Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree (Q1910033)

The author considers a vector field in the plane given by an ordinary differential equation \(dx/dt= X(x)\), where the components of \(X\) are homogeneous polynomials of degree \(n- 1\) and \(n+ 1\). Algebraic conditions for the origin to be a centre and a stable/unstable focus are given. These necessary and sufficient conditions are applied to simple examples (linear, cubic and quintic cases). For some classes of systems with homogeneous nonlinearities criteria for the origin to be a (uniformly isochronous) centre are also given. Finally, some errors in previously published phase portraits of homogeneous cubic systems are provided in the appendix.