Integrability and bifurcation of a three-dimensional circuit differential system

Abstract

<p style='text-indent:20px;'>We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.</p>