ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kashchenko S. A. Local dynamics of aperiodic chains with unidirectional couplings. Izvestiya VUZ. Applied Nonlinear Dynamics, 2026, vol. 34, iss. 1, pp. 9-33. DOI: 10.18500/0869-6632-003197, EDN: KBINPI

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Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-003197
EDN: 
KBINPI

Local dynamics of aperiodic chains with unidirectional couplings

Autors: 
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University
Abstract: 

Chains of N unidirectionally coupled nonlinear first-order equations are considered, where the value of the last element is determined through the first element of the chain. The aim of this work is to investigate the local – in the neighborhood of the zero equilibrium state – dynamics of this system. Critical cases in the problem of equilibrium state stability are identified, and normal forms determining the local behavior of solutions are constructed. A detailed analysis is performed in the simplest cases, where N = 2 and N = 3. The most interesting part of the research concerns the case where the value of N is sufficiently large. It is shown that the critical cases then have infinite dimension.

Methods. The standard research scheme, based on the use of the method of local invariant manifolds and the method of normal forms, turns out to be inapplicable. A special method of infinite-dimensional normalization developed by the author is used. The main results consist in the construction of so-called quasi-normal forms – analogs of normal forms for the infinite-dimensional case. It is important to emphasize that even for sufficiently large values of the number of chain elements N, the quasi-normal forms determining the dynamics of the original system significantly depend on variations in the value of N. Note that for certain values of the system coefficients, its dynamics can be quite complex.

Key words: 
dynamics
differential equation
chain
normal form
stability
Acknowledgments: 
This work was carried out within the framework of a development programme for the Regional Scientific and Educational Mathematical Center of the Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement on provision of subsidy from the federal budget No. 075-02-2025-1636).
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Received: 
04.08.2025
Accepted: 
01.10.2025
Available online: 
15.10.2025
Published: 
30.01.2026
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