Equivariant differential embeddings

Takens [ Dynamical Systems and Turbulence, Lecture Notes in Mathematics, edited by D. A. Rand and L. S. Young (Springer-Verlag, New York, 1981), Vol. 898, pp. 366–381 ] has shown that a dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system diffeomorphic to the original. However, if the original dynamical system is symmetric, it is natural to search for reconstructions that preserve this symmetry. These generally do not exist. We demonstrate that a differential reconstruction of any nonlinear dynamical system preserves at most a twofold symmetry.