Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems

Francisco Braun; Jaume  Llibre; Ana Cristina Mereu

doi:10.3934/dcds.2016029

Article Contents

2016, Volume 36, Issue 10: 5245-5255. Doi: 10.3934/dcds.2016029

This issue Previous Article Positive solutions to indefinite Neumann problems when the weight has positive average Next Article Restrictions on rotation sets for commuting torus homeomorphisms

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.

Received: October 2015

Revised: December 2015

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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 $4$.

Keywords:

Mathematics Subject Classification: Primary: 34A34; Secondary: 34C25, 37C37, 14R15.

Citation:

Related Papers

Cited by

References

[1]	K. Baba and Y. Nakai, A generalization of Magnu's theorem, Osaka J. Math., 14 (1977), 403-409.
[2]	A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations, 157 (1999), 373-413.doi: 10.1006/jdeq.1999.3635.
[3]	A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics 190. Birkhäuser Verlag, Basel, 2000.doi: 10.1007/978-3-0348-8440-2.
[4]	X. Jarque and J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four, J. Differential Equations, 180 (2002), 334-373.doi: 10.1006/jdeq.2001.4065.
[5]	J. Llibre and V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems, J. Differential Equations, 259 (2015), 1649-1662.doi: 10.1016/j.jde.2015.03.009.
[6]	F. Mañosas and J. Villadelprat, Area-preserving normalizations for centers of planar Hamiltonian systems, J. Differential Equations, 179 (2002), 625-646.doi: 10.1006/jdeq.2001.4036.
[7]	T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math., 340 (1983), 140-212.
[8]	M. Sabatini, A connection between isochronous Hamiltonian centres and the Jacobian conjecture, Nonlinear Anal., 34 (1998), 829-838.doi: 10.1016/S0362-546X(97)00604-4.