backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

a simpler dgmres dgmres is an algorithm to solve the drazin inverse solution of large scale and sparse consistent or inconsistent singular linear systems with an arbitrary index in this paper we present a simpler version of this algorithm

existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems abstract this paper presents the existence of si lnikov orbits in two different chaotic systems belong to the class of lorenz systems more exactly in the lu system and in the zhou s system both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z axis by using the undetermined coefficient method the existence of the homoclinic orbit for the zhou s system has been proven also by using the undetermined coefficient method as a result the si lnikov criterion along with some technical conditions guarantees that lu and zhou s systems have both smale horseshoes and horseshoe type of chaos moreover the geometric structures of attractors are determined by these heteroclinic orbits

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

bifurcation analysis of the generalized stretch twist fold flow based on the stretch twist fold flow a generalized stretch twist fold flow is introduced by choosing an appropriate bifurcation parameter hopf bifurcations occur in this system when the bifurcation parameter exceeds a critical value the formulae for determining the direction of the hopf bifurcations and the stability of bifurcating periodic solutions are presented in addition the paper also investigates the bifurcations of the heteroclinic orbits for this system the existence and its associated existing regions are given for two heteroclinic orbits respectively finally some numerical simulations for justifying the theoretical analysis are presented

comment on condition number of drazin inverse and their condition numbers of singular linear systems abstract in this paper we establish the condition number of drazin inverse of a singular matrix a where r a k r a k and k index a by the schur decomposition based on this form the spectral norm and frobenius norm of relative condition number for the drazin inverse and level 2 condition number of the drazin inverse are characterized the sensitivity for the drazin inverse solution of singular systems is also presented we obtain several formulas for the condition number of the drazin inverse by spectral norm instead of the p norm where p is a transformation matrix of the jordan canonical form of a thereby improving the earlier work y wei g wang d wang condition number of drazin inverse and their condition numbers of singular linear systems appl math comput 146 2003 455 467

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

asymptotic expansions of the global error of fixed stepsize methods in his fundamental paper on general fixed stepsize methods skeel 6 studied convergence properties but left the existence of asymptotic expansions as an open problem in this paper we give a complete answer to this question for the special cases of one step and linear multistep methods our proof is shorter than the published ones asymptotic expansions are the theoretical base for extrapolation methods

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

conserved quantities of some hamiltonian wave equations after full discretization hamiltonian pdes have some invariant quantities which would be good to conserve with the numerical integration in this paper we concentrate on the nonlinear wave and schrodinger equations under hypotheses of regularity and periodicity we study how a symmetric space discretization makes that the space discretized system also has some invariants or nearly invariants which well approximate the continuous ones we conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic runge kutta methods or symmetric linear multistep methods for second order systems

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

discretizations integrable systems and computation discretizations and associated numerical computation of solutions of certain integrable systems such as the nonlinear schrodinger equation nls and the sine gordon sg equations with periodic boundary values can lead to instabilities chaotic and spurious results the chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds by using the associated nonlinear spectral transform of the nls equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even for general initial values even though there is no crossing of the unperturbed homoclinic manifolds the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum this paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and runge kutta algorithms for the nls and sg equations in regions of phase space proximate to the homoclinic manifolds

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

invariants and numerical methods for odes the preservation of some invariants is important during the numerical integration of odes in some cases failure to maintain certain invariants leads to physically impossible solutions in other cases to instability some authors report that the long term macroscopic characteristics of a solution are better represented if certain invariants are maintained this paper considers two types of invariants point invariants which are functions along a particular trajectory and differential invariants which are invariant relations between neighboring trajectories such as symplectic invariants conventional numerical methods usually introduce o hp 1 errors in an invariant at each numerical step and these will accumulate over the integration interval two approaches to preventing this accumulation are 1 to find methods that maintain the invariants within round off error or 2 to find methods that maintain another invariant which is no more than o hq different from the invariant satisfied by the ode these approaches will be illustrated with some examples the major objective of this paper is to consider the potential of general methods that is methods that do not depend on the specific differential equation to maintain the invariants

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

conservative numerical methods for solitary wave interactions the purpose of this paper is to show the advantages that represent the use of numerical methods that preserve invariant quantities in the study of solitary wave interactions for the regularized long wave equation it is shown that the so called conservative methods are more appropriate to study the phenomenon and provide a dynamic point of view that allows us to estimate the changes in the parameters of the solitary waves after the collision

a variant algorithm of the orthomin m method for solving linear systems abstract we propose a variant of o rthomin m for solving linear systems a x b it is mathematically equivalent to the original o rthomin m method but uses recurrence formulas that are different from those of o rthomin m they contain alternative expressions for the auxiliary vectors and the recurrence coefficients our implementation has the same computational costs as o rthomin m as a result of numerical experiments on nonsingular linear systems we have confirmed the equivalence of our proposed variant of o rthomin m with the original o rthomin m using finite precision arithmetic numerical experiments on singular linear systems show that our proposed algorithm is more accurate and less affected by rounding errors than the original o rthomin m

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

on existence of homoclinic orbits for some types of autonomous quadratic systems of differential equations abstract the new existence conditions of homoclinic orbits for the system of ordinary quadratic differential equations are founded further the realization of these conditions together with the shilnikov homoclinic theorem guarantees the existence of a chaotic attractor at 3d autonomous quadratic system examples of the chaotic attractors are given

invariant conserving finite difference algorithms for the nonlinear klein gordon equation a formalism for systematicaiiy deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear pdes is presented three algorithms are derived for the nonlinear klein gordon equation nlkge based on the proposed formalism the local conservation laws of the nlkge form the basic starting point in our derivation which hinges essentially on the commutativity of certain finite difference operators such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the pde level with appropriate boundary conditions the proposed algorithms preserve in the discrete sense either the total system energy or the system s linear momentum several variants of the present algorithms and their relation to previously proposed algorithms are discussed an analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms the preservation of energy of the present algorithms for the nlkge can also be viewed as providing a method of stabilization for conditionally stable algo thms for the linear wave equation the computer impiementation of the proposed algorithms with the treatment of the boundary conditions is presented in detail numerical examples are given concerning soliton collisions in the sine gordon equation the double sine gordon equation and the cb phi four equation the numerical results demonstrate that the present algorithms can preserve accurately up to 10 decimal digits the total system energy for a very coarse grid reliable algorithms for josephson junction models which contain dissipation damping mechanisms and driving bias current are obtained as direct by products of the proposed invariant conserving algorithms for the nlkge even though presented mainly for the 1 d case the proposed algorithms are generalizable to the 2 d and 3 d cases and to the case of complex valued nlkge

existence of heteroclinic orbits of the shil nikov type in a 3d quadratic autonomous chaotic system in this paper the existence of heteroclinic orbits of shil nikov type in a three dimensional quadratic autonomous system is proved four heteroclinic orbits and four critical points together constitute two cycles simultaneously the dynamical behaviors of the system are also studied

comment on existence of heteroclinic orbits of the shilʼnikov type in a 3d quadratic autonomous chaotic system j math anal appl 315 2006 106 119 abstract in the referenced paper the existence of heteroclinic orbits of shil nikov type is proved by using the undetermined coefficient method in a three dimensional quadratic autonomous system here we show that the demonstration is incorrect a misuse of a time reversibility property leads the authors to consider an odd expression for the first component of the heteroclinic connection it is evident that this odd expression can never represent the first component of a shil nikov heteroclinic connection an orbit which is necessarily non symmetric

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

explicit finite difference methods for the ew and rlw equations an extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal width ew and regularized long wave rlw equations is reported such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing two of the methods presented here make use of three point fourth order accurate finite difference formulae for the first and second order spatial derivatives two methods are based on the analytical solution of second order ordinary differential equations which have locally exponential solutions and the fourth technique is a standard finite difference scheme a linear stability analysis of the four methods is presented it is shown that for the ew and rlw equations a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations the latter are reported to be more accurate than exponential techniques that employ second order accurate approximations and these in turn are more accurate than the standard explicit method

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

s i lnikov type orbits of lorenz family systems this paper studies the lorenz family system which is known to establish a topological connection among the lorenz chen and lii systems in the parametric space the existence of si inikov heterclinic orbits is proved using an undetermined coefficient method as a consequence the si inikov criterion along with some technical conditions guarantees that the lorenz family system has both smale horseshoes and horseshoe type of chaos it is this heteroclinic orbit that determines the geometric structure of the corresponding chaotic attractor c 2006 elsevier b v all rights reserved

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

general local energy preserving integrators for solving multi symplectic hamiltonian pdes in this paper we propose and investigate a general approach to constructing local energy preserving algorithms which can be of arbitrarily high order in time for solving hamiltonian pdes this approach is based on the temporal discretization using continuous runge kutta type methods and the spatial discretization using pseudospectral methods or gauss legendre collocation methods the local energy conservation law of our new schemes is analyzed in detail the effectiveness of the novel local energy preserving integrators is demonstrated by coupled nonlinear schrodinger equations and 2d nonlinear schrodinger equations with external fields our new schemes are compared with some classical multi symplectic and symplectic schemes in numerical experiments the numerical results show the remarkable long term behaviour of our new schemes

perturbation analysis of generalized saddle point systems we discuss the perturbation analysis of generalized saddle point systems in this paper we give the nonlinear perturbation bounds then derive the condition numbers and analyze the sensitivity of the computed solutions

a note on solving ep inconsistent linear systems in this note we reconsider the problem posed in s l zhang y oyanagi m su gihara necessary and sufficient conditions for the convergence of orthomin k on singular and inconsistent linear systems numer math 87 2000 391 405 we give a stopping criterion for the iterative methods to ep inconsistent linear systems

necessary and sufficient conditions for the convergence of orthomin k on singular and inconsistent linear systems we consider the convergence of orthomin k on singular and inconsistent linear systems criteria for the breakdown of orthomin k are discussed and analyzed moreover necessary and sufficient conditions for the convergence of orthomin k for any right hand side are given and a rate of convergence is provided as well finally numerical experiments are shown to confirm the convergence theorem

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

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

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

comment on silnikov type orbits of lorenz family systems physica a 375 2007 438 446 abstract in the commented paper the authors consider a four parameter family of three dimensional autonomous systems that includes the widely studied systems of lorenz chen and lu their goal is to rigorously prove the existence of silnikov chaos in this lorenz family via the silnikov criterion they use an undetermined coefficient method to demonstrate the presence of heteroclinic orbits in this comment we point out that their analysis is erroneous the odd function they consider for the first component of the silnikov heteroclinic orbit cannot represent such global connection thus the main result of their paper theorem 2 is incorrect

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

time behaviour of the error when simulating finite band periodic waves the case of the kdv equation this paper is devoted to study the error growth of numerical time integrators for n phase or n band quasi periodic in time solutions of the periodic korteweg de vries equation it is shown that the preservation through numerical time integration of conserved quantities of the periodic problem of the equation may be an element to take into account in the selection of the numerical method we explain why the inclusion of these properties of conservation provides a better error propagation in particular we emphasize how the preservation of invariants makes influence in the simulation of some physical parameters of the waves

comment on existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems appl math comput 218 2012 11859 11870 abstract in the commented paper the authors claim to have proved the existence of heteroclinic and homoclinic orbits of silnikov type in two lorenz like systems the so called lu and zhou systems according to them they have analytically demonstrated that both systems exhibit smale horseshoe chaos unfortunately we show that the results they obtain are incorrect in the proof they use the undetermined coefficient method introduced by zhou et al in chen s attractor exists int j bifurcation chaos 14 2004 3167 3178 a paper that presents very serious shortcomings however it has been cited dozens of times and its erroneous method has been copied in lots of papers including the commented paper where a misuse of a time reversibility property leads the authors to use an odd even expression for the first component of the heteroclinic homoclinic connection it is evident that this odd even expression cannot represent the first component of a silnikov heteroclinic homoclinic connection an orbit which is necessarily non symmetric consequently all their results stated in theorems 3 5 are invalid

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions