The Period Function for Hamiltonian Systems with Homogeneous Nonlinearities

Gasull, Armengol; Guillamón, Antoni; Mañosa, Vı́ctor; Mañosas, Francesc

doi:10.1006/jdeq.1997.3296

Cited by 46 publications

(46 citation statements)

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“…in contradiction with (10). This completes the proof for the case when X(x, y) and Y (x, y) are polynomials.…”

Section: Proof Of Theoremsupporting

confidence: 59%

“…After Pleshkan [20] classified the cubic polynomial differential systems with homogeneous nonlinearities and recently isochronous centers of polynomial systems with homogeneous nonlinearities of degree five have been classified in [21]. Several authors (see [4,10,25]) have shown that Hamiltonian systems have no isochronous centers if they have homogeneous nonlinearities. For the Hamiltonian polynomial systems with Hamiltonian of the form H(x, y) = F (x) + G(y) the unique isochronous center is the linear one, see [5].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Isochronicity and linearizability of planar polynomial Hamiltonian systems

Llibre

Romanovski

2015

Journal of Differential Equations

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Abstract. In this paper we study isochronicity and linearizability of planar polynomial Hamiltonian systems. First we prove a theorem which supports a negative answer to the following open question: Do there exist planar polynomial Hamiltonian systems of even degree having an isochronous center? stated by Jarque and Villadelprat in the J. Differential Equations 180 (2002), 334-373. Additionally we obtain some conditions for linearizability of complex cubic Hamiltonian systems.

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“…in contradiction with (10). This completes the proof for the case when X(x, y) and Y (x, y) are polynomials.…”

Section: Proof Of Theoremsupporting

confidence: 59%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Isochronicity and linearizability of planar polynomial Hamiltonian systems

Llibre

Romanovski

2015

Journal of Differential Equations

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“…If the Hamiltonian takes the value h over a solution then it is said that the solution has energy level h. We shall assume that the origin is a nondegenerate center and that H(0, 0)=0. In addition, one can prove (see [6]) that the set of periodic orbits in the period annulus can be parameterized by the energy. Therefore, in the sequel we will denote the periodic orbit in P of energy level h by c h .…”

Section: Definitions and Main Resultsmentioning

confidence: 99%

Area-Preserving Normalizations for Centers of Planar Hamiltonian Systems

Mañosas

Villadelprat

2002

Journal of Differential Equations

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It is well known that a nondegenerate center of an analytic Hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates. This normal form, that we denote by CNF, does not depend on the coordinate transformation. In this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them. For instance, we describe the space of coordinate transformations that bring a Hamiltonian nondegenerate center to its CNF, and we prove that they are all canonical when the center is non-isochronous. We also show that two Hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function.

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“…Hence, the isochronous center problem is equivalent to find a transversal commutator in a punctured neighborhood of the origin. There are only a few families of polynomial differential systems in which a complete classification of the isochronous centers is known, and almost all of them have a polynomial commutator, see for instance [5,10,15,16,18,19]. Moreover, several works are devoted to find polynomial commutators for different families of polynomial differential systems, see [10,18].…”

Section: Definition 3 Two Vector Fields X and Y Commute If Their Liementioning

confidence: 99%