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Regular and Chaotic Dynamics, 2024, Volume 29, Issue 3, Pages 451–473
DOI: https://doi.org/10.1134/S156035472451004X
(Mi rcd1263)
 

Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

Mikhail E. Lebedeva, Georgy L. Alfimovbc

a Nuclera Ltd, One Vision Park, Station Road, Impington, CB24 9NP Cambridge, UK
b Institute of Mathematics with Computer Center, Ufa Scientific Center, Russian Academy of Sciences, ul. Chernyshevskii 112, 450008 Ufa, Russia
c Moscow Institute of Electronic Engineering, Zelenograd, 124498 Moscow, Russia
References:
Abstract: In this paper, we consider the equation uxx+Q(x)u+P(x)u3=0 where Q(x) and P(x) are periodic functions. { It is known that, if P(x) changes sign, a “great part” of the solutions for this equation are singular, i. e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i. e., not singular) on R. For this purpose we consider the Poincaré map P (i. e., the map-over-period) for this equation and analyse the areas of the plane (u,ux) where P and P1 are defined. We give sufficient conditions for hyperbolic dynamics generated by P in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of “numerical evidence”. This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given.
Keywords: Duffing-type equation, periodic coefficients, symbolic dynamics, numerical evidence
Funding agency Grant number
Russian Science Foundation 23-11-00009
The work of GLA was supported by the Russian Science Foundation (Grant No. 23-11-00009).
Received: 04.09.2023
Accepted: 21.03.2024
Document Type: Article
MSC: 34A34, 37B10, 37D05
Language: English
Citation: Mikhail E. Lebedev, Georgy L. Alfimov, “Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients”, Regul. Chaotic Dyn., 29:3 (2024), 451–473
Citation in format AMSBIB
\Bibitem{LebAlf24}
\by Mikhail E. Lebedev, Georgy L. Alfimov
\paper Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients
\jour Regul. Chaotic Dyn.
\yr 2024
\vol 29
\issue 3
\pages 451--473
\mathnet{http://mi.mathnet.ru/rcd1263}
\crossref{https://doi.org/10.1134/S156035472451004X}
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