Abstract
A new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. Through the analysis of the equilibrium points, Lyapunov exponent spectrums, bifurcation diagrams, and Poincaré mapping in this system, the chaotic characteristic of the system is verified. Apart from the research above, an electronic circuit is designed to implement the system. The circuit experimental results are in accordance with the results of numerical simulation, which verify the availability and feasibility of this method.
1. Introduction
Since Lorenz found the first chaotic model in 1963 [1], people have had a great interest to construct the chaotic attractors with different shape and quantity. At present, the existing chaotic attractors could be divided into two types: the multi-scroll chaotic attractors (including double-scroll [2], multi-scroll [3, 4], gird multi-scroll [5], and multi-directional multi-scroll chaotic attractors [6]) and the multi-wing butterfly chaotic attractors (containing two-wing [7–11], four-wing [12–16], multi-wing [17–20], and grid multi-wing [21, 22] butterfly chaotic attractors). The approach had grown pretty mature constructing multi-scroll chaotic attractors. However, it is rarely researched for the method constructing multi-wing and grid multi-wing butterfly chaotic attractors.
In recent years, [17–22] reported the latest research achievement about the multi-wing and grid multi-wing butterfly chaotic attractors. Reference [17] constructed a class of hyperchaotic systems generating -wing butterfly hyperchaotic attractors by coordinate transition and absolute value transition. References [18, 21] proposed the grid multi-wing butterfly chaotic attractors by constructing heteroclinic loops. Reference [7] presented the first and second kinds of Lorenz-type systems to simplify the algebraic form of the Lorenz system while keeping the butterfly structure of Lorenz attractor. Reference [19] constructed a multi-wing butterfly chaotic system based on the first and second kinds of Lorenz-type systems. By designing some piecewise functions to take place of the state variables of the Lorenz system directly, [20, 22] proposed some multi-wing and grid multi-wing butterfly chaotic systems. Though the above literatures proposed some methods constructing multi-wing and grid multi-wing butterfly chaotic attractors, it is very difficult to construct the new multi-wing and grid multi-wing butterfly chaotic attractors via these approaches. Therefore, it is necessary to design the novel methods constructing multi-wing and grid multi-wing butterfly chaotic attractors.
In this paper, through combining a linear coupling controller with a piecewise linear function skillful, a new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. The system is easy to be realized by analog circuits and its algebraic form is also simple.
2. Constructing the Grid Multi-Wing Butterfly Chaotic Attractors via the Nonlinear Coupling Control
The mathematic model of Lorenz chaotic system is shown as follows:where , , and .
Constructing grid multi-wing butterfly chaotic attractors via the nonlinear coupling control, the method is shown as follows.
Step 1. In order to construct gird multi-wing butterfly chaotic attractors easier, the state variables need normalization processing in system (1). The method of the normalization processing is to make the peak value of the all state variables less than one via scale transformation. Let the scaling factors of the state variables , and be , , and , respectively. According to the numerical simulation results of the Lorenz system [1], we can choose , , and . Therefore, system (1) is changed as follows:
Step 2. Two first-order differential equations about state variables and are added based on system (2). The exact approach is as follows: First, the first and second equations are copied and taken as the fourth and fifth equations in system (2), and then the state variable is replaced with in the fourth equation of system (2), and the state variable is used to take place of the state variable in the fifth equation of system (2). Therefore, a five-dimensional autonomous system is obtained:
Step 3. Two linear coupling controllers and are added in the fourth and fifth equations of system (3), respectively. Then system (3) is changed as follows:where the coupling controllers and . is gain parameter.
Step 4. A new piecewise linear function is designed:where . Through replacing the state variables and with the piecewise linear function in system (4), a novel system is obtained as follows:where the system parameter , , , , , , and the nonlinear coupling controllers and where .
Let the system parameters , , , , , , ; that is, , , , , , . Under the action of the nonlinear coupling controllers and , system (6) is able to generate -wing butterfly chaotic attractors, as shown in Figure 1. When , system (6) creates -wing butterfly chaotic attractors as shown in Figure 1(a). When and , system (6) generates -wing butterfly chaotic attractors as shown in Figure 1(b). When and , system (6) creates -wing butterfly chaotic attractors as shown in Figure 1(c), and the time series about the state variable is shown in Figure 1(e). When and , system (6) generates -wing butterfly chaotic attractors as shown in Figure 1(d).
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3. Basic Dynamic Characteristic
3.1. Equilibrium Point
Let ; that is,
According to solution for the equation set (8), the equilibrium points of system (6) are obtained: where , and .
For the equilibrium points , system (6) is linearized and the Jacobian matrix is defined aswhere , , , , and are the coordinates of the equilibrium points , , and .
Substituting the equilibrium points into the characteristic equation , we get the following eigenvalues: , , , , and . , , , and are negative real numbers, and is a positive real number. So the equilibrium points are unstable saddle.
Substituting the equilibrium points into the characteristic equation , we get the following eigenvalues: , , , and . , , and are negative real numbers, and are a pair of conjugate complex eigenvalues with positive real parts. Through computation, we know that the equilibrium points have the same eigenvalues as the equilibrium points . Therefore, the equilibrium points and are unstable saddle-foci of index 2.
3.2. Lyapunov Exponent Spectrum, Bifurcation Diagram, and Poincaré Mapping
Let and ; the Lyapunov exponent of system (6) and its bifurcation diagram which varies with the coefficient and its Poincaré map are shown in Figure 2. The range of values for the coefficient is 0 to 3. Figures 2(a) and 2(b) show that when , system (6) has one positive Lyapunov exponent, so it is in the chaotic state. From Figure 2(c), we can see that the Poincaré mapping spread out from multiple directions. This shows the complicated dynamic behavior of system (6).
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4. Circuit Design of the Grid Multi-Wing Butterfly Chaotic Attractors
According to system (6), the circuit of grid multi-wing butterfly chaotic attractors is designed as shown in Figure 3.
Let and ; the piecewise linear function (7) is changed as follows
The circuit diagram of the piecewise linear function (11) is shown in Figure 4.
In Figures 3 and 4, all the operational amplifiers are selected as UA741CN. Their supply voltage V and saturated voltage V. All the multipliers are of type AD633JN and their gain is 0.1. is the positive pole of the supply voltage ; that is, V. is the negative pole of the supply voltage ; that is, V.
According to Figure 3, the circuit equation can be obtained as follows:
To observe the output wave experimentally, the time scale transformation must be executed for , that is, let and , and (12) can be changed as follows:
Let kΩ, nF, kΩ, and kΩ; according to system (6) and equation (13), we can obtain kΩ, kΩ, kΩ, kΩ, MΩ, kΩ, kΩ, kΩ, kΩ, and kΩ.
In Figure 4, let kΩ. When , , , , and are switched on, the circuit equation can be obtained as follows:
According to the piecewise linear function (11) and equation (14), we can choose kΩ, kΩ, kΩ, and kΩ.
According to Figures 3 and 4, the gird multi-wing butterfly chaotic attractors are obtained via circuit simulation software Multisim 10.0, as shown in Figure 5. When , are switched on and , , and are switched off, the circuit generates -wing butterfly chaotic attractors as shown in Figure 5(a). When , , and are switched on and and are switched off, the circuit creates -wing butterfly chaotic attractors as shown in Figure 5(b). When , , , and are switched on and is switched off, the circuit generates -wing butterfly chaotic attractors as shown in Figure 5(c), and the time series about the state variable is shown in Figure 5(e). When , , , , and are switched on, the circuit creates -wing butterfly chaotic attractors as shown in Figure 5(d).
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From Figures 5 and 1, we can see that the circuit experimental results are in agreement with the results of numerical simulation.
5. Conclusion
A new method is presented to construct grid multi-wing butterfly chaotic attractors via nonlinear coupling control in this paper. A five-dimensional grid multi-wing butterfly chaotic system is constructed via this approach. Through adjusting the nonlinear coupling controllers and the piecewise linear functions, the , , , and -wing butterfly chaotic attractors are obtained. Through the theoretical analysis and numerical simulation, the complex dynamic characteristics of the five-dimensional grid multi-wing butterfly chaotic system are shown. Also, the system has been implemented by designing an electronic circuit.
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this article.
Acknowledgments
This work is supported by the Technology Research Projects of The Chongqing Education Committee (Grant nos. KJ130509, KJ1400410, and KJ130520).