Multifolded torus chaotic attractors: Design and implementation

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
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