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 \(\omega\)=0.03. The initial we used to produce the attractor shown in the following figures are
\(x_0=0\), \(y_0=1\), \(z_0=0\), \(u_0=7\) and \(v_0=0\). 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.

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).

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.