refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

a jacobi davidson method for solving complex symmetric eigenvalue problems we discuss variants of the jacobi davidson method for solving the generalized complex symmetric eigenvalue problem the jacobi davidson algorithm can be considered as an accelerated inexact rayleigh quotient iteration we show that it is appropriate to replace the euclidean inner product in mathbb c n with an indefinite inner product the rayleigh quotient based on this indefinite inner product leads to an asymptotically cubically convergent rayleigh quotient iteration advantages of the method are illustrated by numerical examples we deal with problems from electromagnetics that require the computation of interior eigenvalues the main drawback that we experience in these particular examples is the lack of efficient preconditioners

on a conjugate gradient type method for solving complex symmetric linear systems abstract we consider large sparse linear systems ax b with complex symmetric coefficient matrices a a t which arise e g from the discretization of partial differential equations with complex coefficients for the solution of such systems we present a new conjugate gradient type iterative method csym which is based on unitary equivalence transformations of a to symmetric tridiagonal form an analysis of csym shows that its convergence depends on the singular values of a and that it has both the minimal residual property and constant costs per iteration step we compare the algorithm with other methods for solving large sparse complex symmetric systems

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

kowalevski s analysis of the swinging atwood s machine we study the kowalevski expansions near singularities of the swinging atwood s machine we show that there is an infinite number of mass ratios m m where such expansions exist with the maximal number of arbitrary constants these expansions are of the so called weak painleve type however in view of these expansions it is not possible to distinguish between integrable and nonintegrable cases

on the solution of generalized non linear complex symmetric eigenvalue problems this paper brings an attempt toward the systematic solution of the generalized non linear complex symmetric eigenproblem k o i c 1 2 m 1 i 3 c 2 4 m 2 0 with real symmetric matrices ko c j m j r nxn which are associated with the dynamic governing equations of a structure submitted to viscous damping as laid out in the frame of an advanced mode superposition technique the problem can be restated as k m where k o k t and m m t are complex symmetric matrices given as power series of the complex eigenfrequencies such that if is a solution eigenpair t m 1 and t k the traditional rayleigh quotient iteration and the more recent jacobi davidson method are outlined for complex symmetric linear problems and shown to be mathematically equivalent both with asymptotically cubic convergence the jacobi davidson method is more robust and adequate for the solution of a set of eigenpairs the non linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis thus also presenting cubic convergence two examples help us to visualize some of the basic concepts developed three more examples illustrate the applicability of the proposed algorithm to solve non linear problems in the general case of underdamping but also for overdamping combined with multiple and close eigenvalues

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

existence of exponentially unstable periodic solutions and the non integrability of homogeneous hamiltonian systems abstract in hamiltonian systems with polynomial potential of homogeneous degree characteristic multipliers floquet multipliers permit an explicit expression for straight line periodic solutions by use of ziglin s theorem it is shown that the existence of an exponentially unstable straight line periodic solution implies the non integrability of the system

a fast eigenvalue algorithm for hankel matrices abstract we present an algorithm that can find all the eigenvalues of an n n complex hankel matrix in o n 2 log n operations our scheme consists of an o n 2 log n lanczos type tridiagonalization procedure and an o n qr type diagonalization method

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

a criterion for the non existence of an additional integral in hamiltonian systems with a homogeneous potential abstract a non integrability criterion on the basis of ziglin s theorem is given for two degrees of freedom hamiltonians with a homogeneous potential of integer degree as simple examples the anisotropic kepler problem and the one dimensional newtonian three body problem are proved to be non integrable a self contained proof of ziglin s theorem n dimensional homogeneous potential version is also given

including homoclinic connections and t point heteroclinic cycles in the same global problem for a reversible family of piecewise linear systems global behavior is analyzed in a family of reversible piecewise linear systems the aim is to prove the existence of certain homoclinic orbits and t point cycles a problem including these two different objects as particular cases is constructed this problem leads to a common theorem of existence and local uniqueness the analytical proof of this result is given apart from being a complex task even in piecewise linear systems the proof of the existence of homoclinic connections and t point heteroclinic cycles must be usually carried out in separate ways because they are obviously different dynamical objects despite this some features of the system may narrow the disparities between such global bifurcations and help us to look for alternative methods to analyze them in this work taking advantage of the reversibility and some geometrical features of a piecewise linear version of the michelson system we construct by adding a suitable parameter a global problem that includes homoclinic connections and t point heteroclinic cycles as particular cases moreover this problem leads to a common result for the existence and local uniqueness of these global bifurcations whose proof is also given

the convergence behavior of ritz values in the presence of close eigenvalues abstract a number of results for ritz values are presented these are used to study the local effects in the convergence behavior of ritz values corresponding to a pair of close eigenvalues in the spectrum the local effects that are typical for such a situation are illustrated by numerical examples

a time domain discontinuous galerkin method for mechanical resonator quality factor computations numerical simulations are becoming increasingly important in the design of micromechanical resonators in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses frequency based approaches have been shown to predict these accurately however they require the solution of eigenvalue problems or the inversion of helmholtz type operators which are known to be very difficult for large scale iterative solvers we propose using a time domain approach instead where a broadband input signal is propagated through the system with a local explicit time stepper this is achieved using a new high order discontinuous galerkin dg discretization for the linear elasticity equations in particular a second order formulation with compact dg fluxes and a runge kutta time integrator where the block diagonal mass matrices allow for efficient stable and accurate time stepping our solver scales well on distributed parallel computers even in three spatial dimension and for large problem sizes the resulting output signal is analyzed using a well known filter diagonalization method which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data we validate the properties of our scheme on model problems and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

two sided and alternating jacobi davidson we discuss two variants of a two sided jacobi davidson jd method which have asymptotically cubic convergence for nonnormal matrices and aim to find both right and left eigenvectors these methods can be seen as jacobi davidson analogs of ostrowski s two sided rayleigh quotient iteration rqi some relations between exact and inexact two sided jacobi davidson and exact and inexact two sided rayleigh quotient iteration are given together with convergence rates furthermore we introduce an alternating jacobi davidson process that can be seen as the jacobi davidson analog of parlett s alternating rayleigh quotient iteration the methods are extended to the generalized and polynomial eigenproblem advantages of the methods are illustrated by numerical examples

reversible periodic orbits in a class of 3d continuous piecewise linear systems of differential equations abstract the so called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by kent and elgin in the well known michelson system in this work we perform an analysis of the periodic behavior of a piecewise version of the michelson system where this bifurcation also exists this variant is a one parameter three dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence free systems in the piecewise system the noose bifurcation involves reversible periodic orbits that intersect the separation plane at two or four points this work is focused on those reversible periodic orbits that intersect the separation plane twice rp2 orbits it is established that for every t between 2 and a critical point there exists a unique value of the parameter for which the system has an rp2 orbit with period t moreover this critical value that separates periodic orbits with two or four points of intersection with the separation plane corresponds to an rp2 orbit that crosses the separation plane tangentially it is also proved that in a parameter versus period bifurcation diagram the curve of this family of periodic orbits has a unique maximum point which corresponds to the saddle node bifurcation of periodic orbits that appears in the noose bifurcation

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

on the inverse of generalized λ matrices with singular leading term an algorithm is introduced for the inverse of a matrix given as the truncated series a0 i a1 2a2 i 3a3 4a4 o n 1 with square coefficient matrices and singular leading term a0 moreover a1 may be conditionally singular and no restrictions are made for the remaining terms the result is a matrix given as a unique truncated series of the same error order motivation for this problem is the evaluation of the frequency dependent stiffness matrix of general boundary or macro finite elements in the frame of a hybrid variational formulation that is based on a flexibility matrix f expressed as a truncated power series of the circular frequency copyright 2005 john wiley sons ltd