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.

[1]  Vladimír Janovský,et al.  Numerical applications of equivariant reduction techniques , 1992 .

[2]  Klaus Böhmer,et al.  Direct Methods for Solving Singular Nonlinear Equations , 1999 .

[3]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[4]  A. Mielke,et al.  Stability and Diffusive Dynamics on Extended Domains , 2001 .

[5]  Klaus Böhmer,et al.  Resolving singular nonlinear equations , 1988 .

[6]  A. Spence,et al.  On a reduction process for nonlinear equations , 1989 .

[7]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[8]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[9]  Urs Kirchgraber,et al.  Dynamics Reported : Expositions in Dynamical Systems , 1994 .

[10]  P. Ashwin,et al.  A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square , 1995 .

[11]  Friedrich Stummel,et al.  Diskrete Konvergenz linearer Operatoren. II , 1971 .

[12]  G. Dangelmayr,et al.  Mathematical Tools for Pattern Formation , 1998 .

[13]  Wolf-Jürgen Beyn,et al.  Defining Equations for Singular Solutions and Numerical Applications , 1984 .

[14]  W. Rheinboldt.  Nonlinear systems of equations , 1969, SGNM.

[15]  Wolfgang Hackbusch,et al.  Theorie und Numerik elliptischer Differentialgleichungen , 1986, Teubner Studienbücher.

[16]  P Chossat,et al.  Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle , 1993 .

[17]  Klaus Böhmer,et al.  Branch switching at a corank-4 bifurcation point of semi-linear elliptic problems with symmetry , 1994 .

[18]  P. Ashwin,et al.  Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry , 1996 .

[19]  W. Beyn,et al.  Continuation of Low-Dimensional Invariant Subspaces in Dynamical Systems of Large Dimension , 2001 .

[20]  K. Böhmer,et al.  Numerical Liapunov—Schmidt spectral methods for k-determined problems , 1999 .

[21]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[22]  G. Bisnovatyi-Kogan.  Limiting mass of hot superdense stable configurations , 1968 .

[23]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[24]  Alexander Ostermann,et al.  Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization , 1998, Numerische Mathematik.

[25]  A. Mielke.  Hamiltonian and Lagrangian Flows on Center Manifolds , 1991 .

[26]  H. Keller,et al.  Iterations, perturbations and multiplicities for nonlinear bifurcation problems , 1972 .

[27]  Normal form for Hopf bifurcation of partial differential equations on the square , 1995 .

[28]  E. Allgower,et al.  Exploiting symmetry in boundary element methods , 1992 .

[29]  K. Georg,et al.  Some error estimates for the numerical approximation of surface integrals , 1994 .

[30]  Willy Govaerts,et al.  Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies , 1999, Math. Comput..

[31]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[32]  Y. Kuznetsov,et al.  Continuation of stationary solutions to evolution problems in CONTENT , 1996 .

[33]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[34]  A. Mielke.  Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity , 1988 .

[35]  K. Böhmer,et al.  Path-Following of Large Bifurcation Problems with Iterative Methods , 2000 .

[36]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[37]  J. Rappaz,et al.  Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems , 1994 .

[38]  E. Allgower,et al.  A Complete Bifurcation Scenario for the 2-d Nonlinear Laplacian with Neumann Boundary Conditions on the Unit Square , 1991 .

[39]  L. Collatz,et al.  Vainikko, G., Funktionalanalysis der Diskretisierungsmethoden, Leipzig. BSB B. G. Teubner Verlagsgesellschaft. 1976. 136 S., M 14,—(Teubner-Texte zur Mathematik) , 1978 .

[40]  J. Rappaz,et al.  On numerical approximation in bifurcation theory , 1990 .

[41]  K. Böhmer,et al.  On a Numerical Liapunov-Schmidt Spectral Method and Its Application to Biological Pattern Formation , 2002, SIAM J. Numer. Anal..

[42]  Willy Govaerts.  Stable solvers and block elimination for bordered systems , 1991 .

[43]  Klaus Böhmer,et al.  On the Numerical Analysis of the Imperfect Bifurcation of codim <= 3 , 2002, SIAM J. Numer. Anal..

[44]  André Vanderbauwhede,et al.  Center Manifold Theory in Infinite Dimensions , 1992 .

[45]  Three-Dimensional Steady Capillary-Gravity Waves , 2001 .

[46]  F. Brezzi,et al.  Finite Dimensional Approximation of Non-Linear Problems .3. Simple Bifurcation Points , 1981 .

[47]  Allan D. Jepson,et al.  The Numerical Solution of Nonlinear Equations Having Several Parameters I: Scalar Equations , 1985 .

[48]  Scaling Solution Branches of One-Parameter Bifurcation Problems , 1996 .

[49]  P. Ashwin,et al.  A Hopf bifurcation with Robin boundary conditions , 1994 .

[50]  On Numerical Bifurcation Studies for General Operator Equations , 2000 .

[51]  Johannes Tausch,et al.  Numerical Exploitation of Equivariance , 1998 .

[52]  A. Griewank,et al.  Computation of cusp singularities for operator equations and their discretizations , 1989 .

[53]  P. Ashwin,et al.  A Numerical Bifurcation Function for Homoclinic Orbits , 1998 .

[54]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[55]  Vladimír Janovský,et al.  Computer-aided analysis of imperfect bifurcation diagrams, I. simple bifurcation point and isola formation centre. , 1992 .

[56]  C. Lubich.  On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization , 2001 .

[57]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[58]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[59]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[60]  Hans-Jürgen Reinhardt.  Analysis of Approximation Methods for Differential and Integral Equations , 1985 .

[61]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[62]  H. Stetter.  Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[63]  Branching of rotating waves in a one-parameter problem of steady-state bifurcation with spherical symmetry , 1991 .

[64]  Friedrich Stummel,et al.  Diskrete Konvergenz linearer Operatoren. I , 1970 .

[65]  Eugene L. Allgower,et al.  Exploiting symmetry in applied and numerical analysis : 1992 AMS-SIAM Summer Seminar in Applied Mathematics, July 26-August 1, 1992, Colorado State University , 1993 .

[66]  Roberto Monaco,et al.  Waves and Stability in Continuous Media , 1991 .

[67]  H. Kwatny,et al.  Constructing linear families from parameter-dependent nonlinear dynamics , 1998, IEEE Trans. Autom. Control..

[68]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[69]  K. Böhmer.  Über lineare Differentialgleichungen mit Lösungen von endlicher Wachstumsordnung , 1971 .