Random templex encodes topological tipping points in noise-driven chaotic dynamics

Article Mathematics, Applied

Templex: A bridge between homologies and templates for chaotic attractors

Gisela D. Charo et al.

Summary: The theory of homologies introduces cell complexes to algebraically describe spaces. This article extends the approach to take the action of the flow into account, creating a directed graph called templex that characterizes chaotic attractors accurately. Several well-known chaotic attractors are investigated, and their descriptions in terms of templates are established.

CHAOS (2022)

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Article Mathematics, Applied

Noise-driven topological changes in chaotic dynamics

Gisela D. Charo et al.

Summary: This study examines how noise affects chaotic systems by comparing the topological structure of deterministic and stochastically perturbed versions of the system. The results show that stochastic perturbations can lead to sharp topological transitions in the evolution of the system.

CHAOS (2021)

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Article Mathematics, Applied

Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator

Manuel Santos Gutierrez et al.

Summary: This study establishes a strong connection between data-driven and theoretical approaches to achieving efficient and accurate parameterizations for model reduction. Through perturbation expansions, a general stochastic parameterization of weakly coupled dynamical systems is derived, and it is shown that truncation of expansions is not necessary when coupling is additive. Additionally, the study simplifies unwieldy integrodifferential equations into a multilevel Markovian model and establishes an intuitive connection with a generalized Langevin equation. This research supports the physical basis and robustness of the empirical model reduction (EMR) methodology while highlighting the practical relevance of the perturbative expansion used for deriving the parameterizations.

CHAOS (2021)

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Article Physics, Mathematical