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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.


MSC:
37D45Strange attractors, chaotic dynamics
34C28Complex behavior, chaotic systems (ODE)
34H05ODE in connection with control problems
93B10Canonical structure of systems
37N05Dynamical systems in classical and celestial mechanics
93C10Nonlinear control systems