# American Institute of Mathematical Sciences

May  2013, 33(5): 1835-1856. doi: 10.3934/dcds.2013.33.1835

## Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance

 1 Department of Mathematics - University of Torino, Via Carlo Alberto, 10 - 10123 Torino, Italy, Italy 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received  January 2012 Revised  July 2012 Published  December 2012

In this paper we deal with the existence of unbounded orbits of the map $$\left\{\begin{array}{l} θ_1= θ+\frac{1}{ρ} [u(θ)-l_1(ρ)]+h_1(ρ, θ), ρ_1=ρ-u'(θ)+l_2(ρ)+h_2(ρ, θ), \end{array} \right.$$ 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.
Citation: Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
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