Nonlinear Lie symmetries in bifurcation theory

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. Phys. Lett. A 172 (1993), 361 Supported in part by C.N.R. grant 203.01.48, present address: Math. Inst. Univ. of Utrecht (Netherlands ) Bifurcation theory is by now one of the most useful and used tools in attacking nonlinear problem..