Quasiperiodic perturbations of twodimensional Hamiltonian systems with nonmonotone rotation
The paper deals with quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems with nonmonotone rotation, namely \[\begin{array}{c} \dot{x} = \dfrac{\partial H(x,y)}{\partial y} + \varepsilon g(x,y,\omega_1 t, \omega_2 t,\dots , \omega_m t),\\ \dot{y} = -\dfrac{\partial H(x,y)}{\partial x} + \varepsilon f(x,y,\omega_1 t, \omega_2 t,\dots , \omega_m t),\tag{11} \end{array} \] where \(0 < \varepsilon \ll 1\) is a small positive parameter, the Hamiltonian function \(H\) and the functions \(g\), \(f\) giving rise to perturbations are sufficiently smooth (or analytic) and uniformly bounded, as well as the partial derivatives of order up to 2 in \(G\subset \mathbb{R}^2\) or in \(G\subset \mathbb{R} \times \mathbb{S}^1\). Furthermore, the functions \(g\) and \(f\) are at least continuous and quasiperiodic with respect to \(t\) uniformly with respect to \((x,y)\in G\) with independent frequencies \(\omega_1, \omega_2,\dots , \omega_m\) over the ring of integers. It is assumed that the system (11) with \(\varepsilon =0\) has the domain \(D\) filled by closed phase curves and separated from the separatrices and equilibrium states. The behavior of solutions in neighborhoods of resonance energy levels close to degenerate ones is considered. Conditions for the existence of resonant quasiperiodic solutions (\(m\)-dimensional invariant tori in the extended phase space) are found, and bifurcations in neighborhoods of degenerate resonance energy levels under nonconservative quasiperiodic perturbations is discussed (by quasiperiodic functions the authors mean almost periodic functions with finitely many independent frequencies over the field of rational numbers.) The results obtained are illustrated by an example of the asymmetric Duffing equation given by \[ \ddot{x}+x-1.6 x^2+x^3=\varepsilon [\beta \sin(t)\sin(\sqrt{5} t) + \delta \dot{x}], \] where \(\beta\), \(\delta\) are parameters.
scientific article; zbMATH DE number 7351550