Noncompact drift for relative equilibria and relative periodic orbits

Nonlinearity 10 (1997) 595-616

Peter Ashwin and Ian Melbourne


In the context of equivariant dynamical systems with a compact Lie group $\Gamma$ of symmetries, Field and Krupa have given sharp upper bounds on the drifts associated with relative equilibria and relative periodic orbits. For relative equilibria consisting of points of trivial isotropy, the drifts correspond to tori in $\Gamma$. Generically, these are maximal tori. Analogous results hold when there is a nontrivial isotropy subgroup $\Sigma$, with $\Gamma$ replaced by $N(\Sigma)/\Sigma$.

In this paper, we generalize the results of Field and Krupa to noncompact Lie groups. The drifts now correspond to tori or lines (unbounded copies of R) in $\Gamma$ and generically these are maximal tori or lines. Which of these drifts is preferred, compact or unbounded, depends on $\Gamma$: there are examples where compact drift is preferred (Euclidean group in the plane), where unbounded drift is preferred (Euclidean group in three dimensional space) and where neither is preferred (Lorentz group).

Our results partially explain the quasiperiodic (Winfree) and linear (Barkley) meandering of spirals in the plane, as well as the drifting behavior of spiral bound pairs (Ermakova et al). In addition, we obtain predictions for the drifting of the scroll solutions (scroll waves and scroll rings, twisted and linked) considered by Winfree and Strogatz.

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