# Development of a Family of Chaotic Systems with Infinite Equilibria and Its Application for Image Encryption

Authors: Xiaofeng Li; Yulong Bai; Weishuan Pan; Di Wang; Yong-Jie Ma
Journal: Complexity
 (2022)

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## Abstract

Fourth-order autonomous nonlinear differential equations can exhibit chaotic properties. In this study, we propose a family of fourth-order chaotic systems with infinite equilibrium points whose equilibria form closed curves of different shapes. First, the phase diagrams and Lyapunov exponents (LEs) of the system family are simulated. The results show that the system family has complex phase diagrams and dynamic behaviors. Simulation analysis of the Poincarè mapping and bifurcation diagrams shows that the system has chaotic characteristics. The circuit simulation model is constructed and simulated in Multisim. The circuit simulation results coincide with the numerical simulation results, which verifies the circuit feasibility of the system. Then, based on Lyapunov stability theory and the adaptive control method, the synchronous control of the system with infinite equilibria is designed. Numerical simulation results verify that the system synchronization with the adaptive control method is well. Finally, the synchronous drive system is used for image encryption, the response system is used for decryption, and color image encryption is realized by combining deoxyribonucleic acid (DNA) coding and operating rules. Therefore, this study not only enriched the research on infinite equilibria chaotic systems but also further expanded secure communication technology by combining chaotic synchronization control and DNA coding in image encryption.


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## Body

## 1. Introduction

The study of chaotic systems has gradually become an important part of nonlinear dynamics since 1963 when Professor Lorenz proposed the classic Lorenz system [1]. Different characteristics have been identified in different chaotic systems, such as multi-scroll chaotic oscillators [2], fractional-order chaotic systems [3], three-dimensional chaotic systems [4], and chaotic systems with infinite equilibria [5]. Moreover, the complex dynamic characteristics of chaotic systems can be applied to audio encryption technology [6], image watermarking technology [7], chaotic mask communication [8], and other fields. As the research on chaotic systems has deepened, chaotic systems with infinite equilibrium points have gained attention and the analysis of their complex dynamic behavior has become a representative research direction.Recently, investigating infinite equilibrium points in chaotic systems has attracted the attention of many researchers. In 2015, Gotthans et al. [9] proposed a new class of three-dimensional chaotic system, by analyzing two cases in which the system has infinite equilibrium points and constitutes a circle and a square, respectively. Later, the heart-shaped equilibria chaotic system proposed by Pham et al. [5], the pear-shaped equilibria chaotic system proposed by Aceng et al. [10], and the cloud-shaped curve of the equilibrium points chaotic system proposed by Vaidyanathan et al. [11], all promoted the rapid development of three-dimensional infinite equilibria chaotic system. In 2019, Huynh et al. [12] proposed a new four-dimensional memristor chaotic system with a series of equilibrium points and analyzed the stability interval of a series of equilibrium points. Subsequently, Yang et al. [13] proposed a four-dimensional chaotic system with a boomerang-like equilibrium based on Mobayen’s [14] three-dimensional infinite equilibria chaotic system. Infinite equilibria chaotic systems can show complex dynamic characteristics. However, there are few discussions on the application of chaotic systems with infinite equilibria.The applications of chaotic system mainly includes the following topics: weak signal detection [15], image encryption [13, 16], data encryption [17], voice encryption [14], and chaotic system control [15, 18, 19]. Among these topics, image encryption and the synchronous control of the chaotic systems are still hot topics. There are many research studies on the image encryption of chaotic systems. For example, a double-channel image encryption/decryption algorithm based on neural networks and chaos [20] was designed; Huang et al. [21] proposed a parallel image encryption algorithm based on compressed sensing, which combines chaotic systems. Previous researchers have also studied image encryption algorithms based on DNA coding. Zhang and Liu [22] proposed different DNA coding and decoding rules based on generated chaotic sequences. Wang et al. [16] combined a chaotic system with DNA sequence operations to realize image encryption/decryption.There are many methods for the synchronization and control of chaotic systems, such as active control, adaptive control, and sliding mode control. Based on the proposed Chua’s circuit study, Li and Bo [15] realized a weak signal detection method via chaotic synchronization. Changbiao et al. [23] not only completed the signal tracking of the proposed unified chaotic system by designing an adaptive synovial controller but also realized the synchronous control of different structures. Fu et al. [24] have implemented an encryption algorithm and the double-chaotic system (the chaos synchronization between two chaotic systems). González-Zapata et al. [25] proposed the synchronization of chaotic artificial neurons and its application to secure image transmission. Ahmad et al. [26] proposed the application of multi-switch synchronization control system in secure communication, and Luo et al. [27] applied the multi-switch synchronization control of a memristor chaotic system in image encryption. However, this literature does not involve the keys and the coding is simple; therefore, we can improve the security of the encryption algorithm based on these two points.The main works of this study are as follows:(1)
A family of chaotic systems with unique characteristics, which have closed equilibria with different curves, is proposed.(2)
An adaptive controller is designed. According to Lyapunov stability theory, master and slave systems can be synchronized and the synchronization effect works well.(3)
An image encryption/decryption algorithm is designed. An adaptive controller and DNA coding/encoding are selected to achieve the security requirements of encryption.The remainder of this study is organized as follows: Section2 proposes a class of chaotic systems with unique characteristics. Section 3 describes the phase portraits and dynamical analysis of the selected chaotic system, in which parameter k is 2. In Section 4, a circuital implementation of the new chaotic system is reported. In Section 5, the results for the synchronization of the chaotic system are derived via parameter adaptive control. Section 6 describes image encryption via the synchronization of the chaotic system as an engineering application of our work. Section 7 provides a brief discussion. Section 8 draws the main conclusions.


## 2. A Family of Chaotic Systems with Infinite Equilibria

The chaotic system proposed in reference [9] is shown in equation (1), where g1x,y,z=ay+by2+xz and g2x,y,z=x2+y2. The simulated phase diagram is shown in Figure 1, when a=5, b=3, and the initial conditions x0,y0,z0=0,0,0:(1)x.=z,y.=−zg1x,y,z,z.=g2x,y,z.Figure 1
Chaotic phase diagram of System (1) with (a) = 5, (b) = 3, andx0=000T [9].
(a)(b)The system proposed and studied in this study adds the fourth equation of state and replacesg1x,y,z with g1x,y,z,w=ay+by2+xz−cw−d and g2x,y,z with xk+yk−1. The system equation obtained is shown in thefollowing equation:(2)x.=z,y.=−zay+by2+xz−cw−d,z.=xk+yk−1,w.=−ew−fyz.Here,x, y, z, and w are state variables and a, b, c, d, e, f, and k are system parameters. Note that k is an integer. It is clear that the equilibrium points of the system are x,y,0,0:xk+yk=1. As shown in Figure 2, as k increases, the equation |xk+yk=1 forms a closed curve region with increasing k value.Figure 2
Set of equilibrium points of different shapes of the system whenk=1,2,3,4,5,6.System (2) can behave as chaos fora=5, b=3, c=32, d=20, e=0.01, and f=25; the initial conditions are assigned as x0,y0,z0,w0=0.02,0.02,0.02,0.02. The Runge–Kutta method was used for numerical simulation, and simulation phase diagrams under different integers k were obtained, as shown in Figure 3. In Figures 3a–f of the simulation phase diagrams, the black closed curve is the equilibrium points under different integers k, and it can be seen that the attractor surrounds and connects the equilibria. It shows that the new system has a more complex phase diagram and satisfies at least one of the criteria [28].Figure 3
Phase diagram of the difference in integerk. (a) k=1. (b) k=2. (c) k=3. (d) k=4. (e) k=5. (f) k=6.
(a)(b)(c)(d)(e)(f)According to System (2), the Jacobian matrix (3) of the system family can be obtained. The Lyapunov exponents were obtained by numerical simulation based on the famous Wolf’s algorithm [29], as listed in Table 1.(3)J=0010−z2−az−2byz−ay−by2−2xz+cw+dczkxk−1sgnxkkyk−1sgnyk000−fz−fy−e.Table 1
Lyapunov exponents of different integersk.
EquationsEquilibriaLEsDKYk=1x∗+y∗=1(0.0528, 0, −0.0200, −0.0400)3.8200k=2x∗2+y∗2=1(0.0240, 0, −0.0172, −0.0199)3.3417k=3x∗3+y∗3=1(0.0283 ,0, −0.0130, −0.0225)3.6000k=4x∗4+y∗4=1(0.0234 ,0, −0.0073, −0.0238)3.6765k=5x∗5+y∗5=1(0.0090, 0, −0.0047, −0.0125)3.3440k=6x∗6+y∗6=1(0.0144, 0, −0.0051, −0.0262)3.3549


## 3. Analysis of System Dynamics

We analyze the dynamics of the system at integerk=2. For a=5, b=3, c=32, d=20, e=0.01, and f=25 and initial conditions ic=0.02,0.02,0.02,0.02T, this system has chaotic solutions with the phase diagrams, as shown in Figure 4.Figure 4
Chaotic system simulation two-dimensional phase diagram. (a) yz phase diagram. (b) xz phase diagram. (c) yw phase diagram. (d) zw phase diagram.
(a)(b)(c)(d)


### 3.1. Equilibria Analysis and Lyapunov Exponents

Whenk=2, the Jacobian matrix of the system at the equilibrium points [30] is illustrated in the following equation, where x∗ and y∗ are satisfied when x∗2+y∗2=1:(4)J=001000−5y∗−3y∗2+2002x∗2y∗0000−fy∗−e.Then, the characteristic polynomial of the Jacobian matrix is as follows:(5)detλI−J=0.The characteristic values at the equilibrium points obtained by numerical calculation are as follows:(6)λ1=−0.01,λ2=0,λ3,4=±−6y∗3−10y∗2+40y∗+2x∗.In Table2, I, II, III, IV represent the four quadrants, respectively.Table 2
Eigenvalues of system (2) fork = 2.
Caseλ2,3IA pair of purely imaginaryII−0.996<x<0,0.05<y<1A pair of purely imaginaryII−1<x<−0.996,0<y<0.05One positive real, one negative realIIIOne positive real, one negative realIV0<x<0.996,−1<y<−0.05One positive real, one negative realIV0.996<x<1,−0.05<y<0A pair of purely imaginaryx=0,y=1λ3,4=±4.8989x=−1,y=0λ3,4=±1.4142ix=0,y=−1λ3,4=±7.4833ix=1,y=0λ3,4=±1.4142x=0.996,y=−0.05λ3,4=0x=−0.996,y=0.05λ3,4=0A pair of purely imaginary eigenvalues represents an unstable central equilibrium. And this implies an unstable saddle for pure real eigenvalues. In addition, the four-dimensional Bogdanov–Tarkens equilibrium (λ2=λ3=λ4=0) for k=2 is given.This system has a chaotic solution with Lyapunov exponentsLE1=0.0240, LE2=0, LE3=−0.0172, and LE4=−0.0199, and the Lyapunov exponential is shown in Figure 5.Figure 5
Lyapunov exponential of the system over time.The fractal dimension [17] can be calculated using the following equation:(7)DL=j+1LEj+1∑i=1jLEi=3+LE1+LE2+LE3|LE4|=3.3417.The Lyapunov exponents areLE1>0, LE2=0, LE3<0, and LE4<0, and the dimension is fractal (DL=3.3417). The system shows chaotic behavior.


### 3.2. Bifurcation Diagram and the Poincarè Section Diagram

Poincarè is a measure of the chaotic state that can effectively and directly describe the phase space [31]. The system is chaotic when the Poincarè map is a continuous curve or some dense points. The section diagram in Figure 6 shows an irregular distribution of points. Therefore, we can verify that the system is chaotic.Figure 6
Poincarè diagram of the system. (a) Sectionw = 0 in the yzw phase diagram. (b) Section z = 0 in the xzw phase diagram.
(a)(b)The bifurcation diagram can reflect the periodic or chaotic motion of a system [30]. If the parameters change, the stability of the system will vary. This is illustrated by observing the state changes of the chaotic system through the bifurcation diagram and Lyapunov exponential when the system parameter changes.Fora=5, b=3, c=32, d=20, e=0.01, and f∈20,100 this system has chaotic solutions with the bifurcation diagram and Lyapunov exponential, as shown in Figure 7. Obviously, from the bifurcation diagram and Lyapunov exponent spectrum, we conclude that System (2) exhibits robust chaos [32] in the whole region, except for the periodic motion shown when the maximum Lyapunov exponential zero at f = 50. Therefore, this type of system signal has high scientific research value and can be used in image encryption.Figure 7
Bifurcation diagram and Lyapunov exponential spectrum of the system when parameterf changes. (a) Bifurcation diagram with f. (b) Lyapunov exponential with f.
(a)(b)


### 3.3. Multistability Analysis

Multiple stability is an interesting phenomenon, which usually exists in chaotic systems with infinite equilibrium points. It can be seen from the bifurcation diagram in Figure8. When the parameters a=5, b=3, c=32, d=20, e=0.01, and varying f are in the region of [0, 30], there exist coexisting attractors. A set of initial conditions (0.02 0.02 0.02 0.02) with blue color and another set of initial conditions (0.5 0 0 0) with red color are fixed. Some sample coexisting attractors are presented. For example, when f = 2.5 or f = 12.12, the system produces the coexisting periodic attractors as shown in Figures 9(a) and 9(b). Moreover, the coexisting chaotic attractor and periodic attractor are found in the new System (2) with f = 12.72, as shown in Figure 9(c). In addition, the coexisting chaotic attractors are shown in Figure 9(d) with f = 25.Figure 8
Coexisting bifurcation diagram of the state variable z with respect to the control parameter f.Figure 9
(a) The coexisting periodic attractors in y-z plane. (b) The coexisting periodic attractors in x-z plane. (c) The coexisting chaotic attractor and periodic attractor in x-z plane. (d) The coexisting chaotic attractors in x-z plane.
(a)(b)(c)(d)


## 3.1. Equilibria Analysis and Lyapunov Exponents

Whenk=2, the Jacobian matrix of the system at the equilibrium points [30] is illustrated in the following equation, where x∗ and y∗ are satisfied when x∗2+y∗2=1:(4)J=001000−5y∗−3y∗2+2002x∗2y∗0000−fy∗−e.Then, the characteristic polynomial of the Jacobian matrix is as follows:(5)detλI−J=0.The characteristic values at the equilibrium points obtained by numerical calculation are as follows:(6)λ1=−0.01,λ2=0,λ3,4=±−6y∗3−10y∗2+40y∗+2x∗.In Table2, I, II, III, IV represent the four quadrants, respectively.Table 2
Eigenvalues of system (2) fork = 2.
Caseλ2,3IA pair of purely imaginaryII−0.996<x<0,0.05<y<1A pair of purely imaginaryII−1<x<−0.996,0<y<0.05One positive real, one negative realIIIOne positive real, one negative realIV0<x<0.996,−1<y<−0.05One positive real, one negative realIV0.996<x<1,−0.05<y<0A pair of purely imaginaryx=0,y=1λ3,4=±4.8989x=−1,y=0λ3,4=±1.4142ix=0,y=−1λ3,4=±7.4833ix=1,y=0λ3,4=±1.4142x=0.996,y=−0.05λ3,4=0x=−0.996,y=0.05λ3,4=0A pair of purely imaginary eigenvalues represents an unstable central equilibrium. And this implies an unstable saddle for pure real eigenvalues. In addition, the four-dimensional Bogdanov–Tarkens equilibrium (λ2=λ3=λ4=0) for k=2 is given.This system has a chaotic solution with Lyapunov exponentsLE1=0.0240, LE2=0, LE3=−0.0172, and LE4=−0.0199, and the Lyapunov exponential is shown in Figure 5.Figure 5
Lyapunov exponential of the system over time.The fractal dimension [17] can be calculated using the following equation:(7)DL=j+1LEj+1∑i=1jLEi=3+LE1+LE2+LE3|LE4|=3.3417.The Lyapunov exponents areLE1>0, LE2=0, LE3<0, and LE4<0, and the dimension is fractal (DL=3.3417). The system shows chaotic behavior.


## 3.2. Bifurcation Diagram and the Poincarè Section Diagram

Poincarè is a measure of the chaotic state that can effectively and directly describe the phase space [31]. The system is chaotic when the Poincarè map is a continuous curve or some dense points. The section diagram in Figure 6 shows an irregular distribution of points. Therefore, we can verify that the system is chaotic.Figure 6
Poincarè diagram of the system. (a) Sectionw = 0 in the yzw phase diagram. (b) Section z = 0 in the xzw phase diagram.
(a)(b)The bifurcation diagram can reflect the periodic or chaotic motion of a system [30]. If the parameters change, the stability of the system will vary. This is illustrated by observing the state changes of the chaotic system through the bifurcation diagram and Lyapunov exponential when the system parameter changes.Fora=5, b=3, c=32, d=20, e=0.01, and f∈20,100 this system has chaotic solutions with the bifurcation diagram and Lyapunov exponential, as shown in Figure 7. Obviously, from the bifurcation diagram and Lyapunov exponent spectrum, we conclude that System (2) exhibits robust chaos [32] in the whole region, except for the periodic motion shown when the maximum Lyapunov exponential zero at f = 50. Therefore, this type of system signal has high scientific research value and can be used in image encryption.Figure 7
Bifurcation diagram and Lyapunov exponential spectrum of the system when parameterf changes. (a) Bifurcation diagram with f. (b) Lyapunov exponential with f.
(a)(b)


## 3.3. Multistability Analysis

Multiple stability is an interesting phenomenon, which usually exists in chaotic systems with infinite equilibrium points. It can be seen from the bifurcation diagram in Figure8. When the parameters a=5, b=3, c=32, d=20, e=0.01, and varying f are in the region of [0, 30], there exist coexisting attractors. A set of initial conditions (0.02 0.02 0.02 0.02) with blue color and another set of initial conditions (0.5 0 0 0) with red color are fixed. Some sample coexisting attractors are presented. For example, when f = 2.5 or f = 12.12, the system produces the coexisting periodic attractors as shown in Figures 9(a) and 9(b). Moreover, the coexisting chaotic attractor and periodic attractor are found in the new System (2) with f = 12.72, as shown in Figure 9(c). In addition, the coexisting chaotic attractors are shown in Figure 9(d) with f = 25.Figure 8
Coexisting bifurcation diagram of the state variable z with respect to the control parameter f.Figure 9
(a) The coexisting periodic attractors in y-z plane. (b) The coexisting periodic attractors in x-z plane. (c) The coexisting chaotic attractor and periodic attractor in x-z plane. (d) The coexisting chaotic attractors in x-z plane.
(a)(b)(c)(d)


## 4. Implementation of System Circuit

Electronic circuit synthesis is not only a method to model nonlinear dynamic systems accurately but also a method to test the stability of system structures [11]. Multisim is used to design the circuit diagram of the chaotic system to illustrate the feasibility of the system, as shown in Figure 10. We use the popular design approach based on multipliers and operational amplifiers, which is TL082CD, the supply voltage is ±15V, and the saturation voltage is ±13.5V. The multiplier is AD633 (the output gain is set to 0.1).Figure 10
Circuit diagram of the chaotic system.Under the classical system parameters, the dynamic range of the state variable of System (2) is too small. In order to avoid excessive input errors of the multiplier, the linear transformation of the state variable of the system of equation (8) is performed and then the time scale transformation [33] of τ=τ0t is performed. Finally, the system of equation (9) is obtained.A schematic diagram of the designed circuit is shown in Figure9. When the capacitance values are C1=C2=C3=C4=0.1μF and the resistor values are R12=R13=R14=R15=R16=R17=10kΩ, the other resistor values are shown in equation (10):(8)x,y,z,w⟶0.1x,0.25y,0.05z,0.25w,(9)x.=500z,y.=−250zy−37.5y2z−xz2−1600w−z−4000−z,z.=−200x2+1250y2−20000,w.=−10w−1250yz.The phase diagram obtained by circuit simulation is shown in Figure11. Comparing the circuit simulation phase diagrams in Figure 11 and the numerical simulation phase diagrams in Figure 4 shows that they coincide, which verifies the circuit feasibility of the chaotic system.(10)500=1R1C1⟶R1=20kΩ,250=1R2C2⟶R2=4kΩ,37.5=1R3C2⟶R3=2.6kΩ,1=1R4C2⟶R4=100kΩ,1600=1R5C2⟶R5=625Ω,4000=1R6C2⟶R6=2.5kΩ,200=1R7C3⟶R7=5kΩ,1250=1R8C3⟶R8=800Ω,20000=1R9C3⟶R9=500Ω,10=1R10C4⟶R10=1000kΩ,1250=1R11C4⟶R11=800Ω.Figure 11
Circuit simulation phase diagram. (a) yz phase diagram. (b) xz phase diagram. (c) yw phase diagram. (d) zw phase diagram.
(a)(b)(c)(d)


## 5. Parameter Adaptive Synchronous Control

In this study, the synchronous control of two systems with infinite equilibrium points is studied.The drive system with an infinite equilibrium point is designed as follows:(11)x1.=x3,x2.=−ax2x3−bx22x3−x1x32+cx3x4+dx3,x3.=x12+x22−1,x4.=−ex4−fx2x3.The designed response system is as follows:(12)y1.=y3+u1,y2.=−a^y2y3−b^y22y3−y1y32+c^y3y4+d^y3+u2,y3.=y12+y22−1+u3,y4.=−e^y4−f^y2y3+u4.Here, the adaptive controller isu=u1,u2,u3,u4T and the state error of the drive system and the response system is defined as ei=yi−xii=1,2,3,4. The synchronization state error is calculated by the following equation [34]:(13)e1=y1−x1,e2=y2−x2,e3=y3−x3,e4=y4−x4.Similarly, the estimated error of the calculated parameters is as follows:(14)ea=a−a^,eb=b−b^,ec=c−c^,ed=d−d^,ee=e−e^,ef=f−f^,(15)ea.=−a^.,eb.=−b^.,ec.=−c^.,ed.=−d^.,ee.=−e^.,ef.=−f^.,.wherea^, b^, c^, d^, e^, and f^ are the estimated of parameters. The estimation error of the dynamic parameters can be obtained by equation (15).To achieve adaptive synchronous control, namely, to achievelimt⟶+∞et=limt⟶+∞yt−xt=0, the adaptive controller is designed as follows:(16)u1=−e3−k1e1,u2=a^e2e3+e2x3+e3x2+b^y22y3−x22x3+y1y32−x1x32−c^e3e4+e3x4+e4x3−d^e3−k2e2,u3=−e1y1+x1−e2y2+x2−k3e3,u4=e^e4+f^e2e3+e2x3+e3x2−k4e4.Here,k1, k2, k3, and k4 are normal numbers, and the parameter updating law is defined as follows:(17)a^.=−λ1e22e3+e22x3+e2e3x2−ea,b^.=−λ2e2y22y3−e2x22x3−eb,c^.=λ3e2e3e4+e2e3x4+e2e4x3+ec,d^.=λ4e2e3+ed,e^.=−λ5e42−ee,f^.=−λ6e2e3e4+e2e4x3+e3e4x2−ef.The designed controller and parameter updating rules are inserted into the error equation to obtain the following equation:(18)e1.=−k1e1,e2.=−eae2e3+e2x3+e3x2−eby22y3−x22x3+ece3e4+e3x4+e4x3+ede3−k2e2,e3.=−k3e3,e4.=−eee4−efe2e3+e2x3+e3x2−k4e4.Lyapunov stability theory [16, 35] was used to verify the synchronization of the drive system and response system. The Lyapunov function is constructed as follows:(19)Ve1,e2,e3,e4,ea,eb,ec,ee,ef=12e12+12e22+12e32+12e42+12λ1ea2+12λ2eb2+12λ3ec212λ4ed2+12λ5ee2+12λ6ef2.In equations (17) and (19), factor λi>0i=1,2,3,4,5,6 and the derivative of V yields equation (20), which is obvious for V≥0 and V.<0. Therefore, according to Lyapunov stability theory, the driving system and the response system can realize synchronization.The synchronization is verified by an example. That is, the fourth-order Runge–Kutta algorithm is used for numerical simulation. The system parameters area=5, b=3, c=32, d=20, e=0.01, and f=25. The initial values of the drive system are x10=0.01, x20=0.01, x30=0.01, and x40=0.01. The initial values of the response system are y10=2, y20=4, y30=6, and y40=8. The initial conditions of the parameter estimation are a^=2.5, b^=1.5, c^=35, d^=25, e^=0.02, and f^=20. The control parameters are set as k1=k2=k3=k4=80 and λ1=λ2=λ3=λ4=λ5=λ6=60. The simulation results are shown in Figure 12.(20)V.=e1e1.+e2e2.+e3e3.+e4e4.+1λ1eaea.+1λ2ebeb.+1λ3ecec.+1λ4eded.+1λ5eeee.+1λ6efef.=−k1e12−eae2e2e3+e2x3+e3x2−ebe2y22y3−x22x3+ece2e3e4+e3x4+e4x3+ede2e3−k2e22−k3e32−eee42−efe4e2e3+e2x3+e3x2−k4e42−1λ1eaa⌢.−1λ2ebb⌢.−1λ3ecc⌢.−1λ4edd⌢.−1λ5eee⌢.−1λ6eff⌢,=−k1e12+ea−e2e2e3+e2x3+e3x2−1λ1a⌢.+eb−e2y22y3−x22x3−1λ2b⌢.+ece2e3e4+e3x4+e4x3−1λ3c⌢.−k2e22−k3e32+ede2e3−1λ4d⌢.+ee−e42−1λ5e⌢.+ef−e4e2e3+e2x3+e3x2−1λ6f⌢.−k4e42=−ea2−eb2−ec2−ed2−ee2−ef2−k1e12−k2e22−k3e32−k4e42.Figure 12
Synchronous simulation. (a) Synchronization waveform ofx2 and y2. (b) Synchronization waveform of x4 and y4. (c) Synchronization errors e1, e2, e3, and e4. (d) Identification curves of parameters.
(a)(b)(c)(d)The synchronization error curves between the driving system and the response system are achieved in approximately 0.08 seconds. The system completes the identification of unknown parameters in approximately 0.1 seconds. This shows that the synchronization control of the two systems is realized and that the synchronization effect is good. But in different synchronization systems, the control parameters are selected differently, reflecting different synchronization effects. Therefore, the choice of control parameters should depend on the specific situation.


## 6. Application in Image Encryption

In cryptography, there are some basic concepts [36], such as plaintext representing information to be encrypted, ciphertext representing information processed by an encryption algorithm, an encryption algorithm representing the way plaintext converted to ciphertext, and a decryption algorithm representing the way ciphertext converted to plaintext. There are two principles for encryption in cryptography: diffusion and confusion. Diffusion is the purpose of encrypting plaintext by loading plaintext into ciphertext [16]. However, confusion uses obfuscation to conceal the relationship between plaintext and ciphertext without changing the plaintext, making the plaintext undecipherable. The analysis of the image encryption effect mainly includes histograms, information entropy, sensitivities to keys, correlations, and other measures. The overall structure of the image encryption and decryption process of this study is shown in Figure 13.Figure 13
The overall structure of the image encryption and decryption.


### 6.1. Method of Image Encryption and Decryption Method

In this study, a chaotic adaptive parameter synchronization control system with infinite equilibria is used for image encryption to verify that the system is more secure for cryptography.There are 3-color data matricesR, G, and B of original image I with 512×512×3 pixels. The original image I is denoted as M×N.The detailed steps of image encryption are as follows:Step 1: Preprocess the chaotic sequences.When the initial values of the adaptive parameter system arex10=0.01, x20=0.02, x30=0.01, and x40=0.02, the four-dimensional chaotic sequences of the driving system, which are denoted as x=x1,x2,x3,x4T, can be obtained. To eliminate the transient effects for more random effects, the top 1000 items generated iteratively were removed. Each element (c(i)) in the chaotic sequences was processed by the following equation, and the chaotic sequences were converted into an integer between 0 and 255:(21)ci=ci∗103mod256.Step 2: Generate scrambled sequences. Three chaotic sequences ofx1, x2, and x3 were arranged in ascending order with the sort function to obtain the scrambled matrices R′, G′, and B′.Step 3: The chaotic sequence x4 obtained by iteration is transformed into a two-dimensional matrixC, whose size is M×N. The XOR operation [16] (Table 3) was performed on matrices R′, G′, and B′, and matrix C is obtained according to number 1 in Table 4 (binary code for DNA coding rules [16]) at corresponding positions to conceal plaintext information. The diffusion matrices R′′, G′′, and B′′ are obtained.Step 4: Reconstruct the matricesR′′, G′′, and B′′ to obtain the encrypted image.Table 3
Xor operation.
⊕ATCGATAGCTATCGCGCTAGCGATTable 4
Binary versus DNA coding rules.
01234567A0000111101011010T1111000010100101C1001100111001100G0111011100110011The detailed steps of the image decryption are as follows:Step 1. Preprocessed sequences. Input the initial values of the adaptive parameters system as the keys to obtain the 4-dimensional mixture of the response system. It is denoted asy=y1,y2,y3,y4T. Similarly, the first 1000 items are eliminated, and then each element in the chaotic sequences is processed by equation (21) to convert the chaotic sequences into an integer between 0 and 255.Step 2. Read the 3-color ciphertext matricesR′′, G′′, and B′′ of the encrypted image.Step 3. The resulting chaotic sequencey4 is converted to a two-dimensional matrix C′ with a size of M×N. The XOR operation (Table 3) is performed on the ciphertext matrices R′′, G′′, and B′′, and matrix C′ at the corresponding positions according to number 1 in Table 4 (binary rules for DNA encoding [14]). The R′, G′, and B′ matrices of the ciphertext image are obtained.Step 4. TheR′, G′, and B′ matrices of the ciphertext image are restored by using x1, x2, and x3 chaotic sequences as ascending obfuscations. The R, G, and B matrices are obtained.Step 5. Reconstruct two-dimensional matricesR, G, and B. The decrypted image is obtained.


### 6.2. Experimental Simulation

In this study, the classic image Lena is used as the representative for encryption. The adaptive parameter system selects the above parameters and conducts encryption according to the above steps.Figure14 shows the experimental results. It can be seen that the encrypted image has no similar information to the original image, and it can be visually determined that the encryption and decryption are successful.Figure 14
Encryption results. (a) Original image. (b) Confusion image. (c) Encryption image. (d) Decryption image.
(a)(b)(c)(d)


### 6.3. Histogram Analysis

The histogram intuitively sees the encryption effect. Figure15 is a histogram comparison of original and encrypted images in the RGB channels.Figure15 shows that the histograms of the encrypted image are completely different from those of the original image. The pixel values of ciphertext images are evenly distributed by diffusion, and it is difficult to analyze the relationship with the original image. The hiding effect of the pixel values of the original image is better.Figure 15
Histograms analysis. (a–c) Histograms of the original image in the RGB channels, respectively; (d–f) histograms of the encrypted image in RGB channels, respectively.
(a)(b)(c)(d)(e)(f)


### 6.4. Correlation Analysis

The adjacent pixels of the original image have large correlations in the horizontal, vertical, and diagonal directions because of the coherence of the picture. Image encryption destroys the correlation between the pixel values of the original image. Therefore, correlation analysis can be used as an index of image encryption effect analysis. N pairs of adjacent pixels, which areui,vi, from the image are chosen. The correlation coefficient [37] is calculated by equation (22). The correlation curves are shown in Figure 16, and the correlation coefficients are listed in Table 5.(22)rxy=covu,vDuDv,covu,v=1N∑i−1Nui−Euyi−Ev,Du=1N∑i−1Nui−Eu2,Eu=1N∑i−1Nui.Figure 16
Partial correlations of adjacent pixels. (a) Horizontal correlation of the original image in the R channel, (b) horizontal correlation of the ciphered image in the R channel, (c) vertical correlation of the original image in the G channel, (d) vertical correlation of the ciphered image in the G channel, (e) diagonal correlation of the original image in the B channel, and (f) diagonal correlation of the ciphered image in the B channel.
(a)(b)(c)(d)(e)(f)Table 5
Comparison of correlation coefficients.
CorrelationLenaEncryption imageHorizontal0.978920.02025Vertical0.988610.00243Diagonal0.96279−0.03036The correlations of the original image and the ciphered image show that the pixel correlation can be reduced to zero or close to zero after encryption. After encrypting adjacent pixels, it is concluded that the algorithm has a secure encryption effect. This means that the attacker cannot use correlation to crack the encrypted image.


### 6.5. Information Entropy Analysis

Information entropy reflects the uncertainty of image information [26] and can be calculated by the following equation [37]:(23)H=−∑i=0Lpilog2pi,where L is the number of gray levels of the image and pi is the probability of the pixel value in the entire picture. The theoretical value of the information entropy is 8. It is generally believed that the information entropy of all the encrypted images is quite close to the ideal value of 8, which implies that the encrypted images have good randomness.As Table6 shows, the encryption algorithm performs better than the results in the listed literature, which indicates that this encryption algorithm is competitive in this sense.Table 6
Information entropy comparison.
ImageLenaLiterature [37]Literature [38]Literature [39]Literature [40]Literature [41]This studyInformation entropy6.99517.99747.99707.99797.99827.99897.9994


### 6.6. Key Sensitivity Analysis

Key sensitivity [38] means that a small change in a designed key will lead to decryption failure, and the original image cannot be extracted when the same image is decrypted.The keys used in this study that decrypt the image are they10=0.01, y20=0.02, y30=0.01, and y40=0.02, as shown in Figure 17(a). y0=y0+10−16 is slightly changed and the other keys remained unchanged. The decrypted image is shown in Figure 17.Figure 17
Decryption images. (a) Correct key decryption image, and (b) decrypted images with a small change in y (0).
(a)(b)Figure17(a) shows that the decryption image is completely identical to the original image. However, in Figure 17(b), the decryption image is completely different, which means that the proposed encryption method provides a high decrypt sensitivity.


## 6.1. Method of Image Encryption and Decryption Method

In this study, a chaotic adaptive parameter synchronization control system with infinite equilibria is used for image encryption to verify that the system is more secure for cryptography.There are 3-color data matricesR, G, and B of original image I with 512×512×3 pixels. The original image I is denoted as M×N.The detailed steps of image encryption are as follows:Step 1: Preprocess the chaotic sequences.When the initial values of the adaptive parameter system arex10=0.01, x20=0.02, x30=0.01, and x40=0.02, the four-dimensional chaotic sequences of the driving system, which are denoted as x=x1,x2,x3,x4T, can be obtained. To eliminate the transient effects for more random effects, the top 1000 items generated iteratively were removed. Each element (c(i)) in the chaotic sequences was processed by the following equation, and the chaotic sequences were converted into an integer between 0 and 255:(21)ci=ci∗103mod256.Step 2: Generate scrambled sequences. Three chaotic sequences ofx1, x2, and x3 were arranged in ascending order with the sort function to obtain the scrambled matrices R′, G′, and B′.Step 3: The chaotic sequence x4 obtained by iteration is transformed into a two-dimensional matrixC, whose size is M×N. The XOR operation [16] (Table 3) was performed on matrices R′, G′, and B′, and matrix C is obtained according to number 1 in Table 4 (binary code for DNA coding rules [16]) at corresponding positions to conceal plaintext information. The diffusion matrices R′′, G′′, and B′′ are obtained.Step 4: Reconstruct the matricesR′′, G′′, and B′′ to obtain the encrypted image.Table 3
Xor operation.
⊕ATCGATAGCTATCGCGCTAGCGATTable 4
Binary versus DNA coding rules.
01234567A0000111101011010T1111000010100101C1001100111001100G0111011100110011The detailed steps of the image decryption are as follows:Step 1. Preprocessed sequences. Input the initial values of the adaptive parameters system as the keys to obtain the 4-dimensional mixture of the response system. It is denoted asy=y1,y2,y3,y4T. Similarly, the first 1000 items are eliminated, and then each element in the chaotic sequences is processed by equation (21) to convert the chaotic sequences into an integer between 0 and 255.Step 2. Read the 3-color ciphertext matricesR′′, G′′, and B′′ of the encrypted image.Step 3. The resulting chaotic sequencey4 is converted to a two-dimensional matrix C′ with a size of M×N. The XOR operation (Table 3) is performed on the ciphertext matrices R′′, G′′, and B′′, and matrix C′ at the corresponding positions according to number 1 in Table 4 (binary rules for DNA encoding [14]). The R′, G′, and B′ matrices of the ciphertext image are obtained.Step 4. TheR′, G′, and B′ matrices of the ciphertext image are restored by using x1, x2, and x3 chaotic sequences as ascending obfuscations. The R, G, and B matrices are obtained.Step 5. Reconstruct two-dimensional matricesR, G, and B. The decrypted image is obtained.


## 6.2. Experimental Simulation

In this study, the classic image Lena is used as the representative for encryption. The adaptive parameter system selects the above parameters and conducts encryption according to the above steps.Figure14 shows the experimental results. It can be seen that the encrypted image has no similar information to the original image, and it can be visually determined that the encryption and decryption are successful.Figure 14
Encryption results. (a) Original image. (b) Confusion image. (c) Encryption image. (d) Decryption image.
(a)(b)(c)(d)


## 6.3. Histogram Analysis

The histogram intuitively sees the encryption effect. Figure15 is a histogram comparison of original and encrypted images in the RGB channels.Figure15 shows that the histograms of the encrypted image are completely different from those of the original image. The pixel values of ciphertext images are evenly distributed by diffusion, and it is difficult to analyze the relationship with the original image. The hiding effect of the pixel values of the original image is better.Figure 15
Histograms analysis. (a–c) Histograms of the original image in the RGB channels, respectively; (d–f) histograms of the encrypted image in RGB channels, respectively.
(a)(b)(c)(d)(e)(f)


## 6.4. Correlation Analysis

The adjacent pixels of the original image have large correlations in the horizontal, vertical, and diagonal directions because of the coherence of the picture. Image encryption destroys the correlation between the pixel values of the original image. Therefore, correlation analysis can be used as an index of image encryption effect analysis. N pairs of adjacent pixels, which areui,vi, from the image are chosen. The correlation coefficient [37] is calculated by equation (22). The correlation curves are shown in Figure 16, and the correlation coefficients are listed in Table 5.(22)rxy=covu,vDuDv,covu,v=1N∑i−1Nui−Euyi−Ev,Du=1N∑i−1Nui−Eu2,Eu=1N∑i−1Nui.Figure 16
Partial correlations of adjacent pixels. (a) Horizontal correlation of the original image in the R channel, (b) horizontal correlation of the ciphered image in the R channel, (c) vertical correlation of the original image in the G channel, (d) vertical correlation of the ciphered image in the G channel, (e) diagonal correlation of the original image in the B channel, and (f) diagonal correlation of the ciphered image in the B channel.
(a)(b)(c)(d)(e)(f)Table 5
Comparison of correlation coefficients.
CorrelationLenaEncryption imageHorizontal0.978920.02025Vertical0.988610.00243Diagonal0.96279−0.03036The correlations of the original image and the ciphered image show that the pixel correlation can be reduced to zero or close to zero after encryption. After encrypting adjacent pixels, it is concluded that the algorithm has a secure encryption effect. This means that the attacker cannot use correlation to crack the encrypted image.


## 6.5. Information Entropy Analysis

Information entropy reflects the uncertainty of image information [26] and can be calculated by the following equation [37]:(23)H=−∑i=0Lpilog2pi,where L is the number of gray levels of the image and pi is the probability of the pixel value in the entire picture. The theoretical value of the information entropy is 8. It is generally believed that the information entropy of all the encrypted images is quite close to the ideal value of 8, which implies that the encrypted images have good randomness.As Table6 shows, the encryption algorithm performs better than the results in the listed literature, which indicates that this encryption algorithm is competitive in this sense.Table 6
Information entropy comparison.
ImageLenaLiterature [37]Literature [38]Literature [39]Literature [40]Literature [41]This studyInformation entropy6.99517.99747.99707.99797.99827.99897.9994


## 6.6. Key Sensitivity Analysis

Key sensitivity [38] means that a small change in a designed key will lead to decryption failure, and the original image cannot be extracted when the same image is decrypted.The keys used in this study that decrypt the image are they10=0.01, y20=0.02, y30=0.01, and y40=0.02, as shown in Figure 17(a). y0=y0+10−16 is slightly changed and the other keys remained unchanged. The decrypted image is shown in Figure 17.Figure 17
Decryption images. (a) Correct key decryption image, and (b) decrypted images with a small change in y (0).
(a)(b)Figure17(a) shows that the decryption image is completely identical to the original image. However, in Figure 17(b), the decryption image is completely different, which means that the proposed encryption method provides a high decrypt sensitivity.


## 7. Discussions

The proposed family of chaotic systems enriches the study of chaotic systems with infinite equilibrium points, and on this basis, we can independently study chaotic systems with more unique closed curve equilibrium points. The integer-order chaotic system can also be extended to the fractional-order chaotic system, so that the study of chaotic system has more practical significance. The circuit of the new system is designed and simulated with Multisim. The phase diagram of the circuit simulation is in good agreement with the result of numerical simulation. This proves the feasibility of circuit realization, and chaotic systems can be realized by analog and digital circuit construction in the future work, which endods the value of the industrial practical application of the chaotic system. The designed adaptive synchronization control can achieve synchronization quickly. However, other synchronization control technologies can be expanded to achieve faster synchronization for a better synchronization effect. The combination of synchronous control and DNA encoding in image encryption can effectively improve the security performance of encryption/decryption, which has further research value. Furthermore, extending the synchronization control of chaotic system to the application field can also be reflected in other fields such as fault detection and secure communication. This will give chaos control practical application value and has the necessity of further development research.


## 8. Conclusions

A family of fourth-order chaotic systems with an infinite equilibrium point is proposed in this study. A chaotic system is selected for theoretical analysis and numerical simulation of the Lyapunov exponents spectrum, Poincarè cross section, and bifurcation diagram; the results support that the chaotic system has more complex dynamic characteristics. A circuit schematic diagram is built to verify the circuit stability and realizability of the proposed system. The parameter self-adaptive synchronization of the chaotic system is designed. The simulation results show that the synchronization control of the two systems is realized and the synchronization effects work well. The color image encryption algorithm is designed by using chaotic system synchronization control combined with DNA coding and operating rules. The histogram, information entropy, correlation coefficient of each channel, and key sensitivity are analyzed. Comparing the reference for image encryption algorithms, it is found that the algorithm has a more secure performance.



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*Source: 1250489-2022-03-30.xml*