Some remarks on inverse Jacobi multipliers around Hopf singularities
The authors consider systems of real ordinary differential equations of the form \(\dot{x}=-y+F_1(x,y,z)\), \(\dot{y}=x+F_2(x,y,z)\), \(\dot{z}=\lambda z+F_3(x,y,z)\), where \(\lambda \neq 0\), defined in a neighbourhood of \(0\in \mathbb{R}^3\). The origin is a Hopf singularity for the corresponding vector field \(X\). A function \(V\) is an ``inverse Jacobi multiplier'' (IJM) if it is not locally null and satisfies the linear first-order partial differential equation \(XV=V\)div\(X\). The authors study several properties of IJMs which are relevant to the study of local bifurcations of periodic orbits. When the singularity is a saddle-focus they show that: (i) any two locally smooth and non-flat linearly independent IJMs have the same Taylor expansion; (ii) to any smooth and non-flat \(V\) can be associated exactly one smooth center manifold \(W_c\) of \(X\) such that \(W_c\subset V^{-1}(0)\). They consider also the question whether the properties of the vanishing set \(V^{-1}(0)\) proved in the \(3\)-dimensional case remain valid in higher dimension.
scientific article; zbMATH DE number 6390545
scientific article