Closed Curves of Global bifurcations in Chua's equation: a Mechanism for their Formation

An important task for the understanding of the dynamics of parameterized systems of autonomous ordinary differential equations is the determination of the organizing centres as well as the bifurcations they exhibit (see, e.g. [Guckenheimer & Holmes, 1983; Kuznetsov, 1995; Nayfeh & Balachandran, 1995; Wiggins, 1996] as general references). The combination of analytical and numerical tools is usually needed due to the existence of a complex bifurcation scenario. To complement theoretical results, numerical continuation is done in one parameter (bifurcation diagram, for example, period versus parameter) or in two (or more) parameters (bifurcation set, loci where bifurcations occur in the parameter space). The presence of isolas (isolated closed curves of solution branches) in bifurcation diagrams of periodic orbits has been detected, for example, in relation to Hopf curves (see, e.g. [Doedel et al., 1991]). The appearance of an isola configuration depends on the choice of the bifurcation parameter as well as on the shape of the bifurcation curve. A typical o-shaped isola of periodic orbits appears when moving inside resonance zones close to the tip where the corresponding Arnold’s tongues emerge (see, e.g. Fig. 12(a) in [Algaba et al., 2001]). Other kind of isolas of a certain type of periodic orbits was detected, in an electronic circuit, by [Fernández-Sánchez et al., 1997]. In that work, the mechanism of their formation is shown and their existence is related to cusp bifurcations and to Shil’nikov homoclinic connections. In other works, some kinds of isolas have appeared in relation with homoclinic bifurcations/tangencies. In this way [Hirschberg & Laing, 1995] have shown that primary periodic orbits lie on an infinity of isolas in a neighborhood of a homoclinic tangency to a periodic orbit. Also [Champneys & Rodŕıguez-Luis, 1999] have found, unfolding a nontransverse Shil’nikov–Hopf bifurcation,