Numerical study of periodic orbit properties in a dynamical system with three degrees of freedom

The locations and stability features of the main symmetrical periodic orbits in the potential $$V = tfrac{1}{2}left( {Ax^2 + By^2 + Cz^2 } right) - varepsilon xz^2 - eta yz^2 with sqrt {A:} sqrt {B:} sqrt C = 6:4:3$$ are calculated. Two resonant 1-periodic orbits reveal themselves to be the most important of the system. The third dimension and the additional coupling term have a large effect upon the emergence and stability of p.o. prolongated from the bi-dimensional cases 4∶3 and 2∶1. The existence of three main instability types leads to behaviours much more complicated than in systems with two degrees of freedom. Particularly the presence of complex instability, a new feature with respect to bi-dimensional problems, may produce large instability regions in the set of initial conditions. Some asymptotic curves emanating from unstable orbits are calculated in the four-dimensional space of section. The aspect of such curves is considerably modified when a perturbation is added in the third dimension. The neighbourhood of orbits suffering from complex instability is studied in the space of section and by means of the maximum Lyapunov Characteristic Number technique. It is shown that the motion can deviate far from the vicinity of the p.o. representative point as soon as the orbit is of complex instability. When the perturbation is large enough, the stochasticity produced by this type of instability can be very important.