# Isochronicity of plane polynomial Hamiltonian systems

@article{Gavrilov1997IsochronicityOP, title={Isochronicity of plane polynomial Hamiltonian systems}, author={Lubomir Gavrilov}, journal={Nonlinearity}, year={1997}, volume={10}, pages={433-448} }

We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, isochronous Morse critical points of complex polynomial Hamiltonian functions. Our first result is that if the Hamiltonian function H is a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an isochronous Morse critical point, unless the associate Hamiltonian system is linear, that is to say H is of degree two. Our second result gives a topological obstruction for isochronicity…

## 40 Citations

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

- Mathematics
- 2002

Abstract In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree…

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

- Computer Science, PhysicsInt. J. Bifurc. Chaos
- 2019

The topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions H(x,y) such that the isochronic center lies on the level curve is studied.

Isochronicity for Several Classes of Hamiltonian Systems

- Mathematics
- 1999

Abstract In this paper we study isochronous centers of analytic Hamiltonian systems giving special attention to the polynomial case. We first revisit the potential systems and we show the connection…

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

- MathematicsErgodic Theory and Dynamical Systems
- 2000

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin…

Nonexistence of Isochronous Centers in Planar

- Mathematics
- 2002

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four…

Isochronicity and linearizability of planar polynomial Hamiltonian systems

- Mathematics
- 2015

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 stated by…

Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

- MathematicsJournal of Differential Equations
- 2019

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and…

On the multiplicity of hyperelliptic integrals

- Mathematics
- 2003

Let be an Abelian integral, where H = y2 ? xn+1 + P(x) is a hyperelliptic polynomial of Morse type, ?(t) a horizontal family of cycles in the curves {H = t}, and ? a polynomial 1-form in the…

CENTER AND ISOCHRONOUS CENTER AT INFINITY IN A CLASS OF PLANAR DIFFERENTIAL SYSTEMS

- 2008

In this paper, the conditions of center and isochronous center at the infinity for a class of planar differential systems are studied. By a transformation, we first transform the infinity (the…

Complete hyperelliptic integrals of the first kind and their non-oscillation

- Mathematics
- 2002

Let P(x) be a real polynomial of degree 2g + 1, H = y 2 + P(x) and δ(h) be an oval contained in the level set {H = h}. We study complete Abelian integrals of the form I(h) = ∫ δ(h) (α 0 + α 1 x +...…

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