Forced Symmetry-Breaking as a Mechanism for Bursting

Jeffrey Michael Moehlis, Ph.D. Dissertation, University of California, Berkeley, 2000


Forced symmetry-breaking, or the introduction of small perturbations that reduce the symmetry of a system, can be responsible for complex dynamics in a system that would otherwise behave in a regular manner. This observation is of particular interest because in real physical systems symmetries are rarely exact and symmetry-breaking imperfections must be assumed to be present. This dissertation shows that forced symmetry-breaking can lead to periodic or irregular bursts of very large dynamic range. The results are applied to relevant hydrodynamical systems.

First, a model of large aspect-ratio binary fluid convection which considers the competition between two nearly degenerate modes of opposite parity is studied. The model corresponds to a Hopf bifurcation with broken D_4 symmetry, where the ``interchange'' symmetry between the two modes is weakly broken because of the large but finite aspect-ratio. For open parameter regimes, it is shown that periodic or irregular bursting with very large dynamic range may occur close to threshold. The bursts are found to be associated with global bifurcations involving fixed points and periodic orbits ``at infinity''. Global connections involving finite amplitude states are also present. The intricate sequence of bifurcations that take place is described in several cases, and the robustness of the results to small higher order terms in the amplitude equations is demonstrated.

Second, the effect of resonant temporal forcing on a system undergoing a Hopf bifurcation with D_4 symmetry is studied. The forcing breaks the continuous normal form symmetry of the amplitude equations. For the example which is considered, a type of gluing bifurcation (called a ``supergluing bifurcation'') is found and described. It is also shown that the attracting quasiperiodic solution which exists in the absence of forcing ``wrinkles'' into chaos as the amplitude of the forcing is increased, with the overall behavior governed by the approach of the attractor to solutions at infinity. Windows in which stable periodic solutions exist are also found and are associated with the traversal in parameter space through Arnol'd tongues. In addition, it is shown that bursts related to those found in the model of large aspect-ratio binary fluid convection may arise due to the forcing. Here the onset of bursting can be via an interior crisis or Type I intermittency.

Finally, other mechanisms that lead to behavior that has been called bursting are reviewed and compared to the new mechanism described in this dissertation.