Infinite bifurcations, gaps and bubbles in Hamiltonian systems

Abstract

We study the bifurcations, gaps and bubbles along the characteristics of families of periodic orbits in Hamiltonian systems of two degrees of freedom. Bubbles (closed characteristics) appear when the rotation number along a family does not vary monotonically. There are infinite bifurcations, appearing when the rotation number along a family is rational, n/m. The most important bifurcations have m = 1, or m = 2 (equal period bnd double period bifurcations). In a rotating system the central family of periodic orbits is the family of direct circular orbits of the axisymmetric case. At each bifurcation n/1 we have either a change of stability (if n = odd), or a gap (if n= even) of the central family; this phenomenon we explain theoretically. In cases of infinite transitions from stability to instability along the same family, the successive intervals between bifurcations do not follow the universal Feigenbaum ratio. But we found also several cases of infinite pitchfork period doubling bifurcations, and in all cases the Feigenbaum ratio is consistent with the universal value δ = 8.72. All Feigenbaum sequences in rotating systems are followed by inverse Feigenbaum sequences, i.e. they form infinite bubbles. An explanation of this effect is given. We followed the evolution of the bubbles in some cases. For small perturbations, ϵ, no bubbles (of a certain type) exist; as ϵ increases one bubble appears and grows, then this bubble generates a second order bubble, etc, until an infinite sequence is formed. We found also sequences of infinite bubbles in non-rotating systems and their evolution. Some bubbles are “floating”, i.e. they are not connected to other families. An interesting example of the generation of such a “floating bubble” is presented.