscillation in direct current dc voltage biased extrinsic semiconductors is given for the classical model of the gunn effect in n gaas for semiconductor lengths l larger than a certain minimal value the steady state is linearly unstable for voltages in an interval phi alpha phi omega as l increases the branch of time periodic solutions bifurcating at simple eigenvalues when phi phi alpha turns from subcritical to supercritical and then back to subcritical again for very long semiconductors a quasi continuum of oscillatory modes bifurcates from the steady state at the onset of the instability the bifurcating branch is then described by a scalar reaction diffusion equation with cubic nonlinearity subject to antiperiodic boundary conditions on a subinterval of 0 l for the electron velocity curve we have considered the bifurcation is subcritical which may agree with experimental observations in n gaas an extension o

numerical computation of saddle node homoclinic bifurcation points in two parameter families of vector fields there can be curves in the parameter plane along which orbits homoclinic to hyperbolic equilibria occur such curves can end at a point where there is an orbit homoclinic to an equilibrium undergoing saddle node or transcritical bifurcation convergence and stability results are presented for a method of approximating these special parameter values and their associated homoclinic orbits

the statistics of eigenvector components of random band matrices analytical results by using a supersymmetric formalism the authors calculate analytically all higher moments pq sigma n mod psi n mod 2q generalizing the inverse participation ratio p2 where psi n 1 or n or n stands for the nth component of an eigenvector of a large random matrix with a band structure on this basis they reconstruct the whole probability distribution function of eigenvector components the relation with known numerical results is discussed

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

riesz basis property of serially connected timoshenko beams in this paper we study the riesz basis property of serially connected timoshenko beams with joint and boundary feedback controls suppose that the left end of the whole beam is clamped and the right end is free at intermediate nodes the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous the collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system we prove that the operator determined by the closed loop system has compact resolvent and generates a c 0 semigroup in an appropriate hilbert space we also show that there is a sequence of the generalized eigenvectors of the operator that forms a riesz basis with parentheses hence the spectrum determined growth condition holds therefore if the imaginary axis is not an asymptote of the spectrum then the closed loop system is exponentially stable finally we give a conclusion remark to explain that our result can be applied not

bounded traveling waves of the burgers huxley equation abstract in order to investigate bounded traveling waves of the burgers huxley equation bifurcations of codimension 1 and 2 are discussed for its traveling wave system by reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions heteroclinic solutions and periodic solutions which correspondingly give conditions of existence for solitary waves kink waves and periodic waves three basic types of bounded traveling waves furthermore their evolutions are discussed to investigate the existence of other types of bounded traveling waves such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle focus

the existence of localized vegetation patterns in a systematically reduced model for dryland vegetation abstract in this paper we consider the 2 component reaction diffusion model that was recently obtained by a systematic reduction of the 3 component gilad et al model for dryland ecosystem dynamics gilad et al 2004 the nonlinear structure of this model is more involved than other more conceptual models such as the extended klausmeier model and the analysis a priori is more complicated however the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations moreover we find that the model exhibits a richness of analytically tractable patterns that exceeds that of klausmeier type models our study focuses on the 4 dimensional dynamical system associated with the reaction diffusion model by considering traveling waves in 1 spatial dimension we use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic homoclinic periodic orbits that jump between normally hyperbolic slow manifolds representing various kinds of localized vegetation patterns the basic 1 front invasion patterns and 2 front spot gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model by exploiting the rich nonlinear structure of the model we construct many multi front patterns that are novel both from the ecological and the mathematical point of view in fact we argue that these orbits patterns are not specific for the model considered here but will also occur in a much more general singularly perturbed reaction diffusion setting we conclude with a discussion of the ecological and mathematical implications of our findings

dynamic domain decomposition in approximate and exact boundary control in problems of transmission for wave equations this paper is concerned with dynamic domain decomposition for optimal boundary control and for approximate and exact boundary controllability of wave propagation in heterogeneous media we consider a cost functional which penalizes the deviation of the final state of the solution of the global problem from a specified target state for any fixed value of the penalty parameter optimality conditions are derived for both the global optimal control problem and for local optimal control problems obtained by a domain decomposition and a saddle point type iteration convergence of the iterations to the solution of the global optimality system is established we then pass to the limit in the iterations as the penalty parameter increases without bound and show that the limiting local iterations converge to the solution of the optimality system associated with the problem of finding the minimum norm control that drives the solution of the global problem to a specified target state

on locating connecting orbits abstract an extension of a method for computing connecting orbits that locates such orbits in a systematic manner is presented as an application we compute homoclinic orbits in a singular perturbation problem

uniqueness and stability of periodic bursting solutions a detailed analysis of so called square bursting oscillators is given an interesting feature of these models is that the bursting solution need not be unique or stable for arbitrarily small values of a singular perturbation parameter this is a global phenomenon due to interactions between a homoclinic orbit and other invariant manifolds using geometric singular perturbation methods we characterize the set of parameters for which the bursting solution is uniquely determined and asymptotically stable

numerical computation of connecting orbits in planar piecewise smooth dynamical system abstract in this paper a numerical algorithm for computing the connecting orbits in piecewise smooth dynamical systems is derived and is analyzed a nondegenerate condition for the connecting orbit with respect to its bifurcation parameter is presented to ensure the defining equation being well posed which is a generalization of the melnikov condition for smooth systems the error caused by the truncation of time interval is also analyzed some numerical calculations are carried out to illustrate the theoretical analysis

fast and slow waves in the fitzhugh nagumo equation abstract it is known that the fitzhugh nagumo equation possesses fast and slow travelling waves fast waves are perturbations of singular orbits consisting of two pieces of slow manifolds and connections between them whereas slow waves are perturbations of homoclinic orbits of the unperturbed system we unfold a degenerate point where the two types of singular orbits coalesce forming a heteroclinic orbit of the unperturbed system let c denote the wave speed and e the singular perturbation parameter we show that there exists a c 2 smooth curve of homoclinic orbits of the form c e c connecting the fast wave branch to the slow wave branch additionally we show that this curve has a unique non degenerate maximum our analysis is based on a shilnikov coordinates result extending the exchange lemma of jones and kopell we also prove the existence of inclination flip points for the travelling wave equation thus providing the evidence of the existence of n homoclinic orbits n pulses for the fitzhugh nagumo equation for arbitrary n

a survey on free boundary identification of the truncated boundary in numerical bvps on infinite intervals a free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in fazio siam j numer anal 33 1996 1473 we consider here a survey on recent developments related to the free boundary identification of the truncated boundary the goals of this survey are to recall the reasoning for a free boundary identification of the truncated boundary to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line

the existence and stability of travelling waves with transition layers for some singular cross diffusion systems abstract this paper is concerned with the existence and the stability of travelling waves for a class of quasilinear cross diffusion systems describing two competing or predator prey species firstly by geometric singular perturbation method the existence of travelling waves with transition layers is proved which extends the results of hosono and mimura y hosono m mimura singular perturbation approach to travelling waves in competing and diffusion species models j math kyoto univ 22 1982 435 461 and gardner r a gardner topological methods arising in the study of travelling waves reaction diffusion equations clarendon press oxford 1990 pp 173 198 for non cross diffusion systems and wu y wu the existence of travelling waves for a cross diffusion system with small parameter beijing math 3 1997 74 85 to more general cross diffusion systems applying the stability index method introduced in alexsander et al j alexsander r a gardner c k r t jones a topological invariant arising in the stability analysis of travelling waves j die reine angewandte math 410 1990 167 212 and gardner and jones r a gardner c r k t jones stability of travelling wave solutions of diffusive predator prey systems trans am soc 327 1991 465 524 to the more general eigenvalue problem induced by the quasilinear cross diffusion systems by detailed spectral and topological analysis the travelling waves with transition layers for the cross diffusion systems are proved to be stable which also extends the results of gardner and jones r a gardner c r k t jones stability of travelling wave solutions of diffusive predator prey systems trans am soc 327 1991 465 524 to the more general systems

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

construction and asymptotic stability of structurally stable internal layer solutions we introduce a geometric asymptotic method to treat structurally stable internal layer solutions we consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability proofs of the existence of exact solutions and eigenvalue eigenfunctions are outlined multi layered solutions are constructed by a new shooting method through a sequence of pseudo poincare mappings that do not require the transversality of the flow to cross sections the critical eigenvalues are determined by a coupling matrix that generates the slep matrix the transversality of the shooting method is related to the nonzeroness of the critical eigenvalues an equivalent approach is given to mono layer solutions they can be determined by the intersection of a fast jump surface and a slow switching curve which reduces fenichel s transversality condition to the slow manifold the critical eigenvalue is determined by the angle of the intersection we present three examples the first treats the critical eigenvalues of the system studied by angenent mallet paret peletier the second shows that a key lemma in the slep method may not hold the third is a perturbed activator inhibitor system that can have any number of mono layer solutions some of the solutions can only be found with the new shooting method

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

persistence of overflowing manifolds for semiflow this paper which is a sequel to a previous one 4 by the same authors is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a banach space we consider a c1 semiflow defined on a banach space x that is it is continuous on 0 x and for each t 0 t t x x is c1 and t t t x t t s x for all t s 0 and x x a typical example is the solution operator for a differential equation in 4 we proved that a compact normally hyperbolic invariant manifold m persists under small c1 perturbations in the semiflow we also showed that in a neighborhood of m there exist a center stable manifold and a center unstable manifold that intersect in the manifold m in 4 the compactness and invariance of the manifold m were important assumptions in the present paper we study the more general case where the manifold m is overflowing negatively invariant and the semiflow crosses the boundary transversally or inflowing positively invariant and the semiflow crosses the boundary transversally we do not assume that m is compact or finite dimensional also m is not necessarily an imbedded manifold but an immersed manifold as an example a local unstable manifold of an equilibrium point is an overflowing manifold in brief our main results on the overflowing manifolds may be summarized as follows the precise statements are given in section 2 we assume that the immersed manifoldm does not twist very much locally m is covered by the image under t t of a subset a positive distance away from boundary dt t contracts along the normal direction and does so more strongly than it does along the tangential direction and dt t has a certain uniform continuity in a neighborhood of m if the c1 perturbation t t of t t is sufficiently close to t t then t t has a unique c1 immersed overflowing manifold m nearm furthermore if t t isck and a spectral gap condition holds then m isck similar results for inflowing manifolds are also obtained and given in section 7

laguerre approximation of stable manifolds with application to connecting orbits we present an algorithm based on approximation by laguerre polynomials for computing a point on the stable manifold of a stationary solution of an autonomous system a superconvergence phenomenon means that the accuracy of our results is much higher than the usual spectral accuracy both the theory and the implementation of the method are considered finally as an application of the algorithm we describe a fully spectral approximation of homo and heteroclinic orbits

bifurcation of traveling waves in extrinsic semiconductors abstract we analyze the bifurcation of traveling waves in a standard model of electrical conduction in extrinsic semiconductors in scaled variables the corresponding traveling wave problem is a singularly perturbed nonlinear three dimensional o d e system the relevant bifurcation parameters are the wave speed s and the total current j by means of geometric singular perturbation theory it suffices to analyze a two dimensional reduced problem depending on the relative size of s and a dimensionless small parameter different types of traveling waves exist for 0 s the only waves are fronts corresponding to heteroclinic orbits for s similar fronts but with left and right states reversed exist the transition between these regimes occurs for s o in a complicated global bifurcation involving a hopf bifurcation bifurcation of multiple periodic orbits and heteroclinic and homoclinic bifurcations we present a consistent bifurcation diagram which is confirmed by numerical computations

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

numerical computation of viscous profiles for hyperbolic conservation laws viscous profiles of shock waves in systems of conservation laws can be viewed as heteroclinic orbits in associated systems of ordinary differential equations ode in the case of overcompressive shock waves these orbits occur in multi parameter families we propose a numerical method to compute families of heteroclinic orbits in general systems of ode the key point is a special parameterization of the heteroclinic manifold which can be understood as a generalized phase condition in the case of shock profiles this phase condition has a natural interpretation regarding their stability we prove that our method converges and present numerical results for several systems of conservation laws these examples include traveling waves for the navier stokes equations for compressible viscous heat conductive fluids and for the magnetohydrodynamics equations for viscous heat conductive electrically resistive fluids that correspond to shock wave solutions of the associated ideal models i e the euler resp lundquist equations

smooth schur factorizations in the continuation of separatrices numerical computation of separatrices as general connecting orbits in dynamical systems is performed and their continuation as problem parameters vary is approached via a direct application of smooth block schur factorizations of jacobians and monodromy matrix functions several numerical examples are presented to illustrate the effectiveness of the algorithms used

single and multiple pulse waves for the fitzhugh nagumo the fitzhugh nagumo equations describe the propagation of impulses along a nerve axon we study traveling wave solutions which satisfy a set of ordinary differential equations with a cubic nonlinearity f v v 1 v v a our emphasis is on solutions which tend to zero at pm infty an improved proof of a previous result that there are at least two such solutions is presented further by allowing the middle root v a of the nonlinearity to be a small parameter better estimates of the parameter range where such solutions exist are given in view of recent work of evans fenichel feroe and sil nikov our results imply that there are a great many more bounded traveling waves than was known previously in particular there is an infinite family of periodic waves all traveling with the same speed a reasonable transversality hypothesis introduced by evans fenichel and feroe implies also that there are multiple pulse waves at speeds close to that of a single pulse

targeting functional motifs of a protein family the structural organization of a protein family is investigated by devising a method based on the random matrix theory rmt which uses the physiochemical properties of the amino acid with multiple sequence alignment a graphical method to represent protein sequences using physiochemical properties is devised that gives a fast easy and informative way of comparing the evolutionary distances between protein sequences a correlation matrix associated with each property is calculated where the noise reduction and information filtering is done using rmt involving an ensemble of wishart matrices the analysis of the eigenvalue statistics of the correlation matrix for the lactamase family shows the universal features as observed in the gaussian orthogonal ensemble goe the property based approach captures the short as well as the long range correlation approximately following goe between the eigenvalues whereas the previous approach treating amino acids as characters gives the usual short range correlations while the long range correlations are the same as that of an uncorrelated series the distribution of the eigenvector components for the eigenvalues outside the bulk rmt bound deviates significantly from rmt observations and contains important information about the system the information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate which shows that for the lactamase family the smallest eigenvectors low eigenmodes are highly localized as well as informative these small eigenvectors when processed gives clusters involving positions that have well defined biological and structural importance matching with experiments the approach is crucial for the recognition of structural motifs as shown in lactamase and other families and selectively identifies the important positions for targets to deactivate activate the enzymatic actions

computing connecting orbits via an improved algorithm for continuing invariant subspaces a successive continuation method for locating connecting orbits in parametrized systems of autonomous odes was considered in numer algorithms 14 1997 pp 103 124 in this paper we present an improved algorithm for locating and continuing connecting orbits which includes a new algorithm for the continuation of invariant subspaces the latter algorithm is of independent interest and can be used in contexts different than the present one

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

infinite wavelength analysis for two phase flow a three parameter computer assisted study of global bifurcations abstract a computer assisted analysis of the long wave solutions of a two phase flow model in packed beds is presented based on the numerical study of global bifurcations a time dependent pulsing flow arises spontaneously for relatively high flow rates of gas and liquid through a packed bed a recently developed nonlinear model for this phenomenon invokes a travelling wave assumption and associates the observed wavelike pulses with the existence of long period periodic solutions in the travelling wave frame in parameter space such periodic solutions are located in the neighborhood of certain infinite period global bifurcations such as homoclinic and double heteroclinic loops in order to completely map out the travelling wave flow behavior for a given set of fluids and packing it is necessary to perform a three parameter analysis of these global bifurcations using techniques for the continuation of homoclinic and heteroclinic connections we have examined the relation between homoclinic heteroclinic both codimension 1 and double heteroclinic codimension 2 connections in the model in parameter space the families of these bifurcations interact and undergo further qualitative changes these changes include a crossing of homoclinic connection branches and the birth of a pair of double heteroclinic connections we discuss briefly how double heteroclinic loops apparently arise from a local codimension three singularity where a double zero eigenvalue occurs simultaneously with the vanishing of a higher second order term we also discuss the relation between the nature of the predicted solutions and experimental observations this computational study of global bifurcations provides a concise description of the operating region in which pulsing flow can be observed in packed beds

stabilisation of timoshenko beam system with delay in the boundary control in this paper we consider the exponential stabilisation problem of a timoshenko beam with delay in boundary control suppose that the controller outputs of the forms 1u1 t 1u1 t and 2u2 t 2u2 t where u1 t and u2 t are the inputs of boundary controllers in the past most of the stabilisation results are required j j 0 j 1 2 in the present paper we shall give a new dynamic feedback control law that makes the system exponential stabilisation 0 provided that j j j 1 2 the exponential stabilisation result is proved via test of exact observability of the system

on the semi discretization of optimal control problems for networks of elastic strings global optimality systems and domain decomposition we consider a star graph as an examplary network with elastic strings stretched along the edges the network is allowed to perform out of the plane displacements we consider such networks as being controlled at its simple nodes via dirichlet conditions the objective is to steer given initial data to final target data in a given time t with minimal control costs this problem is discussed in the continuous as well as in the discrete case we discuss an iterative domain decomposition technique and its discrete analogue we prove convergence and show some numerical results

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

spectral stability of shock waves associated with not genuinely nonlinear modes we study viscous shock waves that are associated with a simple mode r of a system ut f u x uxx of conservation laws and that connect states on either side of an inflection hypersurface in state space at whose points r 0 and r 2 0 such loss of genuine nonlinearity the simplest example of which is the cubic scalar conservation law ut u3 x uxx occurs in many physical systems we show that such shock waves are spectrally stable if their amplitude is sufficiently small the proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory well chosen rescalings are crucial for resolving degeneracies by results of zumbrun the spectral stability shown here implies nonlinear stability of these shock waves

efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons a numerical method is set out which efficiently computes stationary z independent two and three dimensional spatiotemporal solitons in second harmonic generating media the method relies on a chebyshev decomposition with an infinite mapping bunching the collocation points near the soliton core known results for the type i interaction are extended and a stability boundary is found by two parameter continuation as defined by the vakhitov kolokolov criteria the validity of this criterion is demonstrated in 2 1 dimensions by simulation and direct calculation of the linear spectrum the method has wider applicability for general soliton bearing equations in 2 1 and 3 1 dimensions

numerical computation of solitary waves in infinite cylindrical domains the numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated rather than solving on the infinite cylinder the equation is approximated by a boundary value problem on a finite cylinder convergence and stability results for this approach are given it is also shown that galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder in addition it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure finally the theoretical predictions are compared with numerical computations in particular post buckling of an infinitely long cylindrical shell under axial compression is considered it is shown numerically that for a fixed spatial truncation the error in the truncation scales with the length of the cylinder as predicted theoretically

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

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

theory and application of a nongeneric heteroclinic loop bifurcation homoclinic and heteroclinic bifurcations from a heteroclinic loop are considered the system under consideration has three parameters two of which are not suitable for generic unfoldings analytical criteria in terms of derivatives to melnikov s functions are given for nongeneric parameters four qualitatively distinct bifurcation diagrams are obtained the result is used to give an explanation of a numerical finding on the generation of traveling pulsing waves in a two phase flow problem

periodically generated propagating pulses certain equations with integral constraints have as solutions time periodic pulses of a fieldlike unknown while a currentlike unknown oscillates periodically with time a general asymptotic theory of this phenomenon the generalized gunn effect has been found recently here we extend this theory to the case of nonlinearities having only one stable zero which is the case for the usual gunn effect in n gaas our ideas are presented in the context of a simple scalar model where the waves can be constructed analytically and explicit expressions for asymptotic approximations can be found

an analytical and numerical study of the bifurcations in a system of linearly coupled oscillators abstract we study a two parameter family of ordinary differential equations in r 4 that governs the dynamics of two coupled planar oscillators each oscillator has a unique periodic solution that is attracting and the uncoupled product system has a unique invariant torus that is attracting the torus persists for weak coupling and contains two periodic solutions when the coupling is linear and conservative one of these in which the oscillators are synchronized persists and is stable for all coupling strengths the other in which the oscillators are radiant out of phase disappears either in a hopf bifurcation or when fixed points appear on the orbit at a critical ratio of the coupling strength to the frequency the out of phase oscillation is unstable except on an open set in the frequency coupling strength plane which contains moderate values of both parameters furthermore there are tori bifurcating from the out of phase solution which means according to the arnol d theory for hopf bifurcations in maps that there may be periodic solutions of arbitrarily large period and chaotic solutions as well numerous other bifurcations occur and there are a number of higher codimension singularities in a large region of the frequency coupling parameter plane stable steady states coexist with stable periodic solutions

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

finite difference schemes on quasi uniform grids for bvps on infinite intervals abstract the classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point the so called truncated boundary a truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach on the other hand the free boundary approach overcomes the need for a priori definition of the truncated boundary in fact in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution in this paper we consider a different way to overcome the introduction of a truncated boundary namely non standard finite difference schemes defined on quasi uniform grids a quasi uniform grid allows us to describe the infinite domain by a finite number of intervals the last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly we apply the proposed approach to the falkner skan model and to a problem of interest in foundation engineering the obtained numerical results are found in good agreement with those available in literature moreover we provide a simple way to improve the accuracy of the numerical results using richardson s extrapolation finally we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

existence and stability of traveling wave solutions for a population genetic model via singular perturbations using singularr perturbation methods the existence and stability of traveling wave solutions for a density dependent selection migration model in population genetics is proved single locus and two alleles are assumed and it is also assumed that the fitnesses of the heterozygotes in the population are close to but below those of the homozygotes unlike previous models this paper does not assume that the population is in hardy weinberg equilibrium

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

a singular approach to discontinuous vector fields on the plane in this paper we deal with discontinuous vector fields on r 2 and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation the regularization process developed by sotomayor and teixeira is crucial for the development of this work

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

a method for locating symmetric homoclinic orbits using symbolic dynamics in this paper we present a method which can identify and locate symmetric homoclinic orbits in a homoclinic tangle formed by the intersecting stable and unstable manifolds of a symmetric 2d map the method consists of a systematic search in parameter space and determination of the order in which these orbits arise using symbolic dynamics each orbit corresponds to a unique sequence and it is computed by iterating the map along the unstable manifold to match a specific symmetry at the middle of the orbit an application of the method to the determination of multibreather solutions of 1d lattices is discussed

dynamic domain decomposition of optimal control problems for networks of strings and timoshenko beams we consider general networks of strings and or timoshenko beams we apply controls at boundary nodes of the network and want to minimize some cost function along part of the structure optimality systems for the entire structure are far too complex to compute in reasonable time in particular in real time applications one wants to reduce the size of the problem thus dynamic decomposition into its physical elements appears to be a natural approach we show how to iteratively decompose the global optimality system into a system related to a substructure then we interpret the local system as an optimality system corresponding to an optimal control problem for the substructure and finally we show convergence of the outer iteration

the gunn effect instability of the steady state and stability of the solitary wave in long extrinsic semiconductors a linear stability analysis of the stationary solution of a one dimensional drift diffusion model used to describe the gunn effect in gaas is performed it is shown that for long semiconductor samples under do voltage bias conditions and small diffusivity the steady state may lose stability via a hopf bifurcation in the limit of infinitely long samples there is a quasicontinuum of oscillatory modes of the equation linearized about the steady state that a acquire positive real part for voltages larger than a certain critical value the linear stability of the solitary wave characteristic of the gunn effect is proved for an idealized electron velocity curve in the zero diffusion limit

analysis of predator prey models with continuous threshold harvesting abstract we formulate an alternative form of threshold harvesting and investigate its properties within the framework of a predator prey model our formulation accounts for economic constraints and is defined as a continuous harvesting function of the predator species theoretical and numerical analysis indicate that our formulation which we call continuous threshold policy ctp can help improve undesirable behavior of the predator prey ecosystem our study includes uniform boundedness of solutions stability of equilibria and periodic orbits bifurcations and heteroclinic orbits

time domain decomposition of optimal control problems for the wave equation we consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time domain decomposition of the corresponding optimality system we develop an iterative algorithm which shows that the decomposed optimality system corresponds to local in time optimal control problems which can be treated in parallel we show convergence of the algorithm finally we provide a time discretization which is reminiscent of an instantaneous control scheme we thereby also contribute to the problem of convergence of such schemes

stability of pulses in nonlinear optical fibers using phase sensitive amplifiers we consider the stability of soliton like pulses propagating in nonlinear optical fibers with periodically spaced phase sensitive amplifiers a situation where the averaged pulse evolution is governed by a fourth order nonlinear diffusion equation similar to the kuramoto sivashinsky or swift hohenberg equations a bifurcation and stability analysis of this averaged equation is carried out and in the limit of small amplifier spacing a steady state pulse solution is shown to be asymptotically stable furthermore both a saddle node bifurcation and a subcritical bifurcation from the zero solution are found analytical results are confirmed using the bifurcation software package auto the analysis provides evidence for the existence of stable pulse solutions for a wide range of parameter values including those corresponding to physically realizable soliton communications systems

a hermite spectral method for the computation of homoclinic orbits and associated functionals we present a spectral method for the computation of homoclinic orbits in ordinary differential equations the method is based on hermite fourier expansions of the complete homoclinic solution and exhibits exponential convergence in addition our method can be used to approximate nonlinear functionals which depend on the complete homoclinic solution this is demonstrated using examples from phase separation dynamics and metastability