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

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

existence of heteroclinic orbits for systems satisfying monotonicity conditions we prove existence of heteroclinic orbits for systems of ordinary differential equations satisfying monotonicity conditions the proof is carried out by means of the implicit function theorem we apply these results to prove existence of travelling waves for the system describing a multicomponent plasma sustained by a laser beam

hamiltonian systems near relative equilibria abstract we give explicit differential equations for the dynamics of hamiltonian systems near relative equilibria these split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit the form of the differential equations that is inherited from the symplectic structure and symmetry properties of the hamiltonian system is analysed and the effects of time reversing symmetries are included the results will be applicable to the stability and bifurcation theories of relative equilibria of hamiltonian systems

isometric embeddings of compact spaces into banach spaces abstract we show the existence of a compact metric space k such that whenever k embeds isometrically into a banach space y then any separable banach space is linearly isometric to a subspace of y we also address the following related question if a banach space y contains an isometric copy of the unit ball or of some special compact subset of a separable banach space x does it necessarily contain a subspace isometric to x we answer positively this question when x is a polyhedral finite dimensional space c 0 or l 1

the existence of transverse homoclinic solutions for higher order equations abstract differential equations are considered which contain a small parameter when the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained the technique follows the work of palmer using exponential dichotomies and the method of lyapunov schmidt

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

degeneracy of the action frequency map a mechanism for homoclinic bifurcation of invariant tori a family of smooth invariant tori of a hamiltonian system can be parameterized by the values of the actions or the frequencies these parameterizations are related by the action frequency map the purpose of this paper is to show that when the action frequency map is degenerate it signals a homoclinic bifurcation remarkably the nonlinear properties of this homoclinic bifurcation to invariant tori are determined by the curvature of the action frequency map a homoclinic angle is also generated which is analogous to a hannay berry phase shift the theory is constructive and so can usefully be combined with computation some implications for quantization and the generation of solitary waves are also discussed

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

degenerate hyperbolic conservation laws with dissipation reduction to and validity of a class of burgers type equations a conservation law is said to be degenerate or critical if the jacobian of the flux vector evaluated on a constant state has a zero eigenvalue in this paper it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles burger s dynamics the case of k fold degeneracy is also treated and it is shown that it leads to a reduction to a quadratically coupled k fold system of burgers type equations validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces thereby capturing both finite and infinite energy solutions the theory is applied to some examples from stratified shallow water hydrodynamics that model the birth of hydraulic jumps

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

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

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

degenerate relative equilibria curvature of the momentum map and homoclinic bifurcation abstract a fundamental class of solutions of symmetric hamiltonian systems is relative equilibria in this paper the nonlinear problem near a degenerate relative equilibrium is considered the degeneracy creates a saddle center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit the surprising result is that the curvature of the pullback of the momentum map to the lie algebra determines the normal form for the homoclinic bifurcation there is also an induced directional geometric phase in the homoclinic bifurcation the backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map the theory is constructive and generalities are given for symmetric hamiltonian systems on a vector space of dimension 2 n 2 with an n dimensional abelian symmetry group examples for n 1 2 3 are presented to illustrate application of the theory

covering a compact set in a banach space by an operator range of a banach space with basis a banach space x has the approximation property if and only if every compact set in x is in the range of a one to one bounded linear operator from a space that has a schauder basis characterizations are given for l p spaces and quotients of l p spaces in terms of covering compact sets in x by operator ranges from l p spaces a banach space x is a l 1 space if and only if every compact set in x is contained in the closed convex symmetric hull of a basic sequence which converges to zero

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

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

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

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

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

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

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

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

multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems abstract sensitivity analysis methods are important tools for research and design with simulations many important simulations exhibit chaotic dynamics including scale resolving turbulent fluid flow simulations unfortunately conventional sensitivity analysis methods are unable to compute useful gradient information for long time averaged quantities in chaotic dynamical systems sensitivity analysis with least squares shadowing lss can compute useful gradient information for a number of chaotic systems including simulations of chaotic vortex shedding and homogeneous isotropic turbulence however this gradient information comes at a very high computational cost this paper presents multiple shooting shadowing mss a more computationally efficient shadowing approach than the original lss approach through an analysis of the convergence rate of mss it is shown that mss can have lower memory usage and run time than lss

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

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

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

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

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

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

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

periodic orbits of hamiltonian systems and symplectic reduction the notion of relative periodic orbits for hamiltonian systems with symmetry is discussed and a correspondence between periodic orbits of reduced and unreduced hamiltonian systems is established variational principles with symmetries are studied from the point of view of symplectic reduction of the space of loops leading to a characterization of reduced periodic orbits by means of the critical subsets of an action functional restricted to a submanifold of the loop space of the unreduced manifold finally as an application it is shown that if the symplectic form has finite integral rank then the periodic orbits of a hamiltonian system on the symplectic manifold admit a variational characterization

relative normal modes for nonlinear hamiltonian systems an estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a hamiltonian system with continuous symmetry is given this result constitutes a generalization to the hamiltonian symmetric framework of a classical result by weinstein and moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium the estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed spatiotemporal isotropy subgroup moreover it is easily computable in particular examples it is interesting to see how in our result the existence of non trivial relative periodic orbits requires generic conditions on the higher order terms of the taylor expansion of the hamiltonian function in contrast with the purely quadratic requirements of the weinstein moser theorem which emphasizes the highly non linear character of the relatively periodic dynamical objects

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

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

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

polynomial normal forms with exponentially small remainder for analytic vector fields a key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms invented by poincare which gives simple forms to which a vector field can be reduced close to the equilibrium in the class of formal vector valued vector fields the problem can be easily solved whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order this paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in r m more precisely we prove that for any vector field in r m admitting the origin as a fixed point with a semi simple linearization the order of the normal form can be optimized so that the remainder is exponentially small we also give several examples of non semi simple linearization for which this result is still true

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

on unitary representability of topological groups we prove that the additive group e k e of an mathcal l infty banach space e with the topology k e of uniform convergence on compact subsets of e is topologically isomorphic to a subgroup of the unitary group of some hilbert space is unitarily representable this is the same as proving that the topological group e k e is uniformly homeomorphic to a subset of ell 2 kappa for some as an immediate consequence preduals of commutative von neumann algebras or duals of commutative c algebras are unitarily representable in the topology of uniform convergence on compact subsets the unitary representability of free locally convex spaces and thus of free abelian topological groups on compact spaces follows as well the above facts cannot be extended to noncommutative von neumann algebras or general schwartz spaces

bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations the existence of periodic solutions are studied for certain differential inclusions applications are given to ordinary differential equations with multivalued perturbations embodying also dry friction problems singularly perturbed differential inclusions are investigated as well

adjoint sensitivity analysis of chaotic dynamical systems with non intrusive least squares shadowing abstract this paper presents a discrete adjoint version of the recently developed non intrusive least squares shadowing nilss algorithm which circumvents the instability that conventional adjoint methods encounter for chaotic systems the nilss approach involves solving a smaller minimization problem than other shadowing approaches and can be implemented with only minor modifications to preexisting tangent and adjoint solvers adjoint nilss is demonstrated on a small chaotic ode a one dimensional scalar pde and a direct numerical simulation dns of the minimal flow unit a turbulent channel flow on a small spatial domain this is the first application of an adjoint shadowing based algorithm to a three dimensional turbulent flow

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

stability of gasless combustion fronts in one dimensional solids for gasless combustion in a one dimensional solid we show a type of nonlinear stability of the physical combustion front if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm provided the linearized operator has no eigenvalues in the right half plane other than zero using the evans function we show that the zero eigenvalue must be simple factors that complicate the analysis are 1 the linearized operator is not sectorial and 2 the linearized operator has good spectral properties only when the weighted norm is used but then the nonlinear term is not lipschitz the result is nevertheless physically natural to prove it we first show that when the weighted norm is used the semigroup generated by the linearized operator decays on a subspace complementary to the operator s kernel by showing that it is a compact perturbation of the semigroup generated by a more easily analyzed triangular operator we then use this result to help establish that solutions stay small in the spatially uniform norm which in turn helps establish nonlinear convergence in the weighted norm

a new method to find homoclinic and heteroclinic orbits a new series method is provided for continuous time autonomous dynamical systems which can find exact orbits as opposed to approximate ones the method can reduce the connecting orbit problem as a boundary value problem in an infinite time domain to the initial value problem it consists of transforming time to the logarithmic scale substituting a power series around each fixed point of interest for each of the unknown functions into the system and equating the corresponding coefficients when solving for the power series coefficients additional parameters are used in order to find the intersections of the unstable manifold and the stable manifold of the equilibria this paper demonstrates how the new method allows to obtain heteroclinic and homoclinic orbits in some well known cases such as nagumo system stretch twist fold flow or mathematical pendulum

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

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