A new integrable system on the sphere

We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.Comment: LaTeX, 15 page

(Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and 1:2 (period-doubling). The twist, which is the derivative of the rotation number with respect to the action, is studied near these bifurcations. When the twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is not satisfied, with interesting consequences for the dynamics. We show that near the saddle-centre bifurcation the twist always vanishes. At this bifurcation a ``twistless'' torus is created, when the resonance is passed. The twistless torus replaces the colliding periodic orbits in phase space. We explicitly derive the position of the twistless torus depending on the resonance parameter, and show that the shape of this curve is universal. For the period doubling bifurcation the situation is different. Here we show that the twist does not vanish in a neighborhood of the bifurcation.Comment: 18 pages, 9 figure

Vanishing Twist in the Hamiltonian Hopf Bifurcation

The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus-focus point. We explicitly calculate the frequency map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy-momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus-focus point in the image of the energy-momentum map. This is equivalent to the vanishing of twist in a Poincar\'e map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the non-linear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian Hopf bifurcation.Comment: 18 pages, 4 figure

The Vanishing Twist in the Restricted Three Body Problem

This paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three-body problem. Near the Lagrangian equilibrium $\mathcal{L}_4$ a twistless torus is created near the tripling bifurcation of the short period family. Decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3/10 and 2/7. This scenario is studied by numerically integrating the regularised Hamiltonian flow, and finding rotation numbers of invariant curves in a two-dimensional Poincar\'{e} map. To corroborate the numerical results the Birkhoff normal form at $\mathcal{L}_4$ is calculated to eighth order. Truncating at this order gives an integrable system, and the rotation numbers obtained from the Birkhoff normal form agree well with the numerical results. A global overview for the mass ratio $\mu \in (\mu_4, \mu_3)$ is presented by showing lines of constant energy and constant rotation number in action space