This work is devoted to the practical application of the geometric theory of dynamical systems to find invariants of a set of symmetric time series. Such series are often encountered in information systems processing large volumes of financial data, where spikes, outliers, and fluctuations are visually similar to each other, which can be explained, for example, by identical seasonal cycles or the type of economic activity. Although absolute values, amplitudes, and deviations may differ, the qualitative behaviour of such series is the same, as in oscillatory physical systems under different initial conditions and parameters. The discovered geometric invariant represents a structuring model that can be applied to the entire group of symmetric time series, providing a foundation for robust interval forecasting and the clustering of structurally similar dynamical systems. A theorem is proved establishing that the curvature of the jet space curve is a complete invariant under the group of affine transformations of the jet space coordinates and rotations. The use of translation, scaling, and rotation transformations applied to the locus of points of integral curves makes it possible to identify series that have the same qualitative behaviour up to a weak violation of symmetry. Experimental validation is performed on 25 daily bank account balance series. The proposed method achieves consistent alignment across all series.
Nikulchev et al. (2026) studied this question.