Group representation theory, bifurcation theory and pattern formation

Abstract

Contents. 1. Symmetry-Breaking Instabilities. 2. The Onset of Instability. 3. Group Action on the Kernel. 4. “Planform” Functions of Hydrodynamics. 5. Computation of the Invariant Bifurcation Equations. 6. Analysis of the Bifurcation Equations. 7. Stability of the Bifurcating Solutions. 8. Stability of the Square and Rectangular Solutions. 9. Stability in the Hexagonal Lattice. 10. Gradient Structure of the Bifurcation Equations. 11. Concluding Remarks. 12. Appendix: The Boussinesq Equations. 1. SYMMETRY-BREAKING INSTABILITIES In this paper the phenomenon of pattern formation is regarded as a “symmetry breaking instability”. For a given range of a parameter X we suppose that a physical system possesses a stable solution invariant under a symmetry group 9, but that as h crosses a critical parameter h, new solutions appear which are invariant only under a subgroup H. The appearance of convection cells in the BCnard problem constitutes a classical example of such a bifurcation phenomenon. Prior to the onset of instability the fluid is motionless and the solution is invariant under the entire group of rigid motions in the plane; but after the onset of instability hexagonal or roll-like cellular motions appear which suggest the existence of stable doubly periodic motions in the plane. An excellent account of the BCnard problem is given in the review articles by L. A. Segel [32], and Palm, Ellingsen, and Gjevck [20]. See also [2], 151, and [9]. Plates showing the hexagonal convection cells discovered by BCnard may be found in the article by Koschmieder [ 121. Other familiar examples of symmetry breaking instabilities are the bifurcation of time periodic solutions from an equilibrium solution (the Hopf bifurcation theorem); the buckling of spheres; the onset of convection in spherical