  We have developed a method for complementing an arbitrary classical dynamical
system to a quantum system using the Lorenz and R\"ossler systems as examples.
The Schr\"odinger equation for the corresponding quantum statistical ensemble
is described in terms of the Hamilton-Jacobi formalism. We consider both the
original dynamical system in the position space and the conjugate dynamical
system corresponding to the momentum space. Such simultaneous consideration of
mutually complementary position and momentum frameworks provides a deeper
understanding of the nature of chaotic behavior in dynamical systems. We have
shown that the new formalism provides a significant simplification of the
Lyapunov exponents calculations. From the point of view of quantum optics, the
Lorenz and R\"ossler systems correspond to three modes of a quantized
electromagnetic field in a medium with cubic nonlinearity. From the
computational point of view, the new formalism provides a basis for the
analysis of complex dynamical systems using quantum computers.
