Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance

Abstract

In this paper, we are concerned with the existence of unbounded orbits of the mapping

\begin{align*} \theta_1&=\theta+2\pi+\frac{1}{\rho}\mu(\theta)+o(\rho^{-1}), \\ \rho_1&=\rho+c-\mu'(\theta)+o(1),\quad\rho\to\infty, \end{align*}

where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, that $\mu(\theta)$ is non-negative (or non-positive) and that $\mu(\theta)$ has finitely many degenerate zeros in $[0,2\pi]$. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large $\rho$. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$ ($F(x)=\int_0^xf(s)\,\mathrm{d} s$) and $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.