TITLE
Geometric and statistical properties induced by separatrix crossings in
 volume-preserving systems 

AUTHORS
Anatoly Neishtadt
	Space Research Institute, Profsoyuznaya 84/32, Moscow 117810,Russia
	aneishta@iki.rssi.ru
Carles Sim\'o 
	Departament de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona
	Gran Via, 585, Barcelona 08007, Spain 
	carles@maia.ub.es
Alexei Vasiliev
	Space Research Institute, Profsoyuznaya 84/32, Moscow 117810,Russia
	valex@iki.rssi.ru

ABSTRACT
We consider a 3D volume preserving system inside the unit sphere. The system
is a small perturbation of an integrable one. Almost all phase trajectories of
the integrable system are closed curves. Under an arbitrarily small non-zero
perturbation all the interior of the sphere, up to a residue of a small 
measure, is apparently a domain of chaotic motion. The phenomenon is described
as a result of jumps in an adiabatic invariant of the system occurring when a 
phase trajectory crosses the 2D separatrix of the unperturbed (integrable)
system. The dynamical properties can be understood as a consequence of
splitting of invariant manifolds under the perturbation. A 2D return map 
generated by the system possesses strong stretching properties. This makes the
dynamics of the system close to hyperbolic. Numerical investigation of the 
statistical properties of the system demonstrates a good agreement with 
theoretical predictions made under the assumption that the system has a big 
ergodic component whose measure tends to the measure of the sphere's interior
as the perturbation tends to zero.  However, the system is not ergodic, and
stable periodic solutions are found, surrounded by stability islands of measure
exponentially small with the perturbation.
