Multifolded torus chaotic attractors: Design and implementation

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013118-2 Yu, Lu, and Chen Chaos 17, 013118 2007 Chua and his colleagues proposed a simple third-order autonomous circuit, 9,10,14 called a folded torus circuit, which can generate a double-folded torus chaotic attractor. Later, many researchers studied further the torus breakdown in a PWL forced van der Pol oscillator and a forced Rayleigh oscillator. As far as we know, previous works on torus breakdown only focused on laboratory measurement and numerical simulation for a torus or a folded torus. 9,10 Therefore, it is very interesting to ask whether the folded torus circuit can be slightly modified so as to generate multifolded torus chaotic attractors. This paper gives a positive answer to this question. More precisely, this paper reports our studies of constructing a general PWL characteristic function to replace the characteristic function used in the folded torus circuit, thereby generating multifolded torus chaotic attractors. In particular, our theoretical analysis reveals that these multifolded torus chaotic attractors can be generated via alternative switchings of two basic linear systems. The theoretical design principle and the underlying dynamic mechanism are then further investigated by analyzing the emerging bifurcation and the stable and unstable subspaces of the two linear systems. Moreover, a novel block circuit diagram is designed for hardware implementation of 3-, 5-, 7-, and 9-folded torus chaotic attractors. Recursive formulas for system parameters and for physical circuit parameters are also rigorously derived, useful for improving hardware implementation. It should be noted that this is the first experimental verification of a 9-folded torus chaotic attractor. The rest of this paper is organized as follows. In Sec. II, a systematic theoretical design approach for generating multifolded torus chaotic attractors is proposed, and some recursive formulas for system parameters are rigorously derived. The underlying dynamic mechanism and emerging bifurcation are then discussed in Sec. III. In Sec. IV, a simple circuit diagram is constructed for experimentally verifying the multifolded torus chaotic attractors. Conclusions are finally drawn in Sec. V. II. THEORETICAL DESIGN OF MULTIFOLDED TORUS CHAOTIC ATTRACTORS In this section, we construct a general PWL characteristic function to replace the characteristic function used in the fold torus circuit for generating multifolded torus chaotic attractors. A. Double-folded torus chaotic attractor Chua and his colleagues proposed a double-folded torus chaotic circuit, 9,10 called a folded torus circuit, described by where ẋ =−gy − x ẏ =−gy − x − z ż = y, 1 FIG. 1. Double-folded torus chaotic attractor. gy − x = m 1 y − x + m 0 − m 1 y − x + x 1 − y − x − x 1 2 2 is a PWL odd function satisfying gx−y=−gy−x. When =15, =1, m 0 =0.1, m 1 =−0.07, and x 1 =1, system 1 has a double-folded torus chaotic attractor as shown in Fig. 1. The maximum Lyapunov exponent of this attractor is 0.0270. System 1 has three equilibrium points: O0,0,0 and P ± ± 17 7 ,0,0 . Linearizing system 1 at equilibrium point O gives the corresponding eigenvalues: O 1 1.4328 and O 2,3 −0.0164±1.0231i, which have the corresponding eigenvectors 0.9985,0.0448,0.0312 T and −0.41940.2829i,−0.6169,0.0097±0.6028i T , respectively. Thus, system 1 has a one-dimensional unstable eigenspace E U x O: 9985 = y 448 = z 312 corresponding to 1 O and a two-dimensional stable eigenspace E S O:37186732x O −25007019y+17452101z=0 corresponding to 2,3 at the neighboring region of O. Similarly, linearizing system 1 at the equilibrium points P ± gets the following eigenvalues: P 1 −1.0145 and P 2,3 0.0172±1.0172i, with the corresponding eigenvectors 0.9989,0.0338,−0.0333 T and 0.32780.3124i,0.6358,0.01060.6249i T , respectively. System 1 has a one-dimensional stable eigenspace Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
013118-3 Multifolded torus chaotic attractors Chaos 17, 013118 2007 x 9989 = y 338 = z E S P P ± : −333 corresponding to 1 and twodimensional unstable eigenspace E S P ± :−19865571x P +10076539y+9931196z=0 corresponding to 2,3 in the neighboring region of P ± . B. Design principles of system parameters Definition 1: An n-folded torus attractor is an attractor that is created via an n-torus breakdown route, where the n-torus is described as a quotient of R n under integral shifts in any coordinate with a zero Lyapunov number and an irrational rotation number. 8–10 To generate multifolded torus chaotic attractors from the folded circuit 1, we introduce a new PWL odd function to replace the characteristic function 2, which is described by gy − x = m N−1 y − x N−1 mi−1 − m + i y − x + x i − y − x − x i , 3 i=1 2 where gx−y=−gy−x and , are real parameters. Note that the above PWL characteristic function gy−x can be recast as follows: y − x x 1 , m 0 y − x if x i y − x x i+1 , 1 i N −2, i gy − x =if m i y − x + m j−1 − m j sgny − xx j j=1 N−1 if y − x x n−1 , m N−1 y − x + m j−1 − m j sgny − xx j , j=1 4 where m i 0iN−1 are the slopes of the segments, or radials in various piecewise subregions, and ±x i x i 0, 1 iN−1 are the switching points. Since g· is an odd function, we only need to determine the positive switching points x i 1iN−1. Obviously, the equilibrium points of system 1 with 3 are ±x i E ,0,0x i E 0,0iN−1, where ±x i E 0iN−1 satisfies the following equation: gx i E =0, 0 i N −1. 5 Substituting 4 into 5, and solving x E i 0iN−1 from 5, yields the recursive formulas of x E i 0iN−1 as follows: = 0 i =0, x E i i j=1 m j − m j−1 x j 6 1 i N −1. m i Denote k i = x E i+1 − x i x E , 7 i − x i E where 1iN−2. Let k i =1 for 1iN−2, thus x i = 1 2 x i+x i+1 for 1iN−2. That is, all internal equilibrium points x E i 0iN−2 are the midpoints of two neighboring switching points. According to 6, we have the recursive formulas of the positive switching points x i 2iN−1 as follows: x i+1 = 2 i j=1m j − m j−1 x j − x i , 8 m i where 1iN−2. C. Multifolded torus chaotic attractors Numerical simulation shows that system 1 with 3 has a large parameter region for chaos generation. Here, assume that =14.5, =1.25, and x 1 =0.75. Then, we can calculate all other switching points x i 2iN−1 from the recursive formulas 8. In the following, two typical cases are first discussed: N=2 and N=5. When N=2, we have gy−x=m 1 y−x + 1 2 m 0−m 1 y−x+x 1 −y−x−x 1 . Let m 0 =0.15 and m 1 =−0.17. Figure 2a shows the PWL characteristic function gx with N=2. For the above set of parameters, system 1 with 3 has a 3-folded torus chaotic attractor, as shown in Fig. 3a. The maximum Lyapunov exponent of this 3-folded torus chaotic attractor is 0.1018. It is clear that there are three tori folded in the chaotic attractor, as shown in Fig. 3a. Obviously, system 1 with 3 for N=2 has three equilibrium points: O0,0,0 and P ± ±1.4118,0,0 denoted by “,” as shown in Fig. 3a. Also, the two switching points are denoted by “,” as shown in Fig. 3a. When N=5, we have gy − x = m 4 y − x 4 + 2 1 m i−1 − m i y − x + x i − y − x − x i . i=1 Let m 0 =−0.17, m 1 =0.15, m 2 =−0.17, m 3 =0.15, and m 4 = −0.17. From 8, we have x 2 =2.45, x 3 =3.95, and x 4 =5.65. Figure 2b shows the PWL characteristic function gx for N=5. Here, system 1 with 3 has a 9-folded torus chaotic attractor, as shown in Fig. 3b. The maximum Lyapunov exponent of this 9-folded torus chaotic attractor is 0.0730. It is clear that there are nine tori folded in the chaotic attractor, 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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