Non-isochronicity of the center in polynomial Hamiltonian systems (Q975260)

The authors study the problem of isochronicity for polynomial Hamiltonian systems with the Hamiltonian \[ H(x,y)=\frac{x^2+y^2}2+\sum_{i+j=m}^na_{ij}x^i y^j, \tag{1} \] where \(x,y,a_{ij}\in \mathbb{R}, \;\sum_{i+j=m}a_{ij}^2\neq 0\), \(m, n \in \mathbb{N}\) and \(n\geq m \geq 3\). Let \(M_k= \sum_{i=0}^k a_{2 k -2 i, 2i} w_{i,k-i} \), where \[ w_{ij}=\frac{(2i)! (2j)! \pi}{i!j!(i+j)! 2^{2i+2j+1}} \] and \(a_{2k-2i,2i}=0\) for \(2k<m\) or \(2k>n\). It has been proven in [\textit{X. Chen, V. G. Romanovski} and \textit{W. Zhang}, Nonlinear Anal., Theory Methods Appl. 68, No.~9A, 2769--2778 (2008; Zbl 1144.34021)] that (1) is non-isochronous if \(M_{2m-2}\leq 0\). In this, paper the authors give a new criterion for non-isochronicity of (1) for the case when \(M_{2m-2}>0\). Applying the criterion they also prove non-isochronicity of some classes of polynomial Hamiltonian systems.