Completely integrable dynamical systems of Hopf–Langford type

Abstract In this work we consider a three-dimensional autonomous system of nonlinear ordinary differential equations, which may be thought of as a generalization of the well-known Hopf–Langford system introduced about forty years ago. This dynamical system turned out to be equivalent to the nonlinear force-free Duffing oscillator. In three special cases, it is found to be completely integrable. To the best of our knowledge, these facts have not been noticed so far in the rich literature on the subject. In the aforementioned three special cases, the general solutions of the respective systems are expressed in explicit analytical form by means of elementary and Jacobi elliptic functions depending on the values of the system parameters. This allowed us to characterize in details the dynamics of the regarded systems.