Manchester Applied Mathematics and Numerical Analysis Seminar Degenerate relative equilibria and homoclinic bifurcation Thomas J. Bridges University of Surrey Wednesday 23 November 2005

Abstract Relative equilibria are a class of solutions of symmetric Hamiltonian systems that model a wide range of physical phenomena. A familiar example is travelling wave solutions. Relative equilibria (RE) arise in n-parameter families where n is related to the dimension of the symmetry group. In this talk the nonlinear problem near a degenerate RE is considered. A new normal form is derived, showing that degeneracy implies a kind of homoclinic bifurcation. A surprising result is that the nonlinearity in the normal form is determined by the curvature of the momentum map. There is a geometric phase shift of the homoclinic orbit along the group, and this phase is determined by the normal form. The theory is constructive and examples include homoclinic bifurcation from periodic orbits, a new formulation of the quasi-periodic saddle-centre bifurcation of Hanssmann, and applications in optics and the theory of water waves. The theory explains properties of water-wave breaking in deep water, and is a mechanism for the bifurcation of a new class of dark solitary waves in shallow water.

References TJB & N.M. Donaldson.   Degenerate periodic orbits and homoclinic torus bifurcation, Phys. Rev. Lett. 95 104301 (2005)   .pdf TJB & N.M. Donaldson.   Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, Preprint (2005)   .pdf TJB.   Superharmonic instability, homoclinic torus bifurcation and water wave breaking, J. Fluid Mech. 505 153-162 (2004)   .pdf TJB.   Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation, Preprint (2006)