In several chemical systems such as the Belousov-Zhabotinsky reaction or the catalysis on platinum surfaces, transitions from meandering spiral waves to more complicated patterns have been observed. Seemingly key to the dynamics of spiral waves is the Euclidean symmetry group SE(N). In this article, it is shown that the dynamics near meandering spiral waves or other patterns is determined by a finite-dimensional vector field that has a certain skew-product structure over the group SE(N). This generalizes our earlier work on center-manifold theory near rigidly rotating spiral waves to meandering spirals. In particular, for meandering spirals, it is much more sophisticated to extract the aforementioned skew-product structure since spatio-temporal rather than only spatial symmetries have to be accounted for. Another difficulty is that the action of the Euclidean symmetry group on the underlying function space is not differentiable, and in fact may be discontinuous. Using this center-manifold reduction, Hopf bifurcations and periodic forcing of spiral waves are then investigated. The results explain the transitions to patterns with two or more temporal frequencies that have been observed in various experiments and numerical simulations.
Bibliographical noteFunding Information:
We are grateful to Herbert Koch who, for relative equilibria, communicated Lemma 6.4 and its proof. B. Sandstede was partially supported by a Feodor-Lynen Fellowship of the Alexander von Humboldt Foundation. Part of this work was done while the first two authors were visiting the Institute of Mathematics and its Applications (Minneapolis, USA). We thank the IMA for providing financial support and a stimulating atmosphere.
- Center manifolds
- Hopf bifurcation
- Noncompact groups
- Periodic forcing
- Spiral waves