Rational first integrals for periodic systems (Q638664)

The authors consider the periodic differential system \[ \dot{x}=f(t,x), \] where \((t,x)\in S^1\times \mathbb{C}^n,\) \(S^1=\mathbb{R}\backslash (\mathbb{N}T),\) \(f(t,x)\in C^r(S^1\times \mathbb{C}^n)\), \(r\geq 1\), \(f(t+T,x)=f(t,x)\), \(f(t,0)\equiv 0.\) Sufficient conditions for non-existence and partial existence of rational first integrals for this system in a neighborhood of the constant solution \(x=0\) are presented. Two examples are given to illustrate the results.