A driven Chen system for a double-torus attractor

Zhengdi Zhang, Yanyan Li & Qinsheng Bi investigated the driven Chen system  [1]

where u is the driving term. This is a 5D system which can be view as a 3D dissipatif system driven by a 2D conservative system. For this reason, the system is mixed in the sense that it is semi-dissipative (or semi-conservative) [2]. The system is investigated along a line of the parameter space defined by a=4, b=3 and =0.03. The initial we used to produce the attractor shown in the following figures are , , , and . The effect of the driving term on the third equation is to transform the spiral present in each wing of the "Lorenz attractor" into a torus as shown in Fig. 1 for c=2. This first attractor is thus characterized by two tori with possible transitions from one to the other.

Fig. 1. Double-torus attractor

For a slightly different value of parameter c=2.05, the transitions from one torus to the other look similar to the transition from one wing to the other in the Lorenz system (Fig. 2).

Fig. 2. Toroidal Lorenz chaos.

For c=2.5, the symmetry of the attractor is broken and a single torus remains (Fig. 3). This is to compare to the asymmetric attractor produced by the Lorenz (or the Chen) system for large R-values.

Fig. 3. Asymmetric toroidal Lorenz chaos