ISOLAS, CUSPS AND GLOBAL BIFURCATIONS IN AN ELECTRONIC OSCILLATOR

The aim of the present work is to describe the bifurcation behaviour o f a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch ; that is, their bifurcation diagram showed an eight-shaped isola, wit h a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its dest ruction in an isola centre with a cusp of periodic orbits. Finally, th e introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifur cation) and to homoclinic connections of the non-trivial equilibria. T he isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil'nikov homoclinic and heteroc linic connections, related to the symmetric and asymmetric periodic or bits, emerge from T-points and end at Shil'nikov-Hopf singularities.