A. E. Rassadin

Higher School of Economics University (Nizhny Novgorod, Russian Federation)

Abstract. The article is devoted to the analysis of dynamical systems in a specially constructed countably-dimensional phase space, namely, in the space of double-sided sequences. It is not only a linear space but also a Banach algebra. Using the multiplication operation in this algebra, the article constructs two examples of dynamical systems in such a space; both systems allow exhaustive investigation. One dynamical system is concentrated, meaning that its dynamic variables depend only on time, while another system is distributed, meaning that its dynamic variables depend not only on time but also on a spatial variable. A concentrated system can be obtained in two ways. The first way is simplifying the Cauchy problem for the Kolmogorov-Petrovsky-Piskunov equation with a periodic initial condition. The second way involves deriving the evolution equation for the countably-dimensional energy in a countably-dimensional system of ordinary differential equations, which is the normal form for the analogue of the Andronov-Hopf bifurcation in the space of double-sided sequences. A distributed dynamic system arises as a result of the spatial discretization of a certain nonlinear integro-differential equation. In this paper general solutions of Cauchy problems for these dynamic systems are constructed using the method of generating functions in the form of Laurent series. The article also provides specific examples of exact solutions of Cauchy problems for these dynamic systems. It is shown how, by analogy with digital signal processing methods, these exact solutions generate a countable set of exact solutions for the dynamic systems under consideration.

Key Words: Fourier scries, commutative algebra with unit, open ring, upsampling

For citation: A. E. Rassadin. On the Integrability of Dynamical Systems in the Space of Double-Sided Sequences. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:2(2026), 28–46. DOI: https://doi.org/10.15507/2079-6900.28.202602.28-46

Information about the author:

Alexander E. Rassadin, Postgraduate student, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12, B. Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID: https://orcid.org/0000-0001-5644-4012, acrassadin@hsc.ru