Spiral wave dynamics under full Euclidean symmetry-breaking: A dynamical system approach - Boily, Patrick (University of Ottawa (Canada), 2006)
Spirals are common in Nature: the snail's shell and the ordering of seeds in the sunflower are amongst the most widely-known occurrences. While these are static, dynamic spirals can also be observed in excitable systems such as heart tissue, retina, certain chemical reactions, slime mold aggregates, flame fronts, etc. The images associated with these spirals are often breathtaking, but spirals have also been linked to cardiac arrhythmias, a potentially fatal heart ailment. In the literature, very specific models depending on the excitable system of interest are used to explain the observed behaviour of spirals (such as anchoring or drifting). Barkley [5] first noticed that the Euclidean symmetry of these models, and not the model itself, is responsible for the observed behaviour. But in experiments, the physical domain is never Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry, LeBlanc and Wulff [48,51] introduced forced Euclidean symmetry-breaking (FESB) in the analysis. To accurately model the physical situation, two basic types of symmetry-breaking perturbations are used: translational symmetry-breaking (TSB) and rotational symmetry-breaking (RSB) terms. LeBlanc and Wulff, [51] and LeBlanc [48] have studied the effects of these individual perturbations, and they have shown that phenomena such as anchoring and quasi-periodic meandering can be explained by FESB. However, these specific perturbations only tell part of the story. In this thesis, the effects of multiple TSB perturbations, as well as those of combined TSB-RSB perturbations are studied and provide a more complete explanation for two aspects of spiral dynamics: anchoring and boundary drifting. Higher co-dimension phenomena are also considered.
Keywords: Mathematics.