Bifurcation of limit cycles from a 4-dimensional center in \(\mathbb R^m\) in resonance \(1:N\)
The authors study how many limit cycles emerge from the periodic orbits of a center when we perturb it inside a given class of differential equations in dimension higher than two. More precisely, given \(m \geq 5\) and for every positive integer \(N \geq 2\) they consider the linear differential center \(\dot{x}=A x\) in \(\mathbb{R}^m\) that has at the origin a singular point with eigenvalues \(\pm i\), \(\pm Ni\) and \(0\) with multiplicity \(m-4\). Hence, this singular point has a 4-dimensional center in resonance \(1:N\). The authors are interested in studying how many limit cycles can bifurcate from the periodic orbits of this center when we perturb it in \(\mathbb{R}^m\) with \(m > 4\) inside the class of polynomial vector fields of the form linear plus a homogeneous nonlinearity of degree \(N\), that is, the perturbed system is \[ \dot{x}= Ax + \varepsilon F(x), \] where every component of \(F(x)\) is a linear polynomial plus a homogeneous polynomial of degree \(N\). Some explicit upper bounds for the number of limit cycles are given using the averaging method that improves and extends previous results obtained for \(\mathbb{R}^4\).
scientific article; zbMATH DE number 6015243