- 93-63 Cicogna G., Gaeta G.
- Nonlinear Lie symmetries in bifurcation theory
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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.