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__Patterns, Nonlinear Dynamics and Applications - PANDA__

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__Fourth meeting: Friday 18th October
2002, DAMTP, Cambridge__

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.