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TMF, 2003, Volume 137, Number 1, Pages 142–152 (Mi tmf1757)  

This article is cited in 16 scientific papers (total in 16 papers)

Asymptotic Behavior of Solutions of Equations of Main Resonance

L. A. Kalyakin

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: We investigate a system of two first-order differential equations that appears when averaging nonlinear systems over fast one-frequency oscillations. The main result is the asymptotic behavior of a two-parameter family of solutions with an infinitely growing amplitude. In addition, we find the asymptotic behavior of another two-parameter family of solutions with a bounded amplitude. In particular, these results provide the key to understanding autoresonance as the phenomenon of a considerable growth of forced nonlinear oscillations initiated by a small external pumping.

Keywords: nonlinear equations, asymptotic behavior, WKB approximation

DOI: https://doi.org/10.4213/tmf1757

Full text: PDF file (236 kB)
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English version:
Theoretical and Mathematical Physics, 2003, 137:1, 1476–1484

Bibliographic databases:


Citation: L. A. Kalyakin, “Asymptotic Behavior of Solutions of Equations of Main Resonance”, TMF, 137:1 (2003), 142–152; Theoret. and Math. Phys., 137:1 (2003), 1476–1484

Citation in format AMSBIB
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\paper Asymptotic Behavior of Solutions of Equations of Main Resonance
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\jour Theoret. and Math. Phys.
\yr 2003
\vol 137
\issue 1
\pages 1476--1484
\crossref{https://doi.org/10.1023/A:1026065025429}
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  • http://mi.mathnet.ru/eng/tmf/v137/i1/p142

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. L. A. Kalyakin, “Resonance Capture in a Nonlinear System”, Theoret. and Math. Phys., 144:1 (2005), 944–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Kalyakin L.A., “Asymptotic analysis of the autoresonance phenomenon”, 2005 International Conference on Physics and Control (PHYSCON), 2005, 616–621  crossref  isi  scopus  scopus
    3. L. A. Kalyakin, “Intermediate asymptotics for solutions to the degenerate principal resonance equations”, Comput. Math. Math. Phys., 46:1 (2006), 79–89  mathnet  crossref  mathscinet  zmath
    4. Kiselev, OM, “The capture into parametric autoresonance”, Nonlinear Dynamics, 48:1–2 (2007), 217  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    5. L. A. Kalyakin, “Asymptotic analysis of autoresonance models”, Russian Math. Surveys, 63:5 (2008), 791–857  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. L. A. Kalyakin, M. A. Shamsutdinov, “Autoresonant asymptotics in an oscillating system with weak dissipation”, Theoret. and Math. Phys., 160:1 (2009), 960–967  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. L. A. Kalyakin, “Metod usredneniya v zadachakh ob asimptotike na beskonechnosti”, Ufimsk. matem. zhurn., 1:2 (2009), 29–52  mathnet  zmath  elib
    8. Shamsutdinov M.A., Kalyakin L.A., Sukhonosov A.L., Khalfina A.A., “Autoresonance excitation of oscillations of domain walls in a ferromagnetic film”, Physics of Metals and Metallography, 108:1 (2009), 8–18  crossref  adsnasa  isi  scopus  scopus
    9. Glebov S., Kiselev O., Tarkhanov N., “Weakly Nonlinear Dispersive Waves under Parametric Resonance Perturbation”, Stud. Appl. Math., 124:1 (2010), 19–37  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. Shamsutdinov M.A., Kalyakin L.A., Kharisov A.T., “Autoresonance in a ferromagnetic film”, Technical Physics, 55:6 (2010), 860–865  crossref  adsnasa  isi  elib  scopus  scopus
    11. L. A. Kalyakin, O. A. Sultanov, M. A. Shamsutdinov, “Asymptotic analysis of a model of nuclear magnetic autoresonance”, Theoret. and Math. Phys., 167:3 (2011), 762–771  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    12. Kalyakin L.A., Sultanov O.A., “Stability of Autoresonance Models”, Differ. Equ., 49:3 (2013), 267–281  crossref  mathscinet  zmath  isi  elib  elib  scopus
    13. O. A. Sultanov, “Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations”, Comput. Math. Math. Phys., 54:1 (2014), 59–73  mathnet  crossref  crossref  isi  elib  elib
    14. L. A. Kalyakin, “Adiabatic approximation in a resonance capture problem”, Ufa Math. J., 9:3 (2017), 61–75  mathnet  crossref  isi  elib
    15. Sultanov O., “Stability and Asymptotic Analysis of the Autoresonant Capture in Oscillating Systems With Combined Excitation”, SIAM J. Appl. Math., 78:6 (2018), 3103–3118  crossref  isi
    16. Kalyakin L.A., “Capture and Keeping of a Resonance Near Equilibrium”, Russ. J. Math. Phys., 26:2 (2019), 152–167  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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