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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
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 P−1 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
Received: 04.09.2023 Accepted: 21.03.2024
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
Linking options:
https://www.mathnet.ru/eng/rcd1263 https://www.mathnet.ru/eng/rcd/v29/i3/p451
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Abstract page: | 66 | References: | 33 |
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