Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators
Most applications of KAM theory have been restricted to smooth dynamical systems. This paper applies KAM theory to analyze the dynamical behavior of Duffing-type equations with impact. The authors consider a special but typical case of suplinear impact oscillators as follows: \[ \begin{cases} \ddot{x} + x^{2n+1} = p(t),\quad & \text{for }x(t)>0, \\ x(t)\geqslant 0\\ \dot{x}(t_0^+)=-\dot{x}(t_0^-), & \text{if } x(t_0) = 0, \end{cases}\eqno{(0.1)} \] where \(n\in {\mathbb N}^+\), and \(p(t)\) is a \(C^5\) periodic function with period \(1\). This paper proves that all solutions of the above equation are bounded, and that there are infinitely many periodic and quasiperiodic solutions. The idea of proving the boundedness is as follows. First, the equation can be described by a Hamiltonian function \(H(\phi, \rho, t)\) in action-angle variables defined on the whole space \({\mathbb T}\times {\mathbb R}^+ \times {\mathbb T}\) by means of transformation theory. The Hamiltonian \(H(\phi, \rho, t)\) is continuous in \(\phi\) but is \(C^5\) smooth in \(\rho\) and \(t\). The authors introduce an equivalent Hamiltonian \({\mathcal H}(\theta, I, \tau)\) through a transformation by exchanging the roles of the variables \(t\) and \(\phi\) outside of a large disk. The new Hamiltonian \({\mathcal H}\) is sufficiently smooth in \(\theta\) and \(I\), and therefore Moser's theorem is applicable and guarantees the existence of arbitrarily large invariant curves. Going back to the original system, every such curve is a base of a time-periodic and flow-invariant cylinder in the extended phase space \((x,\dot{x}, t)\in {\mathbb R}^+\times {\mathbb R} \times {\mathbb R}\), which leads to a bound of these solutions if the uniqueness of the initial value problem holds.
scientific article; zbMATH DE number 5870907
scientific article