A one-parameter family of periodic orbits with frequency $\omega$ and energy $E$ of an autonomous Hamiltonian system is degenerate when $E^{\prime }(\omega )=0$. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of the bifurcating “homoclinic torus” is the homoclinic angle and a new asymptotic formula for it is derived. The theory is constructive, and so is useful for physical applications and in numerics.

• Received 20 June 2005

DOI:https://doi.org/10.1103/PhysRevLett.95.104301