Characterizing isochronous points and computing isochronous sections

Abstract

We consider two-dimensional autonomous systems of differential equations $$\dot{x}=-y+\lambda x+P(x,y),\qquad \dot{y}=x+\lambda y+Q(x,y),$$ where $\lambda$ is a real constant and $P$ and $Q$ are smooth functions of order greater than or equal to two. These systems, so-called centre-focus type systems, have either a centre or a focus at the origin. We characterize the systems with a weak isochronous focus at the origin by means of their radial and azimuthal coefficients. We prove, in this case, the existence of a normalized vector field and an isochronous section which arrives at the origin with defined direction. We also provide algorithms that compute the radial and azimuthal coefficients, terms of normalized vector field and of isochronous section of a system. As applications, we analyze the weak isochronous foci for quadratic systems and for systems with cubic non-linearities, and we give a three-parameter family of Rayleigh equations with four local critical periods.