Invariant surfaces and orbital behaviour in dynamical systems of 3 degrees of freedom. II

Two formal integrals of motion besides the Hamiltonian are found in a system of three degrees of freedom with three unperturbed frequencies ω₁, ω₂, ω₃ which are close to, or exactly at, the double resonance 6:4:3. The integrals are truncated at various orders and they are used to find theoretical invariant surfaces on a surface of section.The theoretical invariant surfaces are then compared with the empirical results. The agreement is improved as the order of the integrals increases from the 5th up to the 11th degree. This is surprising because theoretical considerations indicate that one formal integral breaks down beyond order 6, because of the second resonance 4:3. (However it is possible to construct one further integral by a different method). As expected the agreement is worse when the perturbation becomes large. A check of the constancy of the truncated integrals along various orbits shows that the relative variations become smaller as the order increases; they become larger as the perturbation increases.