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ЖУРНАЛЫ // Russian Journal of Nonlinear Dynamics // Архив

Rus. J. Nonlin. Dyn., 2023, том 19, номер 4, страницы 473–506 (Mi nd869)

Nonlinear physics and mechanics

Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

G. L. Alfimovab, M. E. Lebedevc

a Institute of Mathematics with Computer Center, Ufa Scientific Center, Russian Academy of Sciences, ul. Chernyshevskogo 112, Ufa, 450008 Russia
b Moscow Institute of Electronic Engineering, Shokin square 1, Zelenograd, Moscow, 124498 Russia
c Nuclera Ltd, One Vision Park, Station Road, Impington, Cambridge, CB24 9NP United Kingdom

Аннотация: We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|<\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called “codes” of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a “great part” of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Ключевые слова: Duffing-type equation, periodic coefficients, symbolic dynamics, Smale horseshoe

MSC: 34A34, 37B10, 37D05

Поступила в редакцию: 29.06.2023
Исправленный вариант: 04.09.2023

Язык публикации: английский

DOI: 10.20537/nd231102



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