Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four (Q1599263)

The authors consider centers of polynomial Hamiltonian planar differential systems \((S_H)\), \(x'=-H_y(x,y)\), \(y'=H_x(x,y)\). They prove that, if the Hamiltonian function \(H(x,y)\) is a five-degree polynomial, then the system does not have isochronous centers. This paper also contains some results for Hamiltonians of the type \(H(x,y)=A(x)+ B(x)y+C(x)y^2+ D(x)y^3\), with \(A(x)\), \(B(x)\), \(C(x)\), \(D(x)\) analytic functions. For instance, if \(D(x)=a_0 x^2+b_0 x+c_0\), \(a_0\neq 0\), then, under an additional technical assumption, the period function of any center of \((S_H)\) is unbounded.