  We develop the semiclassical method of complex trajectories in application to
chaotic dynamical tunneling. First, we suggest a systematic numerical technique
for obtaining complex tunneling trajectories by the gradual deformation of the
classical ones. This provides a natural classification of the tunneling
solutions. Second, we present a heuristic procedure for sorting out the least
suppressed trajectory. As an illustration, we apply our technique to the
process of chaotic tunneling in a quantum mechanical model with two degrees of
freedom. Our analysis reveals rich dynamics of the system. At the classical
level, there exists an infinite set of unstable solutions forming a fractal
structure. This structure is inherited by the complex tunneling paths and plays
the central role in the semiclassical study. The process we consider exhibits
the phenomenon of optimal tunneling: the suppression exponent of the tunneling
probability has a local minimum at a certain energy which is thus (locally) the
optimal energy for tunneling. We test the proposed method by comparison of the
semiclassical results with the results of the exact quantum computations and
find a good agreement.
