CDSNS Colloquium, Georgia Tech
Monday 16 April 2007
Orbits homoclinic to periodic solutions and their application in hydrodynamics
Thomas J. Bridges
University of Surrey

Degenerate periodic orbits in Hamiltonian systems provide a mechanism for the creation of orbits homoclinic to periodic solutions. Periodic orbits are degenerate when the energy considered as a function of the frequency has a critical point. The normal form for this bifurcation has an interesting geometric characterisation in terms of the curvature of the energy-frequency map. When the periodic orbits are spatial, say steady solutions of a Hamiltonian PDE, then the bifurcation leads to a class of dark solitary waves, with a geometric phase. It turns out that these observations are a special case of the class of homoclinic orbits generated by degenerate relative equilibria. A new theory for this homoclinic bifurcation is presented. Examples in dynamical systems include a new formulation of the saddle-center bifurcation of invariant tori. There are a range of applications of this bifurcation in hydrodynamics. For illustration, the generation of internal solitary waves by this mechanism will be discussed, and application to the classical water wave problem shows that there is a new pervasive class of steady dark solitary waves in shallow water.

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  • TJB & N.M. Donaldson.   Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves, Phys. Fluids (to appear, 2007)   .pdf

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