The return map for a planar vector field with nilpotent linear part: a direct and explicit derivation (Q1035167)

The author considers the planar system \[ \dot{X}=-Y, \quad \dot{Y}=X^3-Y^3, \] which has a nilpotent linear part. He proves that for \(\epsilon\) small enough the solution of the system above satisfying \(X(0)=\epsilon, Y(0)=0\) first returns to the positive \(X\)-axis at the value \(\tilde{\tilde{X}}\) satisfying \[ \tilde{\tilde{X}}=\epsilon+\sum_{n=1}^\infty X_n\epsilon^{3n+1}, \] which is a convergent series. The coefficients \(X_n\) can be calculated iteratively, and the proof provides an algorithm for calculating iteratively the coefficients \(X_n\).