Parametric resonance in a conservative system of coupled nonlinear oscillators

Johann Maddi, Christophe Coste, and Michel Saint Jean
Phys. Rev. E 105, 054208 – Published 17 May 2022

Abstract

We study a dimer in a periodic potential well, which is a conservative but nonintegrable system. This seemingly simple system exhibits a surprisingly rich dynamics. Using a systematic asymptotic analysis, we demonstrate that the translation mode of the dimer (center of mass motion) may induce a parametric resonance of the oscillatory mode. No external forcing occurs, thus this system belongs to the class of autoparametric systems. When the dimer energy is such that both particles are trapped in neighboring potential wells, we derive the relevant amplitude equations for the eigenmodes (center of mass motion and relative motion) and show that they are integrable. In the opposite limit, when the dimer slides along the external potential so that the center of mass motion is basically a translation, we also exhibit autoparametric amplification of the relative motion. In both cases our calculations provide reliable estimates of the relevant parameters for the autoparametric resonance to appear. Moreover, the comparison between the numerical integration of the actual system and the asymptotic analysis evidences an excellent quantitative agreement.

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  • Received 26 January 2022
  • Accepted 3 May 2022

DOI:https://doi.org/10.1103/PhysRevE.105.054208

©2022 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
  1. Physical Systems
Nonlinear Dynamics

Authors & Affiliations

Johann Maddi, Christophe Coste, and Michel Saint Jean

  • Laboratoire “Matière et Systèmes Complexes” (MSC), UMR 7057 CNRS, Université Paris–Diderot (Paris 7), 75205 Paris Cedex 13, France

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Issue

Vol. 105, Iss. 5 — May 2022

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