Klaus Böhmer

A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square

We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. Using appropriate genericity assumptions (with symmetry), we derive a Taylor series for the reduced equation, where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation. This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. We examine several Hopf bifurcations within the three-parameter space. There are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry

A Complete Bifurcation Scenario for the 2-d Nonlinear Laplacian with Neumann Boundary Conditions on the Unit Square

In this paper we describe the complete scenario of solutions bifurcating from the trivial solution and their stability for a model problem in bifurcation analysis the two-dimensional Laplacian with Neumann boundary conditions on the unit square. We show that at a corank ρ bifurcation point, there are exactly (3 ρ −1)/2 different solution branches bifurcating front the trivial solution curve. Our results are easily extended to other boundary conditions, e.g. Dirichlet or Dirichlet and Neumann along different sides, etc. which can be embedded in the periodic boundary conditions.

On a numerical Liapunov-Schmidt method for operator equations

Let, for a higher singular pointx0 of an operator equationG(x0)=0 and the kernels of the respective derivativesG′(x0) andG′(x0)*, see [1], approximations be available. We present a method to numerically compute the manifolds bifurcating atx0. In particular, the question of convergence of the numerical to the exact solution is studied by proving stability and convergence for solution parts of different order of magnitude. Different approaches are presented and applied to elliptic problems.ZusammenfassungFür einen höheren singulären Punktx0 einer OperatorgleichungG(x0)=0 und die Nullräume der jeweiligen AbleitungenG′(x0) undG′(x0)*, siehe [1], seien Approximatioinen bekannt. Dann definieren wir ein numerisches Verfahren zur Berechnung der inx0 abzweigenden Lösungsmannigfaltigkeiten. Die Frage der Konvergenz der numerischen gegen die exakte Lösung wird studiert durch Nachweis der entsprechenden Stabilitäts- und Konvergenzeigenschaften von Lösungsanteilen verschiedener Größenordnungen. Das Verfahren wird angewandt auf ein elliptisches Problem.

Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry

We perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a PDE with O(2) symmetry to one with SO(2) symmetry. Taking particular care of the consequences of the continuous group action, we concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. In particular, we note that although Chossat (1993) has proved that generic perturbations cause the appearance of quasiperiodic orbits, for the simplest possible flux terms this is not the case. We compare these results with numerical simulations of a Galerkin approximation of the equations.

Regularization and computation of a bifurcation problem with Corank 2

This paper deals with a numerical approximation of a bifurcation problem with corank 2. In the neighborhood of the bifurcation point the nonlinear equation is embedded into an extended system. The regular solution of this system including bifurcation point and null space of the corresponding operator derivative allows approximate computation of the bifurcation point and the null space via general discretization and Newton-like methods. In addition, numerical examples are discussed.ZusammenfassungDie numerische Approximation eines Verzweigungsproblems mit Korang 2 wird hier behandelt. In der Umgebung des Verzweigungspunktes wird die nichtlineare Gleichung in ein erweitertes System eingebettet. Die reguläre Lösung dieses Systems, die den Verzweigungspunkt und den Nullraum des entsprechenden Ableitungsoperators enthält, erlaubt die näherungsweise Berechnung von Verzweigungspunkt und Nullraum von allgemeinen Diskretisierungsmethoden und Newton-ähnlichen Verfahren. Anschließend werden numerische Beispiele diskutiert.

Direct Methods for Solving Singular Nonlinear Equations

A class of Newton-type methods for computing singular points with corank > 1 and index > 0 is presented. The idea is to use a generalized Sherman- Morrison formula to define a regularizing perturbation of the Newton-increment formula. The quadratic convergence of the process is preserved. The proposed approach may be linked to tensor methods, bordered Jacobian techniques, and numerical Liapunov-Schmidt reduction.

Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies

A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on 1-D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification supplies limited qualitative information concerning the imperfect bifurcation diagrams of the detected bifurcation points.

A Nonlinear Discretization Theory

Abstract This paper extends for the first time Schaback’s linear discretization theory to nonlinear operator equations, relying heavily on the methods in Bohmer’s 2010 book. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques. Trial spaces can be arbitrary, including spectral and meshless methods, but have to approximate the solution well, and testing can be weak or strong. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. Some numerical examples are added for illustration.

On Hybrid Methods for Bifurcation and Center Manifolds for General Operators

This presentation uses few basic concepts of numerical functional analysis and approximation theory as the main tools to prove convergence and stability for stationary problems. It applies to a general class of operator equations and general discretization methods. This allows an extension to numerical bifurcation studies, including Hopf bifurcation and center manifold results, for finite difference-, finite element- and spectral methods for general operators. In particular, partial differential equations (PDEs) as reaction-diffusion-systems and Navier-Stokes equations are included. The basic idea is to present an approach as simple as possible but as complex as necessary to cover all these types of problems and their discretizations with reasonably basic concepts. For the first time, the full cycle of qualitative and quantitative results, starting from PDEs via convergent discretization and post-processing back to the bifurcation scenarios in the original equation, is presented. A Г-equi-variant example in biological pattern formation is included. Finally a C ++ — program system with similarly general goals is indicated.

Hartree-Fock Methods a Realization of Variational Methods in Computing Energy Levels in Atoms

In this paper the well known Hartree-Fock methods are interpreted as variational methods. Since good and reliable upper bounds for the lowest eigenvalue of the Schrodinger equation are very important, we discuss the different kinds of numerical errors during the computation and give some hints how to control them.