Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions
Motivated by applications to PDEs equivariant with respect to the affine action of the Euclidean group \(E(2)\), which model the dynamics of spiral waves, the authors study finite dimensional smooth differential systems equivariant with respect to an action of noncompact, finite dimensional Lie groups, near relative equilibria. Introducing some special coordinates respecting the action, the system has a triangular form with the base system independent of other variables. Relative equilibria of the whole system are equilibria of this base system. After that they apply the normal form method to transform the system locally near relative equilibria to some normal form eliminating resonances up to any finite order. All transformations preserve skew product structure and equivariance. Then this normal form is applied to the case of an action of the Euclidean group \(SE(N)\). One of the main points is the interpretation of the normal form results for the initial system since normal form transformations can mix important variables in the group \(SE(N)\), namely, rotations \(R\in SO(N)\), and drift \(S\in \mathbb{R}^n\). This point is clarified in the important case of the group \(SO(2)\). Here the notions of tip motion of the spiral and meandering spiral obtain the precise meaning. Also bifurcations that can occur in the base system are examined and how they can effect to the motion of the group variables in the case of resonances between Hopf and rotation frequencies. They also discuss phenomena in the whole system related with the presence of homoclinic orbits in the base system.
scientific article; zbMATH DE number 1287915