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**On a generalized Lorenz canonical form of chaotic systems.**
*(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.

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 of systems with hyperbolic behavior |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |

34H05 | Control problems involving ordinary differential equations |

93B10 | Canonical structure |

37N05 | Dynamical systems in classical and celestial mechanics |

93C10 | Nonlinear systems in control theory |

### Keywords:

Chaos; Lorenz system; Chen system; canonical form; chaos design; numerical simulation; controlling the Duffin oscillator; bifurcation; chaos synthesis
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\textit{S. Čelikovský} and \textit{G. Chen}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, No. 8, 1789--1812 (2002; Zbl 1043.37023)

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### References:

[1] | DOI: 10.1016/S0375-9601(00)00777-5 · Zbl 0972.37019 |

[2] | Čelikovský S., Kybernetika 30 pp 403– |

[3] | DOI: 10.1142/S0218127499001024 · Zbl 0962.37013 |

[4] | Lü J., Int. J. Bifurcation and Chaos |

[5] | DOI: 10.1103/PhysRevE.50.R647 |

[6] | Ueta T., Int. J. Bifurcation and Chaos 10 pp 1917– |

[7] | Vaněček A., Control Systems: From Linear Analysis to Synthesis of Chaos (1996) |

[8] | Wang X., Contr. Th. Appl. 16 pp 779– |

[9] | DOI: 10.1007/978-1-4612-1042-9 |

[10] | Yang L.-B., ACTA Phys. Sin. 49 pp 1039– |

[11] | Yu X., Int. J. Bifurcation and Chaos 10 pp 1987– |

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