93-63 Cicogna G., Gaeta G.
Nonlinear Lie symmetries in bifurcation theory (16K, plain TEX) Mar 15, 93
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Abstract. We examine the presence of general (nonlinear) time-independent Lie point symmetries in dynamical systems, and especially in bifurcation problems. A crucial result is that center manifolds are invariant under these symmetries: this fact, which may be also useful for explicitly finding the center manifold, implies that Lie point symmetries are inherited by the "reduced" bifurcation equation (a result which extends a known property of linear symmetries). An interesting situation occurs when a nonlinear symmetry of the original equation results in a linear one (e.g. a rotation - typically related to a Hopf bifurcation) of the reduced problem. We provide a class of explicit examples admitting nonlinear symmetries, which clearly illustrate all these points.

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