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This article is cited in 1 scientific paper (total in 1 paper)
Oriented rotatability exponents of solution of autonomous differential systems
A. Kh. Stash Caucasus Mathematical Center, Adyghe State University, 208 Pervomayskaya St., Maikop 385000, Russia
Abstract:
In this paper, the exponents of oriented rotatability of solutions of linear homogeneous autonomous differential systems are fully studied. It is found that for any solution of an autonomous system of differential equations, its strong exponents of oriented rotatability coincide with weak ones. It is also shown that the spectrum of this exponent (i. e., the set of values on nonzero solutions) is naturally determined by the number-theoretic properties of the set of imaginary parts of the eigenvalues of the matrix of the system. This set can contain (unlike the oscillation and wandering exponents) values other than zero and from the imaginary parts of the eigenvalues of the system matrix, moreover, the power of this spectrum can be exponentially large in comparison with the dimension of the space. As a consequence, it is deduced that the spectra of all indicators of the oriented rotatability of autonomous systems with a symmetric matrix consist of a single zero value. In addition, on a set of autonomous systems, relationships were established between the main values of the studied exponents. The obtained results allow us to conclude that the exponents of oriented rotatability, despite its simple and natural definitions, is not an analogs of the Lyapunov exponent in the theory of oscillations.
Key words:
differential equations, autonomous system, exponents of oriented rotatability, exponents of oscillation.
Received: 16.08.2021
Citation:
A. Kh. Stash, “Oriented rotatability exponents of solution of autonomous differential systems”, Vladikavkaz. Mat. Zh., 24:3 (2022), 120–132
Linking options:
https://www.mathnet.ru/eng/vmj830 https://www.mathnet.ru/eng/vmj/v24/i3/p120
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