This investigation deals with the study and development of numerical continuation methods which intrinsically exploit symmetries. The integration of techniques for exploiting symmetry in the solution of linear systems, which have recently been developed by the investigators, into a numerical continuation method will provide a practical and efficient tool for detecting symmetry pattern breaking bifurcations while tracing a branch, and for following bifurcating branches for secondary and multiple bifurcations. The continuation methods which will be studied and developed will concentrate upon the handling of large systems of equations such as those stemming from discretizations of systems of partial differential equations and integral equations. Several techniques for numerically handling bifurcations will be investigated. These include the normal form equations of center manifold theory and the Lyapunov-Schmidt reduction. By applying symmetry reduction methods developed by the investigators, block diagonalizations are obtained which greatly facilitate the bifurcation analysis. Among the applications to be treated will be: nonlinear elliptic equations such as those which model the buckling of shells with symmetries, equations for spherical convection which model planetary atmospheres, Ginzburg-Landau equations modelling the dynamics of travelling waves, and nonlinear boundary integral equations which are discretized via boundary element methods. Integral equations of the latter kind arise in a natural way from exterior boundary value problems. It is anticipated that the programs which will be developed will be disseminated via anonymous ftp in the same manner as the investigator's programs for numerical continuation and symmetry reduction methods have been done.
Numerical continuation for large structured systems, which will be studied in this investigation, has a broad field of important applications. In particular, numerical continuation can be used to computationally simulate how the variation of inherent control parameters causes qualitative changes in a physical system. Such parameters may represent loading, speed, temperature, viscosity, aspect ratio, etc. The determination and analysis of critical values of control parameters is of great significance to the scientist and engineer, because at these values decisive phenomena occur: stability can be lost, structures can break or go into resonance, ingnition takes place, atmospheric flow patterns change, etc. Mathematically, critical values of control parameters in physical models are characterized by bifurcation points. Numerical continuation methods can be used to detect and calculate bifurcation points. The investigation will concentrate on phenomena taking place in regimes which have geometric symmetry structures. The consideration of such structures enhances the scientific understanding, but can also be exploited for more precise and efficient calculations. The computer visualization programs which will be developed will assist researchers and students to simulate, study, and gain insights into the pattern changes of physical phenomena under the variation of parameters.
