Dynamical systems study is fundamental for physical, technical and socio-economic systems management tasks. In this paper the authors have developed a method for complementing an arbitrary classical dynamical system to a quantum system using the Lorenz and Rössler systems as examples. The Schrödinger equation for the corresponding quantum
statistical ensemble is described in terms of the Hamilton — Jacobi formalism. The authors 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. The authors 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 Rössler 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.

Keywords: dynamical system; Lorenz and Rössler systems; quantum system; Hamilton — Jacobi formalism;
Lyapunov exponents; quantum informatics; socio-economic systems management.

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