  We explore the transition from order to chaos for the Bohmian trajectories of
a simple quantum system corresponding to the superposition of three stationary
states in a 2D harmonic well with incommensurable frequencies. We study in
particular the role of nodal points in the transition to chaos. Our main
findings are: a) A proof of the existence of bounded domains in configuration
space which are devoid of nodal points, b) An analytical construction of formal
series representing regular orbits in the central domain as well as a numerical
investigation of its limits of applicability. c) A detailed exploration of the
phase-space structure near the nodal point. In this exploration we use an
adiabatic approximation and we draw the flow chart in a moving frame of
reference centered at the nodal point. We demonstrate the existence of a saddle
point (called X-point) in the vicinity of the nodal point which plays a key
role in the manifestation of exponential sensitivity of the orbits. One of the
invariant manifolds of the X-point continues as a spiral terminating at the
nodal point. We find cases of Hopf bifurcation at the nodal point and explore
the associated phase space structure of the nodal point - X-point complex. We
finally demonstrate the mechanism by which this complex generates chaos.
Numerical examples of this mechanism are given for particular chaotic orbits,
and a comparison is made with previous related works in the literature.

Quantum Physics