Christophe LETELLIER
22/02/2013

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**

[1] **Zhengdi Z., YanYan L. & Qinsheng B.**, Routes to bursting in a periodically driven oscillator, *Physics Letters A*, **377** (13), 975-980, 2013.

[2] **O. Ménard, C. Letellier, J. Maquet, L. Le Sceller & G. Gouesbet**, Analysis of a non synchronized sinusoidally driven dynamical system, *International Journal of Bifurcation & Chaos*, **10** (7), 1759-1772, 2000.