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Article Contents

# 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 $$\left\{\begin{array}{l} θ_1= θ+\frac{1}{ρ} [u(θ)-l_1(ρ)]+h_1(ρ, θ), ρ_1=ρ-u'(θ)+l_2(ρ)+h_2(ρ, θ), \end{array} \right.$$ 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.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B15.

 Citation:

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