##
Noncompact drift for relative equilibria and relative periodic orbits

*Nonlinearity* **10** (1997) 595-616

** Peter Ashwin and Ian Melbourne **

** Abstract **

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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