# A Novel Method for Constructing Grid Multi-Wing Butterfly Chaotic Attractors via Nonlinear Coupling Control

Authors: Yun Huang
Journal: Journal of Electrical and Computer Engineering
 (2016)

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## 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.


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## Body

## 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 2n-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:(1)x˙1=ax2-x1,x˙2=cx1-x2-x1x3,x˙3=x1x2-bx3,where a=10, b=8/3, and c=28.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 x1, x2 and x3 be γ1, γ2, and γ3, respectively. According to the numerical simulation results of the Lorenz system [1], we can choose γ1=16, γ2=23, and γ3=40. Therefore, system (1) is changed as follows:(2)X˙1=aγ2γ1X2-aX1,X˙2=cγ1γ2X1-X2-γ1γ3γ2X1X3,X˙3=γ1γ2γ3X1X2-bX3.Step 2.
Two first-order differential equations about state variablesX4 and X5 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 X1 is replaced with X4 in the fourth equation of system (2), and the state variable X5 is used to take place of the state variable X2 in the fifth equation of system (2). Therefore, a five-dimensional autonomous system is obtained:(3)X˙1=aγ2γ1X2-aX1,X˙2=cγ1γ2X1-X2-γ1γ3γ2X1X3,X˙3=γ1γ2γ3X1X2-bX3,X˙4=aγ2γ1X2-aX4,X˙5=cγ1γ2X1-X5-γ1γ3γ2X1X3.Step 3.
Two linear coupling controllersu1 and u2 are added in the fourth and fifth equations of system (3), respectively. Then system (3) is changed as follows:(4)X˙1=aγ2γ1X2-aX1,X˙2=cγ1γ2X1-X2-γ1γ3γ2X1X3,X˙3=γ1γ2γ3X1X2-bX3,X˙4=aγ2γ1X2-aX4+u1,X˙5=cγ1γ2X1-X5-γ1γ3γ2X1X3+u2,where the coupling controllers u1=hX1-X4 and u2=hX2-X5. h is gain parameter.Step 4.
A new piecewise linear function is designed:(5)fx=x-∑n=-NMsgn⁡x+2n+1+M-N+1,where N,M∈0,1,2,…. Through replacing the state variables X4 and X5 with the piecewise linear function fx in system (4), a novel system is obtained as follows:(6)X˙1=a1X2-a2X1,X˙2=a3X1-X2-a4X1X3,X˙3=a5X1X2-a6X3,X˙4=a1X2-a2fX4+u1′,X˙5=a3X1-fX5-a4X1X3+u2′,where the system parameter a1=aγ2/γ1, a2=a, a3=cγ1/γ2, a4=γ1γ3/γ2, a5=γ1γ2/γ3, a6=b, and the nonlinear coupling controllers u1′=hX1-fX4 and u2′=hX2-fX5(7)fX4=X4-∑n=-N1M1sgn⁡X4+2n+1+M1-N1+1,fX5=X5-∑n=-N2M2sgn⁡X5+2n+1+M2-N2+1,where N1,M1,N2,M2∈0,1,2,….Let the system parametersa=10, b=8/3, c=28, γ1=16, γ2=23, γ3=40, h=5; that is, a1=14.38, a2=10, a3=19.48, a4=27.83, a5=9.20, a6=2.67. Under the action of the nonlinear coupling controllers u1′ and u2′, system (6) is able to generate 2N1+M1+2×N2+M2+2-wing butterfly chaotic attractors, as shown in Figure 1. When N1=M1=N2=M2=0, system (6) creates 4×2-wing butterfly chaotic attractors as shown in Figure 1(a). When N1=M1=M2=0 and N2=1, system (6) generates 4×3-wing butterfly chaotic attractors as shown in Figure 1(b). When N1=N2=1 and M1=M2=0, system (6) creates 6×3-wing butterfly chaotic attractors as shown in Figure 1(c), and the time series about the state variable X4 is shown in Figure 1(e). When N1=N2=M2=1 and M1=0, system (6) generates 6×4-wing butterfly chaotic attractors as shown in Figure 1(d).Figure 1
Grid multi-wing butterfly chaotic attractors and time series of system (6): (a) 4×2; (b) 4×3; (c) 6×3; (d) 6×4; (e) time series about the state variable X4.
(a) (b) (c) (d) (e)


## 3. Basic Dynamic Characteristic

### 3.1. Equilibrium Point

LetX˙1=X˙2=X˙3=X˙4=X˙5=0; that is, (8)a1X2-a2X1=0,a3X1-X2-a4X1X3=0,a5X1X2-a6X3=0,a1X2-a2fX4+u1′=0,a3x1-fx5-a4x1x3+u2′=0.According to solution for the equation set (8), the equilibrium points of system (6) are obtained: (9)En1n21=0,0,0,2n1,2n2,En1n22=A,Aa2a1,A2a2a5a1a6,A+2n1,Aa2a1+2n2,En1n23=-A,-Aa2a1,A2a2a5a1a6,-A+2n1,-Aa2a1+2n2,where A=a1a3a6-a2a6/a2a4a5, n1=-N1+1,-N1,…,0,…,M1 and n2=-N2+1,-N2,…,0,…,M2.For the equilibrium pointsE∗=X1∗,X2∗,X3∗,X4∗,X5∗, system (6) is linearized and the Jacobian matrix is defined as(10)J=-a2a1000a3-a4X3∗-1-a4X1∗00a5X2∗a5X1∗-a600ha10-a2-h0a3-a4X3∗h-a4X1∗0-1-h,where X1∗, X2∗, X3∗, X4∗, and X5∗ are the coordinates of the equilibrium points En1n21, En1n22, and En1n23.Substituting the equilibrium pointsEn1n21=0,0,0,2n1,2n2 into the characteristic equation det⁡J-λI=0, we get the following eigenvalues: λ1=-6.00, λ2=-15.00, λ3=-22.83, λ4=11.83, and λ5=-2.67. λ1, λ2, λ3, and λ5 are negative real numbers, and λ4 is a positive real number. So the equilibrium points En1n21 are unstable saddle.Substituting the equilibrium pointsEn1n22=A,Aa2/a1,A2a2a5/a1a6,A+2n1,Aa2/a1+2n2 into the characteristic equation det⁡J-λI=0, we get the following eigenvalues: λ1=-15.00, λ2=-6.00, λ3,4=0.09±10.20i, and λ5=-13.86. λ1, λ2, and λ5 are negative real numbers, and λ3,4 are a pair of conjugate complex eigenvalues with positive real parts. Through computation, we know that the equilibrium points En1n23 have the same eigenvalues as the equilibrium points En1n22. Therefore, the equilibrium points En1n22 and En1n23 are unstable saddle-foci of index 2.


### 3.2. Lyapunov Exponent Spectrum, Bifurcation Diagram, and Poincaré Mapping

LetN1=N2=1 and M1=M2=0; the Lyapunov exponent of system (6) and its bifurcation diagram which varies with the coefficient a6 and its Poincaré map are shown in Figure 2. The range of values for the coefficient a6 is 0 to 3. Figures 2(a) and 2(b) show that when a6∈0.75,3, 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).Figure 2
Lyapunov exponent spectrum, bifurcation diagram, and Poincaré mapping of system (6): (a) Lyapunov exponent spectrum; (b) bifurcation diagram; (c) Poincaré mapping.
(a) (b) (c)


## 3.1. Equilibrium Point

LetX˙1=X˙2=X˙3=X˙4=X˙5=0; that is, (8)a1X2-a2X1=0,a3X1-X2-a4X1X3=0,a5X1X2-a6X3=0,a1X2-a2fX4+u1′=0,a3x1-fx5-a4x1x3+u2′=0.According to solution for the equation set (8), the equilibrium points of system (6) are obtained: (9)En1n21=0,0,0,2n1,2n2,En1n22=A,Aa2a1,A2a2a5a1a6,A+2n1,Aa2a1+2n2,En1n23=-A,-Aa2a1,A2a2a5a1a6,-A+2n1,-Aa2a1+2n2,where A=a1a3a6-a2a6/a2a4a5, n1=-N1+1,-N1,…,0,…,M1 and n2=-N2+1,-N2,…,0,…,M2.For the equilibrium pointsE∗=X1∗,X2∗,X3∗,X4∗,X5∗, system (6) is linearized and the Jacobian matrix is defined as(10)J=-a2a1000a3-a4X3∗-1-a4X1∗00a5X2∗a5X1∗-a600ha10-a2-h0a3-a4X3∗h-a4X1∗0-1-h,where X1∗, X2∗, X3∗, X4∗, and X5∗ are the coordinates of the equilibrium points En1n21, En1n22, and En1n23.Substituting the equilibrium pointsEn1n21=0,0,0,2n1,2n2 into the characteristic equation det⁡J-λI=0, we get the following eigenvalues: λ1=-6.00, λ2=-15.00, λ3=-22.83, λ4=11.83, and λ5=-2.67. λ1, λ2, λ3, and λ5 are negative real numbers, and λ4 is a positive real number. So the equilibrium points En1n21 are unstable saddle.Substituting the equilibrium pointsEn1n22=A,Aa2/a1,A2a2a5/a1a6,A+2n1,Aa2/a1+2n2 into the characteristic equation det⁡J-λI=0, we get the following eigenvalues: λ1=-15.00, λ2=-6.00, λ3,4=0.09±10.20i, and λ5=-13.86. λ1, λ2, and λ5 are negative real numbers, and λ3,4 are a pair of conjugate complex eigenvalues with positive real parts. Through computation, we know that the equilibrium points En1n23 have the same eigenvalues as the equilibrium points En1n22. Therefore, the equilibrium points En1n22 and En1n23 are unstable saddle-foci of index 2.


## 3.2. Lyapunov Exponent Spectrum, Bifurcation Diagram, and Poincaré Mapping

LetN1=N2=1 and M1=M2=0; the Lyapunov exponent of system (6) and its bifurcation diagram which varies with the coefficient a6 and its Poincaré map are shown in Figure 2. The range of values for the coefficient a6 is 0 to 3. Figures 2(a) and 2(b) show that when a6∈0.75,3, 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).Figure 2
Lyapunov exponent spectrum, bifurcation diagram, and Poincaré mapping of system (6): (a) Lyapunov exponent spectrum; (b) bifurcation diagram; (c) Poincaré mapping.
(a) (b) (c)


## 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.Figure 3
Circuit diagram of grid multi-wing butterfly chaotic attractors.LetN1=N2=M2=1 and M1=0; the piecewise linear function (7) is changed as follows(11)fX4=X4-∑n=-10sgn⁡X4+2n+1,fX5=X5-∑n=-11sgn⁡X5+2n+1+1.The circuit diagram of the piecewise linear function (11) is shown in Figure 4.Figure 4
Circuit diagram of the piecewise linear function (11).In Figures3 and 4, all the operational amplifiers are selected as UA741CN. Their supply voltage E=±15 V and saturated voltage Vsat≈±13.5 V. All the multipliers are of type AD633JN and their gain is 0.1. E+ is the positive pole of the supply voltage E; that is, E+=15 V. E- is the negative pole of the supply voltage E; that is, E-=-15 V.According to Figure3, the circuit equation can be obtained as follows:(12)dX1dτ=1R0C0R15R1X2-R15R2X1,dX2dτ=1R0C0-R1510R3X1X3+R15R4X1-R15R5X2,dX3dτ=1R0C0R1510R6X1X2-R15R7X3,dX4dτ=1R0C0R15R8X2+R15R9X1-R15R10fX4,dX5dτ=1R0C0-R1510R11X1X3+R15R12X1+R15R13X2-R15R14fX5.To observe the output wave experimentally, the time scale transformation must be executed forτ, that is, let τ=τ0t and τ0=10-3, and (12) can be changed as follows:(13)dX1dτ=10-3R0C0R15R1X2-R15R2X1,dX2dτ=10-3R0C0-R1510R3X1X3+R15R4X1-R15R5X2,dX3dτ=10-3R0C0R1510R6X1X2-R15R7X3,dX4dτ=10-3R0C0R15R8X2+R15R9X1-R15R10fX4,dX5dτ=10-3R0C0-R1510R11X1X3+R15R12X1+R15R13X2-R15R14fX5.LetR0=10 kΩ, C0=10 nF, R=10 kΩ, and R15=100 kΩ; according to system (6) and equation (13), we can obtain R1=R8=69.54 kΩ, R2=100 kΩ, R3=R11=3.59 kΩ, R4=R12=51.33 kΩ, R5=1 MΩ, R6=10.87 kΩ, R7=362.32 kΩ, R9=R13=200 kΩ, R10=66.67 kΩ, and R14=166.67 kΩ.In Figure4, let RVI=135 kΩ. When k1, k2, k3, k4, and k5 are switched on, the circuit equation can be obtained as follows:(14)-fX4=-RRX4+RRVIVsatsgn⁡X4-R18R16+R17+R18E--RR·R18R18+R19E++RRVIVsatsgn⁡X4-R18R18+R19E+-RR·R18R16+R17+R18E-=-X4+sgn⁡X4+15R18R16+R17+R18-15R18R18+R19+sgn⁡X4-15R18R18+R19+15R18R16+R17+R18,-fX5=-RRX5+RRVIVsatsgn⁡X5-R18R16+R17+R18E--RR·R18R18+R19E++RRVIVsatsgn⁡X5-R18R18+R19E+-RR·R18R16+R17+R18E-+RRVIVsatsgn⁡X5-R17+R18R16+R17+R18E--RR·R18R18+R19E+=-X5+sgn⁡X5+15R18R16+R17+R18-15R18R18+R19+sgn⁡X5-15R18R18+R19+15R18R16+R17+R18+sgn⁡X5+15R17+R18R16+R17+R18-15R18R18+R19.According to the piecewise linear function (11) and equation (14), we can choose R16=12 kΩ, R17=2 kΩ, R18=1 kΩ, and R19=14 kΩ.According to Figures3 and 4, the gird multi-wing butterfly chaotic attractors are obtained via circuit simulation software Multisim 10.0, as shown in Figure 5. When k1, k3 are switched on and k2, k4, and k5 are switched off, the circuit generates 4×2-wing butterfly chaotic attractors as shown in Figure 5(a). When k1, k3, and k4 are switched on and k2 and k5 are switched off, the circuit creates 4×3-wing butterfly chaotic attractors as shown in Figure 5(b). When k1, k2, k3, and k4 are switched on and k5 is switched off, the circuit generates 6×3-wing butterfly chaotic attractors as shown in Figure 5(c), and the time series about the state variable X4 is shown in Figure 5(e). When k1, k2, k3, k4, and k5 are switched on, the circuit creates 6×4-wing butterfly chaotic attractors as shown in Figure 5(d).Figure 5
The results of circuit experiment: (a)4×2; (b) 4×3; (c) 6×3; (d) 6×4; (e) time series about the state variable X4.
(a) (b) (c) (d) (e)From Figures5 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, the4×2, 4×3, 6×3, and 6×4-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.



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