# American Institute of Mathematical Sciences

October  2016, 36(10): 5245-5255. doi: 10.3934/dcds.2016029

## Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems

 1 Departamento de Matemática, Universidade Federal de São Carlos, Rod. Washington Luís, Km 235 - C.P. 676 - 13565-905, São Carlos, SP 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia 3 Departamento de Física, Química e Matemática, Universidade Federal de São Carlos, 18052-780, S.P., Brazil

Received  October 2015 Revised  December 2015 Published  July 2016

In this paper we completely characterize trivial polynomial Hamiltonian isochronous centers of degrees $5$ and $7$. Precisely, we provide simple formulas, up to linear change of coordinates, for the Hamiltonians of the form $H = \left(f_1^2 + f_2^2 \right)/2$, where $f = (f_1, f_2): \mathbb{R}^2\to \mathbb{R}^2$ is a polynomial map with $\det D f = 1$, $f(0,0) = (0,0)$ and the degree of $f$ is $3$ or $4$.
Citation: Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029
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##### References:
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