Symmetry Invariants of Strange Attractors from Jet-Space Embeddings

Abstract

A method for reconstructing dynamical systems from a scalar time series is proposed, based on replacing delay coordinates with jet-space coordinates—the signal and its time derivatives. A theorem is proved establishing a Lie algebra isomorphism under jet prolongation, which ensures equivariance under the full symmetry group of the original system. Unlike Takens embedding, which provides only topological equivalence, jet-space reconstruction preserves not the visual appearance of the attractor, but its fundamental properties: the symmetry group and the differential structure. This is demonstrated numerically on the Lorenz and Rössler systems, where jet space faithfully reproduces the geometry, group invariants, and correlation dimension, while Takens embedding introduces severe distortions. An additional experiment on a symmetric modification of the Rössler system shows that integrating the reconstructed derivatives in jet space recovers the original signal with near-zero error, whereas the same operation in Takens space fails dramatically. This confirms that jet space preserves the differential structure of the system, making it suitable for data-driven equation identification, in contrast to Takens space, where such identification is inherently biased. The method requires no parameter tuning, applies to systems with arbitrary symmetries.