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
References:
[1]

K. Baba and Y. Nakai, A generalization of Magnu's theorem,, Osaka J. Math., 14 (1977), 403.   Google Scholar

[2]

A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems,, J. Differential Equations, 157 (1999), 373.  doi: 10.1006/jdeq.1999.3635.  Google Scholar

[3]

A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture,, Progress in Mathematics 190. Birkhäuser Verlag, 190 (2000).  doi: 10.1007/978-3-0348-8440-2.  Google Scholar

[4]

X. Jarque and J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four,, J. Differential Equations, 180 (2002), 334.  doi: 10.1006/jdeq.2001.4065.  Google Scholar

[5]

J. Llibre and V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems,, J. Differential Equations, 259 (2015), 1649.  doi: 10.1016/j.jde.2015.03.009.  Google Scholar

[6]

F. Mañosas and J. Villadelprat, Area-preserving normalizations for centers of planar Hamiltonian systems,, J. Differential Equations, 179 (2002), 625.  doi: 10.1006/jdeq.2001.4036.  Google Scholar

[7]

T. T. Moh, On the Jacobian conjecture and the configurations of roots,, J. Reine Angew. Math., 340 (1983), 140.   Google Scholar

[8]

M. Sabatini, A connection between isochronous Hamiltonian centres and the Jacobian conjecture,, Nonlinear Anal., 34 (1998), 829.  doi: 10.1016/S0362-546X(97)00604-4.  Google Scholar

show all references

References:
[1]

K. Baba and Y. Nakai, A generalization of Magnu's theorem,, Osaka J. Math., 14 (1977), 403.   Google Scholar

[2]

A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems,, J. Differential Equations, 157 (1999), 373.  doi: 10.1006/jdeq.1999.3635.  Google Scholar

[3]

A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture,, Progress in Mathematics 190. Birkhäuser Verlag, 190 (2000).  doi: 10.1007/978-3-0348-8440-2.  Google Scholar

[4]

X. Jarque and J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four,, J. Differential Equations, 180 (2002), 334.  doi: 10.1006/jdeq.2001.4065.  Google Scholar

[5]

J. Llibre and V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems,, J. Differential Equations, 259 (2015), 1649.  doi: 10.1016/j.jde.2015.03.009.  Google Scholar

[6]

F. Mañosas and J. Villadelprat, Area-preserving normalizations for centers of planar Hamiltonian systems,, J. Differential Equations, 179 (2002), 625.  doi: 10.1006/jdeq.2001.4036.  Google Scholar

[7]

T. T. Moh, On the Jacobian conjecture and the configurations of roots,, J. Reine Angew. Math., 340 (1983), 140.   Google Scholar

[8]

M. Sabatini, A connection between isochronous Hamiltonian centres and the Jacobian conjecture,, Nonlinear Anal., 34 (1998), 829.  doi: 10.1016/S0362-546X(97)00604-4.  Google Scholar

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