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Quasiperiodic perturbations of two-dimensional Hamiltonian systems. (English. Russian original) Zbl 1394.34088

Differ. Equ. 53, No. 12, 1557-1566 (2017); translation from Differ. Uravn. 53, No. 12, 1607-1615 (2017).
The paper studies quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems of the form \[ \dot x = \frac{\partial H(x,y)}{\partial y}+\varepsilon g(x,y,t), \quad \dot y = \frac{\partial H(x,y)}{\partial x}+\varepsilon f(x,y,t), \] where \(\varepsilon\) is a small positive parameter and the Hamiltonian \(H\) and the functions \(g\) and \(f\) are sufficiently smooth (analytic) and, together with their partial derivatives of order \(\leq 2\), uniformly bounded in a domain \(G\subset \mathbb R^2\) or \(G \times \mathbb S^1\). Additionally, the functions \(g\) and \(f\) are continuous and almost periodic in \(t\) uniformly with respect to \((x,y)\in G\) with rationally independent frequencies. The behavior of the solutions to the considered system in a neighborhood of resonance and nonresonance levels is considered. Conditions for the existence of \(m\)-dimensional resonance tori are derived, and the global behavior of solutions in domains separated from the unpertubed separatrices is discussed. The results of the paper are illustrated by means of the Duffing equation \[ \dot x = y, \quad \dot y = x-x^3 + \varepsilon(p_1 y + F(t)), \] where either \(F(t)=p_2 \sin(\omega_1 t)+p_3 \sin(\omega_2 t)\) or \(F(t)=p_2 \sin(\omega_1 t) \sin(\omega_2 t)\) and \(p_k\), \(k=1,2,3\) are parameters.

MSC:

34C46 Multifrequency systems of ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
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