Response Solutions in Singularly Perturbed, Quasi-Periodically Forced Nonlinear Oscillators

Abstract

For a quasi-periodically forced oscillator, response solutions are quasi-periodic ones having the same frequencies as that of the forcing function. Typically being the most stable or robust ones, they form an important class of oscillatory solutions of the oscillator. Since the introduction of the notion in the 1950 s, response solutions have been extensively studied in regularly perturbed, quasi-periodically forced oscillators with large, small, or zero damping coefficients with recent advances being made toward some singularly perturbed and highly or completely degenerate cases. The aim of the present paper is to make a general investigation toward the existence and stability properties of response solutions in singularly perturbed, quasi-periodically forced oscillators of the normal form

$$\begin{aligned} \left\{ \begin{array}{l} {{\dot{\theta }}}=\omega ,\\ {\dot{z}}=\epsilon ^\alpha A(\epsilon )z+\epsilon ^{\alpha +\beta } f(\theta ,z,\epsilon ),\qquad (\theta ,z)\in {\mathbb {T}}^d \times {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

where $\alpha \in {\mathbb {R}}$ and $\beta >0$ are constants, $\omega \in {\mathbb {R}}^d$ is the forcing frequency vector, $0<\epsilon \ll 1$ is a parameter, and f is of a finite order of smoothness. The normal form includes strongly damped oscillators of the form

$$\begin{aligned} \ddot{x}+\frac{1}{\epsilon } \dot{x}-g(x)=\epsilon ^{\chi _1}f(\omega t),\;\qquad \; x\in {\mathbb {R}}^1 \end{aligned}$$

and damping-free oscillators with large potentials of the form

$$\begin{aligned} \ddot{x}-\frac{1}{\epsilon }g(x)=\epsilon ^{\chi _2}f(\omega t),\;\qquad \; x\in {\mathbb {R}}^1, \end{aligned}$$

where $\chi _1,\chi _2$ are constants. With respect to the normal form, we show the existence of Floquet response tori for all or the majority of sufficiently small $\epsilon >0$ in three typical cases. Not only do our results on response solutions and their stabilities extend some existing ones in both regularly and singularly perturbed cases by allowing finite smoothness of potential and forcing functions, but also they provide new insides to the nature of these solutions, for instance the coexistence of response solutions of both hyperbolic and elliptic types in a given quasi-periodically forced, degenerate nonlinear oscillator.