Cicogna G., Gaeta G.
Nonlinear Lie symmetries in  bifurcation theory
(16K, plain TEX)

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.