Multifrequency oscillations of singularly perturbed systems (Q414043)

The paper is a continuation of the previous work by \textit{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.