Nonlinear Lie symmetries in bifurcation theory

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 also be 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.