Change of the adiabatic invariant at a separatrix in a volume-preserving 3D system

A 3D volume-preserving system is considered. The system differs by a small perturbation from an integrable one. In the phase space of the unperturbed system there are regions filled with closed phase trajectories, where the system has two independent first integrals. These regions are separated by a 2D separatrix passing through non-degenerate singular points. Far from the separatrix, the perturbed system has an adiabatic invariant. When a perturbed phase trajectory crosses the two-dimensional separatrix of the unperturbed system, this adiabatic invariant undergoes a quasi-random jump. The formula for this jump is obtained. If the geometry of the system allows for multiple separatrix crossings, the destruction of adiabatic invariance is possible, leading to chaotic behaviour in the system. An example of such a system is given.