Multifolded torus chaotic attractors: Design and implementation
Multifolded torus chaotic attractors: Design and implementation
Multifolded torus chaotic attractors: Design and implementation
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013118-4 Yu, Lu, <strong>and</strong> Chen Chaos 17, 013118 2007<br />
FIG. 2. PWL characteristic function gx. a Three segments;<br />
b nine segments.<br />
as shown in Fig. 3b. System 1 with 3 for N=5 has nine<br />
equilibrium points: O0,0,0, P ± 91 ±1.6,0,0,<br />
P ± 92 ±3.2,0,0, P ± 93 ±4.8,0,0, <strong>and</strong> P ± 94 ±6.4,0,0 denoted<br />
by “,” as shown in Fig. 3b. The eight switching points are<br />
denoted by “,” as shown in Fig. 3b.<br />
Similarly, system 1 with 3 has a 5-folded <strong>torus</strong> <strong>chaotic</strong><br />
attractor with maximum Lyapunov exponent 0.1151 for<br />
parameters N=3, m 0 =−0.17, m 1 =0.15, m 2 =−0.17, <strong>and</strong> x 2<br />
=2.45. Also, system 1 with 3 has a 7-folded <strong>torus</strong> <strong>chaotic</strong><br />
attractor with maximum Lyapunov exponent 0.0901 for parameters<br />
N=4, m 0 =0.15, m 1 =−0.17, m 2 =0.15, m 3 =−0.17,<br />
x 2 =2.0735, <strong>and</strong> x 3 =3.5735. Obviously, system 1 with 3<br />
can create a maximum 2N−1-folded <strong>torus</strong> <strong>chaotic</strong> attractor<br />
for N1, where every <strong>torus</strong> corresponds to a unique segment<br />
or radial of the PWL characteristic function gx. Furthermore,<br />
theoretical analysis <strong>and</strong> numerical simulation both<br />
show that the slopes of the two radials of the PWL characteristic<br />
function gx must be negative, as depicted in Fig. 2.<br />
III. DYNAMICAL BEHAVIORS OF MULTIFOLDED<br />
TORUS CHAOTIC SYSTEMS<br />
In this section, the dynamical behaviors of multifolded<br />
<strong>torus</strong> <strong>chaotic</strong> systems are further investigated, including<br />
symmetry, bifurcation, eigenspaces, approximation solutions,<br />
<strong>and</strong> system trajectory.<br />
A. Symmetry <strong>and</strong> bifurcation of 3-folded <strong>torus</strong><br />
<strong>chaotic</strong> system<br />
Denote S + =x,y,zy=x−x 1 , S − =x,y,zy=x+x 1 ,<br />
V 0 =x,y,zy−x x 1 , V + =x,y,zy−x−x 1 , <strong>and</strong> V −<br />
=x,y,zy−xx 1 .<br />
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