generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

a chebyshev collocation method for solving two phase flow stability problems this paper describes a chebyshev collocation method for solving the eigenvalue problem that governs the stability of parallel two phase flow the method is based on the expansion of the eigenfunctions in terms of chebyshev polynomials point collocation and the subsequent solution of the resulting generalized eigenvalue problem with the qz algorithm we concentrate on the question how to handle difficulties that arise when these standard techniques are applied to the stability problem of a thin film of liquid that is sheared by a gas after discussing this specific problem in detail it is argued that the method of solution can readily be applied to other two phase flow configurations as well

ruga formation instabilities of a graded stiffness boundary layer in a neo hookean solid we present an analysis of ruga formation instabilities arising in a graded stiffness boundary layer of a neo hookean half space caused by lateral plane strain compression in this study we repres

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

numerical solution of non self adjoint sturm liouville problems and related systems this paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of non self adjoint sturm liouville problems we consider even order problems with equally divided separated boundary conditions the method can find the eigenvalues in a rectangle and in a left half plane it combines the argument principle with the compound matrix method using the magnus expansion in some cases the computational cost of compound matrices can be reduced by transforming to a second order vector sturm liouville problem we study the asymptotics of the solutions of the ode for large absolute values of the eigenvalue parameter in order to calculate the eigenvalues in a left half plane the method is applied to the orr sommerfeld equation and other examples

perturbed euler top and bifurcation of limit cycles on invariant casimir surfaces abstract analytical perturbations of the euler top are considered the perturbations are based on the poisson structure for such a dynamical system in such a way that the casimir invariants of the system remain invariant for the perturbed flow by means of the poincare pontryagin theory the existence of limit cycles on the invariant casimir surfaces for the perturbed system is investigated up to first order of perturbation providing sharp bounds for their number examples are given

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

averaging theory at any order for computing periodic orbits abstract we provide a recurrence formula for the coefficients of the powers of e in the series expansion of the solutions around e 0 of the perturbed first order differential equations using it we give an averaging theory at any order in e for the following two kinds of analytic differential equation d x d k 1 e k f k x d x d k 0 e k f k x a planar polynomial differential system with a singular point at the origin can be transformed using polar coordinates to an equation of the previous form thus we apply our results for studying the limit cycles of a planar polynomial differential systems

instability investigation on fluid loaded pre stretched cylindrical membranes this paper discusses the evaluation of instabilities on the quasi static equilibrium path of fluid loaded pre stretched cylindrical membranes and the switching to a secondary branch at a bifurcation point the membrane is represented by only the in plane stress components for which an incompressible isotropic hyperelastic material model is assumed the free inflation problem yields governing equations and boundary conditions which are discretized by finite differences and solved by a newton raphson method an incremental arclength cubic extrapolation method is used to find generalized equilibrium paths with different parametrizations limit points and bifurcation points are observed on the equilibrium path when fluid level is seen as the controlling parameter an eigen mode injection method is employed to switch to a secondary equilibrium branch at the bifurcation point a limit point with respect to fluid level is observed for a partially filled membrane when the aspect ratio length radius is high whereas for smaller aspect ratios the limit point with respect to fluid level is observed at over filling pre stretch is observed to have a stiffening effect in the pre limit zone and a softening effect in the post limit zone

a modified tau spectral method that eliminated spurious eigenvalues abstract a modified tau spectral method is presented which eliminates the spurious eigenvalues produced by the usual tau method the modified tau method essentially involves an appropriate factorization of the differential operators in the eigenvalue problem it is developed for eigenvalue problems posed as single differential equations or as systems of such equations several eigenvalue problems are solved using both the usual and modified chebyshev tau methods including the orr sommerfeld stability equation for plane poiseuille flow the convergence of the modified tau method is shown to be at least as rapid as that of the usual tau method the use of the tau coefficients in indicating convergence is also discussed

evaluation of eigenfunctions from compound matrix variables in non linear elasticity ii sixth order systems we show how the compound matrix method can be extended to give eigenfunctions as well as eigenvalues to bifurcation problems in non linear elasticity the non trivial boundary conditions create some difficulties and we find that sixth order systems for elasticity problems will require a shooting method for two functions of two unknown parameters over and above the calculations required for comparable problems in fluids

a legendre spectral element method for eigenvalues in hydrodynamic stability a legendre polynomial based spectral technique is developed to be applicable to solving eigenvalue problems which arise in linear and nonlinear stability questions in porous media and other areas of continuum mechanics the matrices produced in the corresponding generalised eigenvalue problem are sparse reducing the computational and storage costs where the superimposition of boundary conditions is not needed due to the structure of the method several eigenvalue problems are solved using both the legendre polynomial based and chebyshev tau techniques in each example the legendre polynomial based spectral technique converges to the required accuracy utilising less polynomials than the chebyshev tau method and with much greater computational efficiency

nonlinear euler buckling the buckling of hyperelastic incompressible cylindrical tubes of arbitrary length and thickness under compressive axial load is considered within the framework of nonlinear elasticity analytical and numerical methods for bifurcation are developed using the exact solution of wilkes for the linearized problem within the stroh formalism using these methods the range of validity of the euler buckling formula and its first nonlinear corrections are obtained for third order elasticity the values of the geometric parameters tube thickness and slenderness where a transition between buckling and barrelling is observed are also identified

axisymmetric bifurcations of thick spherical shells under inflation and compression incremental equilibrium equations and corresponding boundary conditions for an isotropic hyperelastic and incompressible material are summarized and then specialized to a form suitable for the analysis of a spherical shell subject to an internal or an external pressure a thick walled spherical shell during inflation is analyzed using four different material models specifically one and two terms in the ogden energy formulation the gent model and an i1 formulation recently proposed by lopez pamies we investigate the existence of local pressure maxima and minima and the dependence of the corresponding stretches on the material model and on shell thickness these results are then used to investigate axisymmetric bifurcations of the inflated shell the analysis is extended to determine the behavior of a thick walled spherical shell subject to an external pressure we find that the results of the two terms ogden formulation the gent and the lopez pamies models are very similar for the one term ogden material we identify additional critical stretches which have not been reported in the literature before

the center problem via averaging method the averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center in this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields

on the eversion of compressible elastic cylinders we consider the eversion problem for compressible hyperelastic isotropic thick walled cylinders we give a new strain energy function for a highly compressible material that admits an exact solution to the problem with point wise end conditions previously only averaged loads could be accounted for secondly we investigate the bifurcation problem for a wide range of material models we find qualitively similar results to those obtained for incompressible materials while thin walled cylinders can be everted into other right circular cylinders if the wall thickness exceeds some critical value bifurcation into non cylindrical shapes is possible

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

congruence method for global darboux reduction of finite dimensional poisson systems a new procedure for the global construction of the casimir invariants and darboux canonical form for finite dimensional poisson systems is developed this approach is based on the concept of matrix congruence and can be applied without the previous determination of the casimir invariants recall that their prior knowledge is unavoidable for the standard reduction methods thus requiring either the integration of a system of pdes or solving some equivalent problem well the opposite in the new congruence method both the darboux coordinates and the casimir invariants arise simultaneously as the outcome of the reduction algorithm in fact the congruence algorithm proceeds only in terms of matrix algebraic transformations and direct quadratures thus avoiding the need of previously integrating a system of pdes and therefore improving previously known approaches physical examples illustrating different aspects of the theory are provided

the origin and nature of spurious eigenvalues in the spectral tau method the chebyshev tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts even when all true eigenvalues of the problem are known to have negative real parts we explain the origin and nature of the spurious eigenvalues in an example problem the explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem

general theory of small elastic deformations superposed on finite elastic deformations using tensor notations a general theory is developed for small elastic deformations of either a compressible or incompressible isotropic elastic body superposed on a known finite deformation without assuming special forms for the strain energy function the theory is specialized to the case when the finite deformation is pure homogeneous when two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials the potential functions are used to solve the problem of the infinitesimally small indentation by a spherical punch of the plane surface of a semi infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force free plane surface

prym map and second gaussian map for prym canonical line bundles abstract we show that the second fundamental form of the prym map lifts the second gaussian map a of the prym canonical bundle we prove by degeneration to binary curves that a is surjective for the general point c a of r g for g 20

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

a spectral galerkin method for the coupled orr sommerfeld and induction equations for free surface mhd we develop and test spectral galerkin schemes to solve the coupled orr sommerfeld and induction equations for parallel incompressible mhd in free surface and fixed boundary geometries the schemes discrete bases consist of legendre internal shape functions supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions the orthogonality properties of the basis polynomials solve the matrix coefficient growth problem and eigenvalue eigenfunction pairs can be computed stably at spectral orders at least as large as p 3000 with p independent roundoff error accuracy is limited instead by roundoff sensitivity due to non normality of the stability operators at large hydrodynamic and or magnetic reynolds numbers re rm 4x10 4 in problems with hartmann velocity and magnetic field profiles we employ suitable gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error an alternative approach which involves approximating the forms by means of legendre gauss lobatto quadrature at the 2p 1 precision level is found to yield equal eigenvalues within roundoff error as a consistency check we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller than 10 5 for the least stable mode in free surface flow at re 3x10 4 moreover we confirm that the computed normal modes satisfy an energy conservation law for free surface mhd with error smaller than 10 6 the critical reynolds number in free surface mhd is found to be sensitive to the magnetic prandtl number pm even at the pm o 10 5 regime of liquid metals

numerical solution of two point boundary value problems abstract this paper presents an original formulation of two point boundary value and eigenvalue problems expressed as a system of first order equations the fundamental difference between the new method and other methods based on a first order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first order equations which are hence never regarded as an initial value problem the consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations furthermore the global character of the integral conditions nonlocality combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by wilkinson the properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability the proposed method is conceptually simpler and numerically more convenient than existing initial value methods while still retaining all the advantages of a formulation based on a first order system

inverse jacobi multiplier as a link between conservative systems and poisson structures some aspects of the relationship between conservativeness of a dynamical system namely the preservation of a finite measure and the existence of a poisson structure for that system are analyzed from the local point of view due to the flow box theorem we restrict ourselves to neighborhoods of singularities in this sense we characterize poisson structures around the typical zero hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non vanishing first jet by connecting with the classical poincar e center problem from the global point of view we connect the property of being strictly conservative the invariant measure must be positive with the existence of a poisson structure depending on the phase space dimension finally weak conservativeness in dimension two is introduced by the extension of inverse jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed examples including lotka volterra systems quadratic isochronous centers and non smooth oscillators are provided

two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems we develop the compound matrix method and the chebyshev tau method to be applicable to linear and nonlinear stability problems for convection in porous media in a natural way it is shown how to obtain highly accurate answers to problems which may be stiff and spurious eigenvalues are avoided a detailed analysis is provided for a porous convection problem of much current interest namely convection with a horizontally varying temperature gradient

schiffer variations and abelian differentials abstract deformations of compact riemann surfaces are considered using a cech cohomology sliding overlaps approach cocycles are calculated for conformal cutting and regluing deformations at zeros of abelian differentials deformations fixing the periods of a differential and deformations splitting zeros are considered a second order deformation expansion is presented for the riemann period matrix a complete deformation expansion is presented for abelian differentials schiffer s kernel function approach for deformations of a green s function is followed

numerical solution of eigenvalue problems using spectral techniques abstract two algorithms based on spectral chebyshev and pseudospectral chebyshev methods are presented for solving difficult eigenvalue problems that are valid over connected domains coupled through interfacial conditions to demonstrate the applicability of these methods we have examined the eigenvalue problems that describe the linear stability of two superposed newtonian and inelastic power law fluids in plane poiseuille flow for a selected range of parameters both algorithms provide accurate results and the pseudospectral code appears to be more efficient in handling linear stability problems

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

a second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter abstract we deal with nonlinear t periodic differential systems depending on a small parameter the unperturbed system has an invariant manifold of periodic solutions we provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation in the end two applications are done the key tool for proving the main result is the lyapunov schmidt reduction method applied to the t poincare andronov mapping

periodic orbits in analytically perturbed poisson systems abstract analytical perturbations of a family of finite dimensional poisson systems are considered it is shown that the family is analytically orbitally conjugate in u r n to a planar harmonic oscillator defined on the symplectic leaves as a consequence the perturbed vector field can be transformed in the domain u to the lagrange standard form on the latter use can be made of averaging theory up to second order to study the existence number and bifurcation phenomena of periodic orbits examples are given ranging from harmonic oscillators with a potential and duffing oscillators to a kind of zero hopf singularity analytic normal form

numerical solution of eigenvalue problem using the compound matrix method abstract an algorithm based on the compound matrix method is presented for solving difficult eigen value problems details are given for systems of ordinary differential equations of fourth order that are valid over connected domains coupled through interfacial conditions as an example we examine the linear stability of two superposed fluids in plane poiseuille flow and consider both interfacial and shear modes the orr sommerfeld system that describes linear stability is solved for a selected range of parameters

removal of spurious modes encountered in solving stability problems by spectral methods a technique based on the galerkin approximation is developed to remove spurious roots arising when chebyshev spectral methods are used to solve eigenvalue problems in hydrodynamic stability the derivation of galerkin chebyshev approximations is explained and numerical results for the orr sommerfeld equations of plane poiseuille flow and a blasius profile are presented in tables and compared with those obtained by the method of zebib 1984 it is pointed out that the present method does not increase the size of the algebraic system to be solved

numerical methods for higher order sturm liouville problems we review some numerical methods for self adjoint and non self adjoint boundary eigenvalue problems

siegel metric and curvature of the moduli space of curves we study the curvature of the moduli space mg of curves of genus g with the siegel metric induced by the period map j mg ag we give an explicit formula for the holomorphic sectional curvature of mg along a schiffer variationp for p a point on the curve x in terms of the holomorphic sectional curvature of ag and the second gaussian map finally we extend the kahler form of the siegel metric as a closed current on m g and we determine its cohomology class as a multiple of

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

nonlinear stability of the one domain approach to modelling convection in superposed fluid and porous layers studies of the nonlinear stability of fluid porous systems have been developed very recently a two domain modelling approach has been adopted in previous works but was restricted to specific configurations the extension to the more general case of a navier stokes modelled fluid over a porous material was not achieved for the two domain approach owing to the difficulties associated with handling the interfacial boundary conditions this paper addresses this issue by adopting a one domain approach where the governing equations for both regions are combined into a unique set of equations that are valid for the entire domain it is shown that the nonlinear stability bound in the one domain approach is very sharp and hence excludes the possibility of subcritical instabilities moreover the one domain approach is compared with an equivalent two domain approach and excellent agreement is found between the two