Homoclinic bifurcations in Chuaʹs circuit

In this paper we study the possible relationship between the Birth of the Double Scroll [L.O. Chua et al., IEEE-CAS 33(11) (1986) 1073] and the homoclinic bifurcations in the traditional Chuaʹs equations. Using a one-dimensional Poincaré map we determine the existence of secondary symmetric homoclinic orbits of Shilʹnikov type, born with the Chuaʹs attractor, connecting unstable and stable manifolds of the trivial equilibrium point. In addition, taking into account the presence of other homoclinic orbits for the asymmetric attractor and heteroclinic orbits for the symmetric attractor (connecting unstable and stable manifold of the non-trivial equilibrium points), we suggest a hypothesis about the Birth of Double Scroll structure on the (α,β) plane.