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A Remark on the Onset of Resonance Overlap

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Abstract

Chirikov’s celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigorous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring, respectively, the distance to resonance overlap and nonintegrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov’s criterion.

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Change history

  • 29 October 2023

    The numbering issue has been changed to 4-5.

Notes

  1. It is a matter of definition whether our analysis in the regime where \(|\mu|\ll 1\) may indeed be called “Chirikov’s criterion”: on the one hand, this regime is eligible to being analyzed with classical, perturbative tools, as we show below and contrary to the standard domain of application of the criterion; on the other hand, it does correspond to the general idea of overlapping eyes of resonance.

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ACKNOWLEDGMENTS

We have greatly benefitted from discussions with Cristel Chandre, Philip Morrison and Tere Seara, and from many suggestions of the referees.

Funding

M. Guardia has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 757802). This work is part of the grant PID-2021-122954NB-100 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. M. Guardia is also supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. This work is also supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). This work is also supported by the project of the French Agence Nationale pour la Recherche CoSyDy (ANR-CE40-0014).

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Authors and Affiliations

  1. Université Paris-Dauphine — CEREMADE (UMR 7534), Place du Maréchal de Lattre de Tassigny, Cedex 16, Paris, 75775, France

    Jacques Fejoz

  2. Observatoire de Paris — IMCCE (UMR 8028), 77 avenue Denfert-Rochereau, 75014, Paris, France

    Jacques Fejoz

  3. CNRS, PSL Research University, 60 Rue Mazarine, 75006, Paris, France

    Jacques Fejoz

  4. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via, 585, 08007, Barcelona, Spain

    Marcel Guardia

  5. Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193, Bellaterra, Spain

    Marcel Guardia

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  1. Jacques Fejoz
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Correspondence to Jacques Fejoz or Marcel Guardia.

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37J40

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Fejoz, J., Guardia, M. A Remark on the Onset of Resonance Overlap. Regul. Chaot. Dyn. 28, 578–584 (2023). https://doi.org/10.1134/S1560354723040056

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