Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations (Q2075111)

The author studies the effect of oscillatory perturbations on asymptotically autonomous Hamiltonian systems \[ \begin{cases} \dot{x}=\tfrac{\partial H}{\partial y}(x,y,t),\\ \dot{y}=-\tfrac{\partial H}{\partial x}(x,y,t)+F(x,y,t), \end{cases} \] satisfying \[ \lim_{t\to\infty}H(x,y,t)=H_0(x,y),\quad \lim_{t\to\infty}F(x,y,t)=0 \] uniformly on compact subsets of \({\mathbb R}^2\), with \[H_0(x,y)=\frac{x^2+y^2}{2}+O(\sqrt{x^2+y^2}^3).\] Assuming that the perturbations preserve the trivial equilibrium and satisfy a resonance condition, the behavior of the perturbed trajectories near the equilibrium is studied. Depending on the structure of the perturbations, various different long-term behaviors (regimes) are possible. Among them are a phase locking and a phase drifting. In addition, the paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both of the above regimes. Here, the stability analysis requires a combination of averaging techniques and the construction of Lyapunov functions.