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 canonical - also called 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 F\(_0\) is located at the origin of the state space and point F\(_1\) is located at (-1,0,0). Singular point F\(_0\) is a saddle-focus point with two complex conjugated eigenvalues with a positive real parts and point F\(_1\) is also a saddle-focus point but with complex eigenvalues with negative real parts. Such a configuration with two singular 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.

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].