On the number of effective integrals in galactic models

Three different numerical techniques are tested to determine the number of integrals of motion in dynamical systems with three degrees of freedom.

1. The computation of the whole set of Lyapunov Characteristic Exponents (LCE).
2. The triple sections in the configurations space.
3. The Stine-Noid box-counting technique.

These methods are applied to a triple oscillator with coupling terms of the third order. Cases are found for which one effective integral besides the Hamiltonian subsists during a very long time. Such orbits display simultaneously chaotic and quasi-periodic motion, according to which coordinates are considered. As an application, the LCE procedure is applied to a triaxial elliptical galaxy model. Contrary to similar 2-dimensional systems, this 3-dimensional one presents noticeable zones in the phase space without any non-classical integral.