Numerical integration of an excitable reaction-diffusion ͑RD͒ system on a sphere is presented. The evolution of counterrotating double spiral waves on this manifold is studied and it is shown that tips of the spiral can either perform a meandering motion or rigidly rotate around a fixed center, depending on the system control parameter. This transition in dynamics is also illustrated by considering the phase plane of the solutions of the RD system. ͓S1063-651X͑97͒03410-7͔

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A direct method for finding the mode interaction point of a symmetry breaking Hopf bifurcation and a symmetry preserving Hopf bifurcation in problems with ℤ2-symmetry is developed. It has been shown that the mode interaction point corresponds to an isolated solution of an extended system. The existence of this solution relies on the occurrence of the mode interaction point and this is interpreted in the context of the mode interaction, using centre manifold reduction. A numerical example with the symmetry group O(2), which has a branch of ℤ2-symmetric nontrivial steady state solutions, is considered to provide clarification of the method.