Patterns, Nonlinear Dynamics and Applications - PANDA

Jonathan Dawes

The dynamics of mode interactions: patterns, bifurcations and symmetry

 
Abstract: Bifurcation theory attempts to understand qualitative changes in the dynamics of nonlinear systems, often modelled by sets of ordinary different equations. These qualitative changes are often linked to changes in stability of equilibria for the system, as eigenvalues cross the imaginary axis. Generically, marginally stable equilibria have either a single zero eigenvalue or a single purely imaginary pair. The detailed understanding of more complex (and in one sense, degenerate) bifurcations than the generic saddle-node or Hopf cases reveals phenomena which link together different generic cases. Linear degeneracies, where more eigenvalues lie on the imaginary axis than expected, are termed mode interactions. More complicated dynamics are possible since the centre manifold at such a bifurcation is of higher dimension.

In the presence of symmetry, multiple eigenvalues occur naturally. Mode interactions in the sense of the non-symmetric theory still occur, leading to bifurcation problems which are thus made doubly-complicated. Such behaviour relies only on the symmetry of the underlying problem and is therefore not restricted to a single physical system. In this talk I will illustrate some of the resulting complexities using three examples drawn from recent work, motivated by pattern formation problems in thermal convection under the influence of rotation and a magnetic field.