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We have developed a method for complementing an arbitrary classical dynamicalsystem to a quantum system using the Lorenz and R\"ossler systems as examples.The Schr\"odinger equation for the corresponding quantum statistical ensembleis described in terms of the Hamilton-Jacobi formalism. We consider both theoriginal dynamical system in the position space and the conjugate dynamicalsystem corresponding to the momentum space. Such simultaneous consideration ofmutually complementary position and momentum frameworks provides a deeperunderstanding of the nature of chaotic behavior in dynamical systems. We haveshown that the new formalism provides a significant simplification of theLyapunov exponents calculations. From the point of view of quantum optics, theLorenz and R\"ossler systems correspond to three modes of a quantizedelectromagnetic field in a medium with cubic nonlinearity. From thecomputational point of view, the new formalism provides a basis for theanalysis of complex dynamical systems using quantum computers.

The study of Lorenz and R\"ossler strange attractors by means of quantum theory