Multifolded torus chaotic attractors: Design and implementation

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013118-4 Yu, Lu, and Chen Chaos 17, 013118 2007 FIG. 2. PWL characteristic function gx. a Three segments; b nine segments. as shown in Fig. 3b. System 1 with 3 for N=5 has nine equilibrium points: O0,0,0, P ± 91 ±1.6,0,0, P ± 92 ±3.2,0,0, P ± 93 ±4.8,0,0, and P ± 94 ±6.4,0,0 denoted by “,” as shown in Fig. 3b. The eight switching points are denoted by “,” as shown in Fig. 3b. Similarly, system 1 with 3 has a 5-folded torus chaotic attractor with maximum Lyapunov exponent 0.1151 for parameters N=3, m 0 =−0.17, m 1 =0.15, m 2 =−0.17, and x 2 =2.45. Also, system 1 with 3 has a 7-folded torus chaotic attractor with maximum Lyapunov exponent 0.0901 for parameters N=4, m 0 =0.15, m 1 =−0.17, m 2 =0.15, m 3 =−0.17, x 2 =2.0735, and x 3 =3.5735. Obviously, system 1 with 3 can create a maximum 2N−1-folded torus chaotic attractor for N1, where every torus corresponds to a unique segment or radial of the PWL characteristic function gx. Furthermore, theoretical analysis and numerical simulation both show that the slopes of the two radials of the PWL characteristic function gx must be negative, as depicted in Fig. 2. III. DYNAMICAL BEHAVIORS OF MULTIFOLDED TORUS CHAOTIC SYSTEMS In this section, the dynamical behaviors of multifolded torus chaotic systems are further investigated, including symmetry, bifurcation, eigenspaces, approximation solutions, and system trajectory. A. Symmetry and bifurcation of 3-folded torus chaotic system Denote S + =x,y,zy=x−x 1 , S − =x,y,zy=x+x 1 , V 0 =x,y,zy−x x 1 , V + =x,y,zy−x−x 1 , and V − =x,y,zy−xx 1 . Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
013118-5 Multifolded torus chaotic attractors Chaos 17, 013118 2007 FIG. 3. Multifolded torus chaotic attractors. a 3-folded torus; b 9-folded torus. FIG. 4. Bifurcation diagram of 3-folded torus system 1 with 3 for N =2, m 0 =0.15, and m 1 =−0.17. a Parameter =1.25; b parameter =14.5. Note that system 1 is invariant under the transformation x,y,z→−x,−y,−z; that is, system 1 is symmetric about the origin. The symmetry persists for all values of the system parameters. Moreover, V = ẋ x + ẏ y + ż 2 = m 0 −1 x,y,z V 0 m 1 −1 x,y,z V ± , where V is the unit volume of the flow of system 1. Therefore, system 1 with 3 for N=2 is dissipative in V 0 for m 0 −10 and in V ± for m 1 −10. Assume that m 0 0 and m 1 0. Then V changes its sign at =1; = that is, 0 x,y,z V 0 1 0 x,y,z V ± V 0 =1 0 x,y,z V 0 1. 0 x,y,z V ± Theoretical analysis shows that system 1 with 3 for N =2 has a periodic repeller, which coexists with an attracting torus for 1. However, system 1 with 3 for N=2 has a periodic attractor, which coexists with a repelling torus for 1. When =1, neither repeller nor attractor is observed as expected. Instead, a Hopf bifurcation occurs at =1. 8 System 1 with 3 for N=2, m 0 =0.15, and m 1 =−0.17 has a wide parameter region for a 3-folded torus chaotic attractor. Figures 4a and 4b show the bifurcation diagrams of parameters with =1.25 and with =14.5, respectively. According to the bifurcation diagrams in Fig. 4, it is clear that system 1 with 3 for N=2 has three different tori folded in a chaotic attractor. B. Dynamical behaviors of a 3-folded torus chaotic system From the discussion in Sec. II C, system 1 with 3 for N=2 has three equilibriums: O0,0,0 and P ± ±1.4118,0,0. Linearizing system 1 with 3 for N=2 at equilibrium point O gives Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
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