Twistless tori near low order resonances

In this paper we investigate the behaviour of the twist near low order resonances of a periodic orbit or an equilibrium of a hamiltonian system with two degrees of freedom. Namely, we analyse the case when a Hamiltonian has multiple eigenvalues (the hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case when the system possesses a periodic orbit, which multipliers equal to $1$ (the saddle-centrе bifurcation) or $-1$ (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of an equilibrium under consideration we prove the existence of the “twistless” torus for sufficiently small values of the bifurcation parameter. The explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived.