HYPERBOLIC-TYPE GENERALIZED LORENZ CHAOTIC SYSTEM AND ITS CANONICAL FORM

Abstract

This paper shows that a large class of chaotic systems, introduced in (Čelikovský and Vaněček, 1994), (Vaněček and Čelikovský, 1996) as the generalized Lorenz system, can be further generalized to the hyperbolic-type generalized Lorenz system. While the generalized Lorenz system unifies both the famous Lorenz system and new Chen's system (Ueta and Chen, 1999), (Chen and Ueta, 2000), the hyperbolic-type generalized Lorenz system introduced here is in some way complementary to it. Such a complementarity is especially clear when considering the canonical form of the generalized Lorenz system obtained in (Čelikovský and Chen, 2002), where the canonical form is characterized by the eigenvalues of the linearized part together with a key parameter τ ∈ (–1, ∞). The analogous canonical form of the hyperbolic-type generalized Lorenz system introduced here corresponds to the case of, while τ = –1 is a single special case. This new class of chaotic systems is then analyzed, both analytically and numerically, showing its rich variety of dynamical behaviours, including bifurcation and chaos. Moreover, an algorithm for transforming the hyberbolic-type generalized Lorenz system into its canonical form, as well as its inverse scheme, are presented.