Multifolded torus chaotic attractors: Design and implementation

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013118-6 Yu, Lu, and Chen Chaos 17, 013118 2007 TABLE I. Parameters a ij , b ij ,andc ij 1i, j3. a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1.0299 −0.7954 0.3863 0.0549 −0.0424 0.0206 0.0333 −0.0257 0.0125 b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 −0.0299 0.7954 −0.3863 −0.0549 1.0424 −0.0206 −0.0333 0.0257 0.9875 c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 −0.0469 0.4557 0.6980 −0.0315 0.0391 0.9075 0.0602 −1.1345 0.0078 Ẋ = m 0 − m 0 0 m 0 − m 0 −1 0 0 x y z A 0X, where X=x,y,z T . The corresponding characteristic equation of O is 3 + m 0 1− 2 + − m 0 =0. 9 10 Let =14.5, =1.25, m 0 =0.15, and m 1 =−0.17. Solving 10 gives 1 2.0592, 2,3 −0.0171±1.1489i, and the corresponding eigenvectors are v 1 0.9981,0.0532,0.0323 T and Denote 0.2440 v 2 = w 2 ± w 3 i 0.0086 − 0.6328 − 0.4492 ± − 0.5816 i. 0 = m 1 − m 1 0 x z Ẋ m 1 − m 1 −1 y A 1X, 0 0 11 where X=x,y,z T , and its corresponding eigenvalues are ¯ 1−2.3269, ¯ 2,30.0159±1.1506i. The corresponding eigenvectors are and Denote v¯1 0.9980,0.0559,− 0.03 0.2258 v¯2 = w¯ 2 ± w¯ 3i 0.0080 0.6279 0.4696 ± 0.5780 i. 0 0.9981 0.2440 − 0.4492 P = v 1 ,w 1 ,w 2 0.0532 0.0086 − 0.5816 . 0.0323 − 0.6328 0 Thus, the solution of linear system 9 with initial value X 0 is XtP s 11 0 0 0 s 22 − s 32 P−1 X 0 0 s 32 s 22 t 11 t 12 t 13 t 21 t 22 t 23 t 31 t 32 t 33 X 0, where s 11 =e 2.0592t , s 22 =e −0.0171t cos1.1489t, s 32 =e −0.0171t sin1.1489t, and t ij =a ij e 2.0592t +b ij e −0.0171t cos1.1489t+c ij e −0.0171t sin1.1489t1i, j3, in which a ij ,b ij ,c ij 1i, j3 are given in Table I. Therefore, the unstable and stable subspaces of system 9 are E U =Spanv 1 and E S =Spanw 2 ,w 3 , respectively. Figure 5a shows the stable and unstable subspaces of linear system 9. Note that system 9 is the linearized system of the 3-folded torus system 1 with 3 and N=2 at equilibrium point O. The unstable and stable eigenspaces of system x 9981 = y 532 = z 323 1 with 3 and N=2 corresponding to O are and 4600456x−3553172y+1725591z=0, respectively. Linearizing system 1 with 3 at equilibrium points P ± gives FIG. 5. Stable and unstable subspaces E S and E U of the PWL system 1 with 3 for N=2. a In V 0 ; b in V ± . Downloaded 22 Mar 2007 to 144.214.40.14. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
013118-7 Multifolded torus chaotic attractors Chaos 17, 013118 2007 TABLE II. Parameters ā ij , b¯ij, and c¯ij 1i, j3. ā 11 ā 12 ā 13 ā 21 ā 22 ā 23 ā 31 ā 32 ā 33 1.0363 −0.8419 −0.3619 0.0580 −0.0472 −0.0203 −0.0312 0.0253 0.0109 b¯11 b¯12 b¯13 b¯21 b¯22 b¯23 b¯31 b¯32 b¯33 −0.0363 0.8419 0.3619 −0.0580 1.0472 0.0203 0.03123 −0.0253 0.9891 c¯11 c¯12 c¯13 c¯21 c¯22 c¯23 c¯31 c¯32 c¯33 0.0461 −0.4281 0.7368 0.0295 −0.0378 0.9104 0.0635 −1.1379 −0.0083 0.9980 0.2258 0.4696 P¯ = v¯1,w¯ 1,w¯ 2 0.0559 0.0080 0.5780 . − 0.03 0.6279 0 The solution of linear system 11 with initial value X 0 is XtP¯ s¯11 0 0 t¯12 t¯13 0 s¯22 − s¯32 P¯ −1 X 0 t¯11 t¯21 t¯22 t¯23 0 , 0 s¯32 s¯22 t¯31 t¯32 t¯33X where s¯11 =e −2.3269t , s¯22 =e 0.0159t cos1.1506t, s¯32 =e 0.0159t sin1.1506t, and t¯ij =ā ij e −2.3269t +b¯ije 0.0159t cos1.1506t+c¯ij e 0.0159t sin1.1506t1i, j3, in which ā ij ,b¯ij,c¯ij 1i, j3 are given in Table II. Moreover, the stable and unstable subspaces of system 11 are E S =Spanv¯1 and E U =Spanw¯ 2,w¯ 3, respectively. Figure 5b shows the stable and unstable subspaces of linear system 11. Note that system 11 is the linearized system of the 3-folded torus system 1 with 3 and N=2 at equilibrium point P ± . The stable and unstable eigenspaces of system x 1 with 3 and N=2 corresponding to P ± are = z −300 9980 = y 559 and −9073155x+7371546y+3168890z=0, respectively. According to 10 and the Routh-Hurwitz theorem, the equilibrium point O0,0,0 is stable if and only if m 0 1− 0, m 0 0, m 0 0. x Similarly, the equilibrium point P ± 1 m 0 −m 1 m 1 ,0,0 is stable if and only if m 1 1− 0, m 1 0, m 1 0. Referring to 10, denote p=− m 0 2 1− 2 3 , q= 2m 0 3 1− 3 27 − m 01− 3 −m 0 , and =− m 0 4 1− 3 27 − m 0 2 2 1− 2 108 + m 0 2 2 1− 6 + 3 27 + m 0 2 2 2 4 . Based on the classical formula of the solutions of cubic equations, solving 10 gives 1 =− m 01− + 3 − q 2 + + 3 − q 2 − and 2,3 =− m 01− 3 − 1 2 3 − q 2 + + 3 − q 2 − ± 3 2 i 3 − q 2 + − 3 − q 2 − 1 ± 2 i. Theoretical analysis and numerical simulations show that the PWL system 1 with 3 for N=2 can generate a 3-folded torus chaotic attractor under the conditions of 1 0, 1 0, and 2 0. That is, the equilibrium point O is a one-dimensional unstable saddle point. Similarly, denote p¯ =− m 1 2 1− 2 3 , q¯ = 2m 1 3 1− 3 27 − m 11− 3 −m 1 , and ¯ =− m 1 4 1− 3 27 − m 1 2 2 1− 2 108 + m 1 2 2 1− 6 + 3 27 3 + m 1 2 2 2 4 , and the eigenvalues are ¯ 1=− m 11− 3 + 3 − q¯2 + ¯ + 3 − q¯2 − ¯ and ¯ 2,3=− m 11− 3 − 1 2 3 − q¯2 + ¯ + 3 − q¯2 − ¯ ± 3 2 i 3 − q¯2 + ¯ − 3 − q¯2 − ¯ ¯ 1±¯ 2i. Theoretical analysis and numerical simulations show that the PWL system 1 with 3 and N=2 can create a 3-folded torus chaotic attractor under the conditions of ¯ 10, ¯ 1 0, and ¯ 20. That is, the equilibrium points P ± are twodimensional unstable saddle points. Therefore, to generate a 3-folded torus chaotic attractor from 1 with 3 and N=2, one may assume that 0, ¯ 0, 1 0, 12 ¯ 1 0, 1 0, ¯ 1 0. Remark 1: The double-folded torus chaotic system 1 with 2 and the 3-folded torus chaotic system 1 with 3 have the same dynamical equation. The main difference lies in their different parameters , , m 0 , and m 1 . More importantly, different parameters correspond to different stable and unstable subspaces, which finally lead to different dynamical behaviors. Furthermore, different parameters may have different intrinsic dynamic mechanisms. It is clear that the double-folded torus chaotic attractor is nonsymmetrical about the origin O, as shown in Fig. 1; however, the 3-folded torus chaotic attractor is symmetrical about the origin O, as shown in Fig. 3a. C. Dynamical behaviors of multifolded torus chaotic systems The linear systems in the three domains V 0 and V ± of R 3 are Ẋ=AX−E, where X=x,y,z T , A=A 0 , and E=O for X V 0 , and A=A 1 and E= P ± for XV ± . When XV ± , let X=X− P ± , so that Ẋ=A 1 X. Thus, the dynamical behaviors of the 3-folded torus system 1 with 3 and N=2 are completely determined by the two basic linear systems 9 and 11. That is, the 3-folded torus chaotic attractor is generated via alternative switchings of the two basic linear systems 9 and 11. Figure 5 shows the stable and unstable subspaces E S and E U of the PWL system 1 with 3 and N=2. Figure 5a shows the stable and unstable eigenspaces of system 1 with 3 and N=2 corresponding to O in V 0 , where the arrows denote the directions of the flow. Figure 5b shows the stable and unstable eigenspaces of system 1 with 3 and N=2 corresponding to P ± in V ± , where the arrows denote the 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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