Reconstruction of chaotic systems in invariant jet space

Abstract

Takens' theorem is the gold standard for attractor reconstruction from time series, but it guarantees only topological equivalence and does not preserve metric or group properties such as symmetries. We show that switching from delay-coordinate space to jet space (signal and its derivatives) allows one to exactly preserve the symmetry group of the original system. This statement is rigorously justified by a theorem on the isomorphism of Lie algebras under jet prolongation. Numerical experiments on the Lorenz and Rössler systems confirm that jet-space reconstruction preserves geometry and symmetries, whereas Takens embedding distorts them. As quantitative metrics we use a variational elastic energy functional and the correlation dimension. It is shown that jet-space reconstruction not only outperforms Takens embedding but in some cases yields more accurate estimates of invariants than projections of the original system. The proposed approach provides a coordinate-invariant criterion for the classification of strange attractors and can serve as a basis for detecting hidden attractors.