Single Resonance near a Singularity

Abstract

The perturbation of a regular integrable Hamiltonian may lead to bifurcations in the phase portrait. Let us consider the case of only one resonance and the simple case of one degree of freedom. Let H₀ be the Hamiltonian of a differential rotator:

$$ H_0 = \frac{1} {2}\mathcal{V}_1^o (x_1^2 + y_1^2 ) + \frac{1} {8}\mathcal{V}_{11}^o (x_1^2 + y_1^2 )^2 $$

(9.1)

and let us assume that $\mathcal{V}_1^o$ and $\mathcal{V}_{11}^o$ have opposite signs, e.g. $\mathcal{V}_1^o < 0$ < 0 and $\mathcal{V}_{11}^o < 0$ > 0. In that case, H₀ has a minimum on a circle of radius $\sqrt { - 4 - \nu _1^o /\nu _{11}^o }$ on which the direction of motion changes. This is the classical case of a twist mapping and the Poincaré-Birkhoff theorem predicts that, when the rotator is perturbed, new centers and saddle points may appear in the phase portrait near the place where the frequency of the undisturbed rotator is zero (see [63]).

Since the considered Hamiltonian system has one degree of freedom, the separatrices meet, forming a well-defined structure without the possibility of chaotic motions.