Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems
Abstract
The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and Lp-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.
Related
- Lyapunov-Schmidt reduction for periodic orbits of implicit discrete dynamical systems
- A sharp embedding theorem for topological dynamical systems
- Entropy Scales in Topological Dynamical Systems
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- CONTINUOUS VERSUS DISCRETE STRUCTURES -DISCRETE EMBEDDING FORMALISMS AND CONTINUALIZATION METHODS
BibTeX
@misc{marszewska2026topological,
title = {{Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems}},
author = {Marta Marszewska and Justyna Signerska and Paweł Dłotko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15260},
eprint = {2609.15260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15260},
}
RIS
TY - UNPB
TI - Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems
AU - Marta Marszewska
AU - Justyna Signerska
AU - Paweł Dłotko
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15260
UR - https://arxiv.org/abs/2609.15260
AB - The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and L_p-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.
ER -