A New Route to Chaos: Generation of Spiral Characteristics

Abstract

The best known route to chaos in Hamiltonian systems of two degrees of freedom is the Feigenbaum sequence of period doubling bifurcations (Feigenbaum 1978, Coullet and Tresser 1978). As one parameter h (e.g. the energy) changes, a stable family of periodic orbits becomes unstable (at h=h₁) and then a double period family bifurcates, which is stable. The characteristic of this family, that gives the coordinate x (intersection of the periodic orbit with the x-axis) as a function of h, is directed to the right, i.e. towards larger h. At a larger value of the parameter (h=h₂) the double period family becomes unstable and generates a period-4 family and so on. The intervals (h₁h₂), (h₂h₃)… decrease approximately geometrically with universal ratio δ=8.72 (Bennettin et al. 1980). Thus at the limit of the sequence h₁,h₂,h₃… we have an infinity of unstable periodic orbits, that generate a large degree of chaos.