1992 The Genesio-Tesi system

While investigating the algebraic conditions for getting chaos in dynamical systems, Roberto Genesio and Alberto Tesi proposed a new three-dimensional system producing chaotic solutions [1]. The system results from the jerk equation

which can be rewritten as the three-dimensional system

where a and b are the bifurcation parameters. This system has two singular points : point F₀ is located at the origin of the state space and point F₁ is located at (-1,0,0). Singular point F₀ is saddle-focus with two complex conjugated eigenvalues with a positive real parts and point F₁ is also saddle-focus but with complex eigenvalues with negative real parts. Such a configuration with two fixed points is very similar to the configuration of the Rössler system [2]. A chaotic attractor is obtained, for instance, when a=0.446 and b=1.1 as shown in Fig. 1.

Fig. 1. Chaotic attractor solution to the Genesio-Tesi system and a first-return map to a Poincaré section.

Forgetting the layered structure of its map, the chaotic attractor is structured by a smooth unimodal map as always observed after a period-doubling cascade. Such a cascade is evidenced by the bifurcation diagram shown in Fig. 2 (for b=1.1). The chaotic attractor is topologically equivalent to the spiral Rössler attractor as shown in [3].

Fig. 2. Bifurcation diagram of the Genesio-Tesi system.