Unbounded solutions and periodic solutions of perturbed isochronous hamiltonian systems at resonance

In this paper we deal with the existence of unbounded orbits of the map $$ \left\{\begin{array}{l} \theta_1\theta+\d{1}{\rho}[\mu(\theta)-l_1(\rho)]+h_1(\rho, \theta),\\[3mm] \rho_1=\rho-\mu'(\theta)+l_2(\rho)+h_2(\rho, \theta), \end{array} \right. $$ where $\mu$ is continuous and $2\pi$-periodic, $l_1$, $l_2$ are continuous and bounded, $h_1(\rho, \theta)=o(\rho^{-1})$, $h_2(\rho, \theta)=o(1)$, for $\rho\to+\infty$. We prove that every orbit of the map tends to infinity in the future or in the past for $\rho$ large enough provided that $$[\liminf_{\rho\to+\infty}l_1(\rho), \limsup_{\rho\to+\infty}l_1(\rho)]\cap Range(\mu)=\emptyset$$ and other conditions hold. On the basis of this conclusion, we prove that the system $ Jz'=\nabla H(z)+f(z)+p(t)$ has unbounded solutions when $H$ is positively homogeneous of degree 2 and positive. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this system.