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A driven Chen system for a double-torus attractor

Christophe LETELLIER
22/02/2013

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

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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.

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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).

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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.

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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.

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