Non-convergence of formal integrals of motion

We consider formal integrals of motion in 2D Hamiltonian dynamical systems, calculated with the normal form method of Giorgilli (1979 Comput. Phys. Commun. 16 331). Three different non-integrable and one integrable systems are considered. The time variation DI of the formal integral I is found as a function of the order of truncation N of the integral series. An optimal order of truncation is found from the minima of the variations DI. The level lines of the integral I, representing theoretical invariant curves on a Poincare surface of section, are compared with the real invariant curves. When chaos is limited, excellent agreement is found between the theoretical and the real invariant curves, if the order of truncation is close to the optimal order. The agreement is poor (a) far from the optimal order and (b) when chaos is pronounced. The optimal order, calculated as a function of the distance R from the origin, decreases when R increases. The decrease is rather smooth in the 1:1 resonance, but it has abrupt steps in the case of a higher order (4:3) resonance. In the case of an integrable Hamiltonian, a formal integral IF is found which is a function of the exact integral I and of the Hamiltonian, given as power series of the canonical variables. The series converges only within a domain of convergence. The radius of convergence along a particular direction is calculated with the d'Alembert and Cauchy methods. The theoretical invariant curves agree with the real invariant curves only within the domain of convergence of IF. In the case of non-integrable Hamiltonians, we calculate 'pseudo-radii of convergence' that tend to zero as the order of truncation N increases.

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