On calculating the dimension of invariant sets of dynamic systems


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This work investigates numerical approaches for estimating the dimension of invariant sets onto which the trajectories of dynamic systems ``wind'', with a focus on fractal and correlation dimensions. While the classical fractal dimension becomes computationally challenging in spaces of dimension greater than two, the correlation dimension offers a more efficient and scalable alternative. We develop and implement a computational method for evaluating the correlation dimension of large discrete point sets generated by numerical integration of differential equations. An analogy is noted between this approach and the Richardson--Kalitkin method for estimating the error of a numerical method. The method is tested on two representative systems: a conservative system whose orbit lies on a two-dimensional torus, and the Lorenz system, a canonical example of a chaotic flow with a non-integer attractor dimension. In both cases, the estimated correlation dimensions agree with theoretical predictions and previously reported results. The developed software provides an effective tool for analysing invariant manifolds of dynamical systems and is suitable for further studies, including those involving reversible difference schemes and high-dimensional systems.

Sobre autores

Viktor Kadrov

RUDN University

Email: vmkadrov@yandex.ru
ORCID ID: 0009-0008-9394-4874

Student of Department of Computational Mathematics and Artificial Intelligence

Rússia, 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Mikhail Malykh

RUDN University; Joint Institute for Nuclear Research

Autor responsável pela correspondência
Email: malykh-md@rudn.ru
ORCID ID: 0000-0001-6541-6603
Scopus Author ID: 6602318510
Researcher ID: P-8123-2016

DSc., Head of Department of Computational Mathematics and Artificial Intelligence of RUDN University;
Senior Researcher of Joint Institute for Nuclear Research

Rússia, 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

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