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

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Abstract

In this paper, we consider the equation \(u_{xx}+Q(x)u+P(x)u^{3}=0\) 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 \(\mathbb{R}\). For this purpose we consider the Poincaré map \(\mathcal{P}\) (i. e., the map-over-period) for this equation and analyse the areas of the plane \((u,u_{x})\) where \(\mathcal{P}\) and \(\mathcal{P}^{-1}\) are defined. We give sufficient conditions for hyperbolic dynamics generated by \(\mathcal{P}\) 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.

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Notes

  1. We conjecture that the conditions of smoothness for \(Q(x)\) and \(P(x)\) in Proposition 2 are excessive for existence of singular solutions.

  2. By “\(+\)” and “\(-\)” sign we mean strict inequalities \(a_{mn}>0\), \(a_{mn}<0\) to be satisfied.

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Funding

The work of GLA was supported by the Russian Science Foundation (Grant No. 23-11-00009).

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Authors and Affiliations

  1. Nuclera Ltd, One Vision Park, Station Road, Impington, CB24 9NP, Cambridge, UK

    Mikhail E. Lebedev

  2. Moscow Institute of Electronic Engineering, Zelenograd, 124498, Moscow, Russia

    Georgy L. Alfimov

  3. Institute of Mathematics with Computer Center, Ufa Scientific Center, Russian Academy of Sciences, ul. Chernyshevskii 112, 450008, Ufa, Russia

    Georgy L. Alfimov

Authors
  1. Mikhail E. Lebedev
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  2. Georgy L. Alfimov
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Correspondence to Mikhail E. Lebedev or Georgy L. Alfimov.

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MSC2010

34A34,37B10, 37D05

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Lebedev, M.E., Alfimov, G.L. Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients. Regul. Chaot. Dyn. 29, 451–473 (2024). https://doi.org/10.1134/S156035472451004X

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