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Periodic solutions from symmetry. (English) Zbl 0685.34047

The paper deals with autonomous systems of time-evolution ODE’s exhibiting symmetry properties (“equivariance”) under the action of an orthogonal representation T of a compact Lie group G. It is shown that if G has (under T) a maximal isotropy subgroup with minimal fixed space, then the problem has, in the generic case, stationary or periodic solutions; the solutions are truly periodic if the maximal isotropy subgroup is either SO(2) or SU(2). This result is extended to the case of non-minimal fixed spaces, obtaining a sort of generalization of the classical Poincaré-Bendixson theorem. Various simple examples are also considered.
Reviewer: G.Cicogna

MSC:

34C25 Periodic solutions to ordinary differential equations
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