Numerical Continuation for Nonlinear Problems with Symmetry Structures

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DMS 9870274AbstractNumerical Continuation for Nonlinear Problems with Symmetry StructuresKurt Georg and Eugene L. Allgower This investigation deals with the study and development of numericalcontinuation methods which intrinsically exploit symmetries. Theintegration of techniques for exploiting symmetry in the solution oflinear systems, which have recently been developed by theinvestigators, into a numerical continuation method will provide apractical and efficient tool for detecting symmetry pattern breakingbifurcations while tracing a branch, and for following bifurcatingbranches for secondary and multiple bifurcations. The continuationmethods which will be studied and developed will concentrate upon thehandling of large systems of equations such as those stemming fromdiscretizations of systems of partial differential equations andintegral equations. Several techniques for numerically handlingbifurcations will be investigated. These include the normal formequations of center manifold theory and the Lyapunov-Schmidt reduction. By applying symmetry reduction methods developed by theinvestigators, block diagonalizations are obtained which greatlyfacilitate the bifurcation analysis. Among the applications to betreated will be: nonlinear elliptic equations such as those whichmodel the buckling of shells with symmetries, equations for sphericalconvection which model planetary atmospheres, Ginzburg-Landauequations modelling the dynamics of travelling waves, and nonlinearboundary integral equations which are discretized via boundary elementmethods. Integral equations of the latter kind arise in a natural wayfrom exterior boundary value problems. It is anticipated that theprograms which will be developed will be disseminated via anonymousftp in the same manner as the investigator's programs for numericalcontinuation and symmetry reduction methods have been done.Numerical continuation for large structured systems, which will bestudied in this investigation, has a broad field of importantapplications. In particular, numerical continuation can be used tocomputationally simulate how the variation of inherent controlparameters causes qualitative changes in a physical system. Suchparameters may represent loading, speed, temperature, viscosity,aspect ratio, etc. The determination and analysis of critical valuesof control parameters is of great significance to the scientist andengineer, because at these values decisive phenomena occur: stabilitycan be lost, structures can break or go into resonance, ingnitiontakes place, atmospheric flow patterns change, etc. Mathematically,critical values of control parameters in physical models arecharacterized by bifurcation points. Numerical continuation methodscan be used to detect and calculate bifurcation points. Theinvestigation will concentrate on phenomena taking place in regimeswhich have geometric symmetry structures. The consideration of suchstructures enhances the scientific understanding, but can also beexploited for more precise and efficient calculations. The computervisualization programs which will be developed will assist researchersand students to simulate, study, and gain insights into the patternchanges of physical phenomena under the variation of parameters.