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We mean the autoresonance is a considerable growth phenomenon for the energy of forced oscillations
of the nonlinear system while the forcing influence is small.
It was discovered independently by Veksler (1943) and MacMillan (1944).
They proposed to use the autoresonance in assists for the acceleration of elementary particles up to relativistic velocities.
The mathematical models were investigated little as yet.
The known results are based on either numerical experiments, or primal physical analysis.
the nonlinearity of the system, in particular, the dependence on energy of the oscillation period lead to the
difficulties in analytical studying.
Our purpose is an asymptotic analysis of the autoresonance model,
where the amplitude of forcing oscillations is a natural small parameter.
Formally, in simplest case, the problem is reduced to two-dimensional hamilton system of nonlinear equations.
For this system we study the forced oscillations near the stable equilibrium.
The forcing influence is represented by the rapid oscillations with a small amplitude and
slowly varying frequency. We investigate what kind of the conditions imply the trajectory of the system
goes away from the initial equilibrium on the distance of order unity.
The main mathematical results are concerned to studying the asymptotic solution of the principal resonance equations.
These equations describe the process of amplitude growth for the forced oscillations in first order of approximation.
Problems
The main purpose is a finding of increasing solutions that describe the initial stage of the autoresonance
on coupled oscillators. In 2004:
Determine the role of the initial data in appearing self-phasing and autoresonance.
- Obtain the criterion of the appearing autoresonance.
This requires the asymptotic solution of the principal resonance equations and
finding the relation for the constants of asymptotics at different infinities.
- Investigate the nonlinear resonance effects in the system of three weakly coupled nonlinear oscillators.
Construct the asymptotisc for the solution of principal resonance equations system of three degrees of freedom at the infinity.
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Results
We began to study the above posed problems in 2002.
Results of 2003:
- The problem of passing one degree of freedom hamilton system through the nonlinear resonance was considered.
The first members of asymptotics were constructed.
- The conditions of the autoresonance appearing in this system were obtained.
In rigorous formulation these conditions are related to investigation of the asymptotics of the principal resonance equations.
- The conditions of the autoresonance appearing on subharmonics and the corresponding time scales were obtained.
We establish that the autoresonance is impossible on the higher subharmonics.
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Publications
- L.A. Kalyakin,
Asymptotic solution of the problem of threshold effect for the principal resonance equations,
Dif. uravn., V. 40, Is. 6. 2004. P. 731-739. (in Russian)
- Kalyakin L.A., Autoresonance in the dynamical system, Contemporary mathematics and its applications. V. 5. Asymptotic methods of functional analysis. Publishing by Georgian Academy of Sciences, Institute of cybernetics, Tbilisi, 2003. P. 79-108. (in Russian)
- Kalyakin L.A. Asymptotics for the solutions of the principal resonance equations. Abstracts of the Int. conference "Differential Equations and Related Topics", Moscow, 2004. P. 94.
- Garifullin R.N. Construction of asymptotic solutions of the autoresonance problem. Dokl. RAS, v.398, Is.3, 2004. p.306-309
- Garifullin R.N. Construction of asymptotic solutions of the autoresonance problem. Abstracts of the Int. conference "Differential Equations and Related Topics", Moscow, 2004. P. 69. (in Russian)
- L.A. Kalyakin and Yu.Yu. Bagderina.Asimptotics of the bounded on infinity solutions of quadratic principal resonance equations. Math. Notes, Vol. 78, Is. 1, 2005. P. 85-97 (accepted).
- L.A. Kalyakin and Yu.Yu. Bagderina.Asimptotics of the solution of averaged equations for coupled oscillators system. Prikladnaya i fundamentalnaya matematika, (accepted).
- Kalyakin L.A. Analysis of mathematical model of autoresonance. Lecture in regional school-conference. Ufa, BSU, 2004.
- Garifullin R.N. Study of the increase for nonlinear equation solutions depending on the initial data.
Proc. of regional school-conference for students, postgraduate and young scientists>. BSU, Ufa, 2003, p.189-195.
- Kalyakin L.A., Asymptotics for the solutions of the principal resonance equations, Teor. Matem. Fiz., 2003. V. 137, № 1. P. 142-152. (in Russian)
- Kalyakin L.A., Averaging in autoresonance model, Matem. zametki, 2003. V. 73, Is. 3. P. 449-452. (in Russian)
- L.A. Kalyakin, Asymptotics for the solutions of the principal resonance equations at infinity, Dokl. RAS, 2003. V. 388, Is. 3. P. 305-308. (in Russian)
- L.A. Kalyakin, Justification of Asymptotic Expansions for the Principal
Resonance Equation, Proc. of the Steklov Inst. of Math., 2003, v.1, S108-S122.
- R.N. Garifullin, Asymptotic analysis of subharmonic autoresonance model,
Proc. of the Steklov Inst. of Math., 2003, v.1, S75-S83.
- L.A. Kalyakin, Asymptotic Analysis of an autoresonance model.
Russain J. of Math. Phys., 2002, v.9, n2, 84-95.
- Л.А. Калякин, Asymptotic analysis of the autoresonance model. Dokl. RAS, 2001. V. 378, № 5. P. 594-597. (in Russian)
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