*(English)*Zbl 1043.37023

Summary: This paper shows that a large class of systems, as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [*S. Čelikovský* and *S. Vaněček*, Kybernetika 30, 403–424 (1994; Zbl 0823.93026) and Control systems. From linear analysis to synthesis of chaos, London: Prentice Hall (1996; Zbl 0874.93006)], and also covers the so-called Chen system, recently introduced in [*G. Chen* and *T. Ueta*, ibid. 9, 1465–1466 (1999; Zbl 0962.37013) and ibid. 10, 1917–1931 (2000)].

Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some other closely related topics are studied and discussed, too.

##### MSC:

37D45 | Strange attractors, chaotic dynamics |

34C28 | Complex behavior, chaotic systems (ODE) |

34H05 | ODE in connection with control problems |

93B10 | Canonical structure of systems |

37N05 | Dynamical systems in classical and celestial mechanics |

93C10 | Nonlinear control systems |