Christophe LETELLIER
07/03/2015

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 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 saddle-focus with two
complex conjugated eigenvalues with a positive real parts and point F_{1} 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.**

[1] **R. Genesio & A. Tesi**, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, *Automatica*, **28** (3), 531-548, 1992.

[2] **O. E. Rössler**, An equation for continuous chaos, *Physics Letters A*, **57** (5), 397-398, 1976.

[3] **E. Mendes & C. Letellier**, Displacement in the parameter space versus spurious solution of discretization with large time step, *Journal of Physics A*, **37**, 1203-1218, 2004.