Multifrequency oscillations of singularly perturbed systems. (English. Russian original) Zbl 1277.34056
Differ. Equ. 48, No. 1, 19-25 (2012); translation from Differ. Uravn. 48, No. 1, 21-26 (2012).
The paper is a continuation of the previous work by V. R. Bukaty [Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 1992, No. 2, 110–112 (1992; Zbl 0760.34041)]
Consider a singularly perturbed system of differential equations. Assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic, i.e., all of its eigenvalues have nonzero real parts. Under some general assumptions, the authors study the existence of invariant manifolds of the system homeomorphic to tori of various dimensions. Such manifolds, which are called “invariant tori”, support multifrequency oscillations. For this aim, the authors derive the so-called determining equation. Every positive solution of the determining equation corresponds to an invariant torus.
It is shown that there is an \(m\)-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here \( m\) is the number of pure imaginary eigenvalues. In addition, in the degenerate case, the authors derive conditions for the coexistence of two- and three-dimensional invariant tori.
Consider a singularly perturbed system of differential equations. Assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic, i.e., all of its eigenvalues have nonzero real parts. Under some general assumptions, the authors study the existence of invariant manifolds of the system homeomorphic to tori of various dimensions. Such manifolds, which are called “invariant tori”, support multifrequency oscillations. For this aim, the authors derive the so-called determining equation. Every positive solution of the determining equation corresponds to an invariant torus.
It is shown that there is an \(m\)-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here \( m\) is the number of pure imaginary eigenvalues. In addition, in the degenerate case, the authors derive conditions for the coexistence of two- and three-dimensional invariant tori.
Reviewer: E. V. Shchetinina (Samara)
MSC:
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C46 | Multifrequency systems of ordinary differential equations |
Citations:
Zbl 0760.34041References:
| [1] | Bukaty, V.R., On the Existence of Invariant Tori of Singularly Perturbed Systems, Vestnik St. Peterburg. Univ. Math. Ser. 1, 1992, vol. 2(8), pp. 110–112. · Zbl 0760.34041 |
| [2] | Hale, J.K., Integral Manifolds of Perturbed Differential Systems, Ann. of Math., 1961, vol. 73, no. 3, pp. 496–531. · Zbl 0163.32804 · doi:10.2307/1970314 |
| [3] | Bibikov, Yu.N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii (Multifrequency Nonlinear Oscillations and Their Bifurcations), Leningrad: Leningrad. Univ., 1991. · Zbl 0791.34032 |
| [4] | Bibikov, Yu.A., Kurs obyknovennykh differentsial’nykh uravnenii (Course of Ordinary Differential Equations), Moscow, 1991. |
| [5] | Bautin, N.N., On Periodic Solutions of a System of Differential Equations, Prikl. Mat. Mekh., 1954, vol. 18, no. 1, p. 128. |
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