Geodesic
Capture and holding of resonance far from equilibrium
Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 64-76
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Capture into resonance occurs in nonlinear oscillating systems. The study of mathematical models of this phenomenon is a part of a modern theory of nonlinear oscillations. The known result in this field were obtained by averaging method in the asymptotic regime with a small parameter. In this way, an initial stage of the capture into resonance was studied in details.The matter of this approach is an asymptotic passage to a model equation of mathematical pendulum kind. In the present work we consider an asymptotic construction at long time, which describes a slow evolution of a solution captured into resonance. The main aim is to determine a time interval, during which the resonance is held. The problem is reduced to studying a perturbation of a model equation of pendulum type. Our main success is the description of the time interval, in which the resonance is captured, and the description is given in terms of the data in the original problem. Formally we consider a nonlinear oscillating system with a small perturbation. The perturbation is described by an external pumping with a prescribed slowly changing frequency. For the solutions captured into the resonance, we consider asymptotics with respect to the small parameter. We write out an equation, the solution to which allows us to find the time of the capturing into resonance.
Keywords: nonlinear oscillations, small parameter, asymptotics, capture in resonance.
Mots-clés : perturbation 
@article{UFA_2018_10_4_a5,
     author = {L. A. Kalyakin},
     title = {Capture and holding of resonance far from equilibrium},
     journal = {Ufa mathematical journal},
     pages = {64--76},
     year = {2018},
     volume = {10},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a5/}
}
TY  - JOUR
AU  - L. A. Kalyakin
TI  - Capture and holding of resonance far from equilibrium
JO  - Ufa mathematical journal
PY  - 2018
SP  - 64
EP  - 76
VL  - 10
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a5/
LA  - en
ID  - UFA_2018_10_4_a5
ER  - 
%0 Journal Article
%A L. A. Kalyakin
%T Capture and holding of resonance far from equilibrium
%J Ufa mathematical journal
%D 2018
%P 64-76
%V 10
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a5/
%G en
%F UFA_2018_10_4_a5
L. A. Kalyakin. Capture and holding of resonance far from equilibrium. Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 64-76. http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a5/

[1] L. A. Kalyakin, “Asymptotic analysis of autoresonance models”, Russ. Math. Surv., 63:5 (2008), 791–857 | DOI | MR | Zbl

[2] A.I. Neishtadt, A.A. Vasiliev, A. V. Artemyev, “Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum”, Regular and Chaotic Dynamics, 18:6 (2013), 691–701 | DOI | MR

[3] N. N. Bogolyubov, Yu. A. Mitropol'sky, Asymptotic methods in the theory of non-linear oscillations, Gordon and Breach, New York, 1961 | MR | MR | Zbl

[4] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Springer, Berlin, 1997 | MR | MR | Zbl

[5] A. I. Neishtadt, “Passage through a separatrix in a resonance problem with a slowly-varying paramete”, J. Appl. Math. Mech., 39:4 (1975), 594–605 | DOI | MR

[6] O. M. Kiselev, S. G. Glebov, “An asymptotic solution slowly crossing the separatrix near a saddle-centre bifurcation point”, Nonlinearity, 16 (2003), 327–362 | DOI | MR | Zbl

[7] A. Neishtadt, A. Vasilliev, “Phase change between separatrix crossing in slow-fast Hamiltonian systems”, Nonlinearity, 18 (2005), 1393–1406 | DOI | MR | Zbl

[8] O. M. Kiselev, “Oscillations near a separatrix in the Duffing equation”, Proc. Stekl. Inst. Math., 281, no. Suppl. 1 (2013), 82–94 | DOI | MR

[9] K. S. Golovanivsky, “Autoresonant Acceleration of Electrons at Nonlinear ECR in a Magnetic Field Which is Smoothly Growing in Time”, Physica Scripta, 22 (1980), 126–133 | DOI

[10] K. S. Golovanivsky, “The Gyromagnetic Autoresonance”, IEEE Transactions on plasma science, PS-1 1:1 (1983), 28–35 | DOI

[11] G. M. Zaslavskii, R. Z. Sagdeev, Introduction to nonlinear physics. From the pendulum to turbulence and chaos, Nauka, M., 1977 (in Russian)

[12] M. I. Rabinovich, D. I. Trubetskov, Introduction into oscillation and wave theory, Regul. Chaot. Dyn., Izhevsk, 2000 (in Russian)

[13] B. V. Chirikov, “The passage of a nonlinear oscillating system through resonance”, Sov. Phys. Dokl., 4 (1959), 390–394 | MR | Zbl

[14] L. A. Kalyakin, “Averaging in the autoresonance model”, Math. Notes, 73:3 (2003), 414–418 | DOI | MR | Zbl

[15] A. P. Itin, A. I. Neishtadt, A. A. Vasiliev, “Capture into resonance in dynamics of a charged partice in magnetic field and electrostatic wave”, Physica D, 141:4 (2000), 281–296 | DOI | MR | Zbl

[16] A. I. Neishtadt, “Capture into resonance and scattering on resonances in two-frequency systems”, Proc. Steklov Inst. Math., 250 (2005), 183–203 | MR | Zbl

[17] A. I. Neishtadt, “Averaging, passage through resonances, and capture into resonance in two-frequency systems”, Russ. Math. Surv., 69:5 (2014), 771–843 | DOI | DOI | MR | Zbl