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Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2024, Volume 26, Number 1, Pages 32–43
DOI: https://doi.org/10.15507/2079-6900.26.202401.32-43
(Mi svmo876)
 

Mathematics

On the Minkowski dimension of some invariant sets of dynamical systems

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

National Research University – Higher School of Economics in Nizhny Novgorod
References:
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.
Keywords: Minkowski dimension, set coverage, $\varepsilon$-distinguishable set, Andronov-Hopf bifurcation
Funding agency Grant number
Russian Science Foundation 23-71-30008
Received: 09.01.2024
Accepted: 27.02.2024
Document Type: Article
UDC: 515.163
MSC: 57N10
Language: Russian
Citation: S. V. Zelik, O. V. Pochinka, A. A. Yagilev, “On the Minkowski dimension of some invariant sets of dynamical systems”, Zhurnal SVMO, 26:1 (2024), 32–43
Citation in format AMSBIB
\Bibitem{ZelPocYag24}
\by S.~V.~Zelik, O.~V.~Pochinka, A.~A.~Yagilev
\paper On the Minkowski dimension of some invariant sets of dynamical systems
\jour Zhurnal SVMO
\yr 2024
\vol 26
\issue 1
\pages 32--43
\mathnet{http://mi.mathnet.ru/svmo876}
\crossref{https://doi.org/10.15507/2079-6900.26.202401.32-43}
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