Abstract
In this paper we study equivariant systems on the plane. We first give some criterions for outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show a Z_3 equivariant system to have 13 limit cycles.
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