Multifolded torus chaotic attractors: Design and implementation

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013118-10 Yu, Lu, and Chen Chaos 17, 013118 2007 FIG. 8. Equivalent circuit of adjustable inductance and capacitance. where dv C C1 1 =−fv C2 − v C1 dt dv C2 C 2 =−fv C2 − v C1 − i L dt L di L dt = v C2, N−1 16 fv C2 −v C1 =G N−1 v C2 −v C1 + 1 2 G i−1 −G i v C2 −v C1 +E i −v C2 −v C1 −E i . According to 16, we have dv C1 dt dv C2 dt dRi L dt =− 1 RC 2 C 2 C 1 Rfv C2 − v C1 i=1 =− 1 RC 2 Rfv C2 − v C1 + Ri L = 1 RC 2 R 2 C 2 L v C2. 17 Comparing 1 with 17, we get the following equivalent relationships: 0 = RC 2 , = t , = C 2 = 14.5, t 0 C 1 = R2 C 2 L = 1.25, x = v C1 , y = v C2 , z = Ri L , V BP V BP V BP x i = E i , G = 1 18 V BP R , G i = m i G0 i N −1, fv C2 − v C1 = 1 gy − x, R where V BP =1 V, and 1 0 = 1 RC 2 is the time-scale transformation factor. Let R=1 k. From 18, we have C 1 =1.29 nF, C 2 =18.75 nF, and L=15 mH. The subcircuitry N S in Fig. 7 is the subtraction generator and its output is v C2 −v C1 . The subcircuitry N R in Fig. 7 is the generator of the PWL characteristic function fv C2 −v C1 and its input and output satisfy the condition I N = fv C2 −v C1 . Moreover, we can rigorously calculate the theoretical values of all resistors in N R by using the recursive formulas. 25 The operational amplifier is selected as type TL082, and the supply voltage of electrical source is ±E C =±15V. Thus, the saturating voltage of the operational amplifier is E sat =14.3V. In the following, we calculate the theoretical values of all resistors in N R for the experimental confirmation of the 3-, 5-, 7-, and 9-folded torus chaotic attractors. To generate a 3-folded torus chaotic attractor, we rigorously calculate the theoretical values of the resistors in N R , based on the given parameters in Sec. II, as follows: G 0 = m 0 R = 0.15 mS, G 1 = m 1 = − 0.17 mS, R E 1 = x 1 V BP , r 1 = R 12 =−G 1 R 2 = 0.34, R 11 r 2 = R 22 = E sat − 1 = 18.07, R 21 E 1 r 3 = R 32 1+r 2 =− R 31 R 2 G 1 − G 0 − 1 = 28.79. 19 Similarly, we can rigorously calculate the theoretical values of the resistors in N R for the 5-, 7-, and 9-folded torus chaotic attractors based on the given parameters in Sec. II, as follows: G 0 =−0.17 mS, G 1 =0.15 mS, G 2 =−0.17 mS, E i =x i V BP i=1,2, r 1 =0.34, r 2 =4.84, r 3 =8.12, r 4 =19.1, and r 5 =28.79 for a 5-folded torus chaotic attractor; G 0 =0.15 mS, G 1 =−0.17 mS, G 2 =0.15 mS, G 3 =−0.17 mS, E i =x i V BP i=1,2,3, r 1 =0.34, r 2 =3.00, r 3 =5.25, r 4 =6.90, r 5 =9.78, r 6 =18.07, and r 7 =28.79 for a 7-folded torus chaotic attractor; G 0 =−0.17 mS, G 1 =0.15 mS, G 2 =−0.17 mS, G 3 =0.15 mS, G 4 =−0.17 mS, E i =x i V BP 1i4, r 1 =0.34, r 2 TABLE III. Status of all switches, ratios of resistors r n = R n2 R n1 1n9, and number of folded tori. K 1 K 2 K 3 K 4 K 5 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 Number of tori on on off off off 0.34 18.1 28.8 3 on on on off off 0.34 4.84 8.12 19.1 28.8 5 on on on on off 0.34 3.00 5.25 6.90 9.78 18.1 28.8 7 on on on on on 0.34 1.53 2.95 3.62 4.66 4.84 8.12 19.1 28.8 9 Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
013118-11 Multifolded torus chaotic attractors Chaos 17, 013118 2007 TABLE IV. Status of switches, resistors R n2 =r n R n1 1n9, and number of folded tori. K 1 K 2 K 3 K 4 K 5 R 12 R 22 R 32 R 42 R 52 R 62 R 72 R 82 R 92 Number of tori on on off off off 3.4 181 28.8 3 on on on off off 3.4 48.4 8.12 191 28.8 5 on on on on off 3.4 30.0 5.25 69.0 9.78 181 28.8 7 on on on on on 3.4 15.3 2.95 36.2 4.66 48.4 8.12 191 28.8 9 =1.53, r 3 =2.95, r 4 =3.62, r 5 =4.66, r 6 =4.84, r 7 =8.12, r 8 =19.1, and r 9 =28.79 for a 9-folded torus chaotic attractor. B. Circuit implementation In the circuit design, we select all operational amplifiers shown in Fig. 7 to be type TL082. The supply voltage of the electrical source is ±E C = ±15 V, and the saturating voltages of the operation amplifiers are E sat =14.3 V. Moreover, all resistors in Fig. 7 are exactly adjustable resistors or potentiometers. Note also that the capacitances C 1 , C 2 , and inductance L are adjustable, as shown in Fig. 8. Therefore, one can adjust the real parameter values of and by tuning the capacitances C 1 , C 2 , and inductance L. For the PWL function generator N R in Fig. 7, all unknown resistors R n2 1n9 can be rigorously calculated by using a formula similar to 19 as shown in Tables III and IV. FIG. 9. Experimental observations of the multifolded torus chaotic attractors. From left to right: a 3-folded torus, where x=0.86V/div and y=0.6V/div; b 5-folded torus, where x=1.25V/div and y=0.64V/div; c 7-folded torus, where x=1.25V/div and y=0.64V/div; d 9-folded torus, where x=1.6V/div and y=0.9V/div. 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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