S. V. Zelik, O. V. Pochinka, A. A. Yagilev

Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of $\varepsilon$-distinguishable set of graph points.

Key Words: Minkowski dimension, set coverage, ε-distinguishable set, Andronov-Hopf bifurcation

For citation: S. V. Zelik, O. V. Pochinka, A. A. Yagilev. On the Minkowski dimension of some invariant sets of dynamical systems. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 26:1(2024), 32–43. DOI: https://doi.org/10.15507/2079-6900.26.202401.32-43

Information about the authors:

Sergey V. Zelik, D. Sci. (Phys.-Math.), Chief Researcher at the International Laboratory of Dynamic Systems and Applications, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-4884-5040, s.zelik@surrey.ac.uk.

Olga V. Pochinka, D. Sci. (Phys.-Math.), Head of the International Laboratory of Dynamic Systems and Applications, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-6587-5305, olga-pochinka@yandex.ru.

Andrey A. Yagilev, Student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0009-0008-5088-8075, agilevandrej@gmai.com.