Multifolded torus chaotic attractors: Design and implementation

More documents

Recommendations

Info

013118-8 Yu, Lu, and Chen Chaos 17, 013118 2007 For N2, we can transform the multifolded torus system 1 with 3 into two basic linear systems 9 and 11 by using a simple linear transformation. In the following, we only select N=5 as an example to explain how to construct the linear transformation. When N=5, construct the following linear transformation: FIG. 6. Equilibrium points and their corresponding eigenspaces of system 1 with 3 and N=2. directions of the flow. Let t be the flow generated by 1 with 3 and N=2, and X 0 be the initial value that is near O above T 0 but not on l 0 , where T 0 and l 0 are the stable and unstable eigenspaces of system 1 with 3 and N=2 corresponding to O, respectively. Because 1 0, t,X 0 moves forward with respect to the x axis while rotating clockwise around l 0 E U , as shown in Figs. 5a and 6. After long enough time, t,X 0 will hit S + and then enter V + . Due to the relative positions of P + and l + , t,X 0 continues to move forward while rotating around l + E S , as shown in Figs. 5b and 6, where l + and T + are the stable and unstable eigenspaces of system 1 with 3 and N=2 corresponding to P + , respectively. Because ¯ 10, t,X 0 increases its magnitude of oscillation and eventually returns to V 0 . Then, t,X 0 moves backward around l 0 . After a long time, it eventually hits S − and then enters V − . Then, t,X 0 moves further backward while rotating around l − , which is the stable eigenspace of system 1 with 3 and N=2 corresponding to P − . Since ¯ 10, t,X 0 increases its magnitude of oscillation and finally returns to V 0 . Because 1 0, it decreases its magnitude of oscillation and returns to the original neighboring region of O. The trajectory t,X 0 is repeatedly stretched and folded for infinitely many times and finally forms a 3-folded torus chaotic attractor. Figure 6 shows the relative positions of all equilibrium points O, P ± and their eigenspaces l 0 , T 0 , l ± , and T ± . We can get the approximation formulas of the eigenspaces as follows: l 0 : x 9981 = y 532 = z 323 , T 0 : 4600456x − 3553172y + 1725591z =0, l ± : x 1.4118 9980 = y 559 = z − 300 , T ± : − 9073155x 1.4118 + 7371546y + 3168890z =0. Moreover, the flow t,X 0 is completely determined by the two basic linear systems 9 and 11. Therefore, the flow t,X 0 follows the approximation solution of 9 in V 0 and the approximation solution of 11, as verified by using a linear transformation X=X− P ± in V ± . In particular, the vector field of system 1 with 3 and N=2 is symmetrical about the origin. 94 − P − x − y − x 4 93 X − P − − x 4 x − y − x 3 92 X − P − − x 3 x − y − x 2 91 X − P − − x 2 x − y − x 1 X =X X x − y x 1 91 X − P + x 1 x − y x 2 92 X − P + x 2 x − y x 3 93 X − P + x 3 x − y x 4 94 X − P + x − y x 4 . 13 Since m 0 =−0.17, m 1 =0.15, m 2 =−0.17, m 3 =0.15, and m 4 = −0.17, using the linear transformation 13, system 1 with 3 and N=5 becomes = x − y x 1 A 1 X or x 2 x − y x 3 Ẋ or x − y x 4 A 0 X x 1 x − y x 2 or x 3 x − y x 4 . 14 Therefore, the 9-folded torus system 1 with 3 and N=5 is also generated via alternative switchings of the two basic linear systems 9 and 11. Similarly, the underlying dynamic mechanism of the 9-folded torus system 1 with 3 and N=5 is the same as that of the 3-folded torus system 1 with 3 and N=2. In the following, we compare the dynamical mechanisms of the 3-folded torus system 1 with 3 for N=2 with that of Chua’s double-scroll system. 8–10 It is noticed that the magnitude of the real part Re 2,3 0.0171 or Re¯ 2,30.0159 of the pair of complex eigenvalues of system 9 or system 11 is less than that of the imaginary part Im 2,3 1.1489 or Im¯ 2,31.1506 of the pair of complex eigenvalues of system 9 or system 11 by two quantity levels O10 2 . Moreover, the magnitude of the real part characterizes the increasing or damping speed of the amplitude of the oscillator. However, the magnitude of the imaginary part characterizes the frequency of the oscillation. For the 3-folded torus system 1 with 3 and N=2, the increasing or damping speed of the amplitude of the oscillator is relatively much smaller than the frequency of the oscillator. Therefore, the 3-folded torus system 1 with 3 and N=2 can easily form increased or damped oscillations in every PWL region. That is, every PWL region can generate a torus. The familiar Chua’s circuit is described by 8 Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
013118-9 Multifolded torus chaotic attractors Chaos 17, 013118 2007 FIG. 7. Circuit diagram of multifolded torus chaotic attractors. ẋ = „y − fx… ẏ = x − y + z ż =−y, 15 where fx=m 1 x+ 1 2 m 0−m 1 x+x 1 −x−x 1 . When =10, =15, m 0 =− 1 7 , and m 1= 2 7 , system 15 has a double-scroll attractor. 8 Denote V¯ 0=x,y,zx x 1 , V¯ +=x,y,zxx 1 , and V¯ −=x,y,zx−x 1 . When XV¯ 0, system 15 has a unique equilibrium point O and the corresponding eigenvalues are 1 2.4777, 2,3 −1.0246±2.7566i. When XV¯ + or XV¯ −, system 15 has a unique equilibrium point P¯ + m 1−m 0 m 1 x 1 ,0,− m 1−m 0 m 1 x 1 or P¯ −− m 1−m 0 m 1 x 1 ,0, m 1−m 0 m 1 x 1 and the corresponding eigenvalues are ¯ 1−4.3290,¯ 2,3 0.2359±3.1376i. Obviously, when XV¯ 0, the magnitudes of the real part Re 2,3 1.0246 and the imaginary part Im 2,3 2.7566 are in the same quantity level O1. Thus, the increasing or damping speed of the amplitude of the oscillator is relatively much bigger than the frequency of the oscillator. Hence, system 15 cannot create a scroll in V¯ 0. However, when XV¯ +or XV¯ −, the magnitude of the real part Re¯ 2,3 0.2359 is less than that of the imaginary part Im¯ 2,3 3.1376 by one quantity level O10. Thus, the increasing or damping speed of the amplitude of the oscillator is relatively much smaller than the frequency of the oscillator. Therefore, system 15 can generate a scroll in V¯ + or V¯ −. Remark 2: In conclusion, all multifolded torus chaotic systems have the same dynamical mechanism; that is, all multifolded torus chaotic attractors are generated via alternative switchings between two basic linear systems 9 and 11. Moreover, all PWL regions of the characteristic function gx with a positive slope correspond to the linear system 9, and the PWL regions of the characteristic function gx with a negative slope correspond to the linear system 11. In particular, the two outer regions correspond to the negative slope. However, the dynamical mechanism of the multifolded torus chaotic attractors is very different from that of Chua’s double-scroll. 8 IV. CIRCUIT IMPLEMENTATION FOR MULTI-FOLDED TORUS CHAOTIC ATTRACTORS This section designs a novel circuit to experimentally verify the multifolded torus chaotic attractors. A. Fundamental principle of circuit design Based on the above theoretical analysis, a block circuit diagram is designed for generating multifolded torus chaotic attractors, as shown in Fig. 7. This circuit is described by Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
Page 1 and 2: CHAOS 17, 013118 2007 Multifolded t
Page 3 and 4: 013118-3 Multifolded torus chaotic
Page 7: 013118-7 Multifolded torus chaotic

Multifolded torus chaotic attractors: Design and implementation

Create successful ePaper yourself

Delete template?

Save as template?