Caption: Figure 3: Chaotic attractors
of system (1) with [[beta].sub.1] = 8.25, [[beta].sub.2] = 0.8, [[beta].sub.3] = 12.25, and [[beta].sub.4] = 0.2: (a) in [x.sub.1] - [x.sub.2] - [x.sub.3] space, (b) in [x.sub.1] - [x.sub.2] - [x.sub.4] space, (c) in [x.sub.1] - [x.sub.3] - [x.sub.4] space, and (d) in [x.sub.2] - [x.sub.3] - [x.sub.4] space.
of master system (26) are shown in Figure 1.
Chen, "Impulsive control of chaotic attractors
in nonlinear chaotic systems," Applied Mathematics and Mechanics, vol.
The threshold value is the associated dimension of the chaotic attractor
. The embedded dimension is the minimum embedded dimension which ensures that the topological structures of chaotic attractors
can be unfolded completely.
 solved the ultimate boundary problem of more existing chaotic attractors
and hyperchaotic attractors and got the numerical solutions of corresponding bounds.
The two- and three-dimensional chaotic attractors
with Matlab simulation of the modified Lorenz-like chaotic system (2) are shown in Figure 1.
Yorke, "The dimension of chaotic attractors
," in The Theory of Chaotic Attractors
Since Lorenz found the first chaotic attractor
in a smooth three-dimensional autonomous system, considerable research interests have been made in searching for the new chaotic attractors
These systems can generate one-directional (1D) n-scrolls, two-directional (2D) n x m grid scroll, and three-directional (3D) nxmxl grid scroll chaotic attractors
by adding breakpoints in the PWL function and increasing the number of PWL functions into the nonlinear system [7, 8].
The NCS exhibits complex and abundant dynamics behaviors; see Figure 3 where chaotic attractors
Caption: FIGURE 3: The chaotic attractors
of system (3) with a = 10, b = 40, c = 2.5, k = 2, h = 2, and l = 1.
By comparing black dots of Figure 5(a) and red dots of Figure 5(a), one can notice that system (1a), (1b), and (1c) displays coexistence of period-6-oscillations and chaotic attractors
in the range 3.182 < b < 3.188 and coexistence of period-3-oscillations and chaotic attractors
in the range 3.188 [less than or equal to] b < 3.2012.