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.

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

CAPIETTO, Anna;DAMBROSIO, Walter;
2013-01-01

Abstract

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.
2013
33
1835
1856
Hamiltonian system; Resonance; Periodic solution; Unbounded solution
A. Capietto; W. Dambrosio; T. Ma; Z. Wang
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/129139
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