Separation of motions in a neighborhood of a semistable cycle (Q2480466)

This paper consider an arbitrary \(C^\infty\) system \(\dot x=F(x)\) of ordinary differential equations in \(\mathbb{R}^n\), \(n\geq 2\), that has a periodic trajectory \(L_0\) of the type of a simple saddle-node. The authors introduce the two-dimensional vector field \(F_0\) that is the restriction of the original system to the center manifold \(W^c(L_0)\) of the cycle \(L_0\), and consider the simplest form to which the vector field \(F_0\) can be reduced in some sufficiently small neighborhood \(U\subset W^c(L_0)\) of the cycle \(L_0\). The paper shows that there exists local coordinates \((r,\psi): | r| \leq r_0\), \(r_0=\text{const}>0\), \(0\leq \psi \leq 2\pi \pmod {2\pi}\) in \(U\) in which the field \(F_0\) takes the form \(F_0 = (r^2 + \alpha r^3) \partial /\partial r + \partial /\partial \psi\), where \(\alpha\in \mathbb{R}\) is some constant.