# Dynamic Analysis and Robust Control of a Chaotic System with Hidden Attractor

Authors: Huaigu Tian; Zhen Wang; Peijun Zhang; Mingshu Chen; Yang Wang
Journal: Complexity
 (2021)

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## Abstract

In this paper, a 3D jerk chaotic system with hidden attractor was explored, and the dissipativity, equilibrium, and stability of this system were investigated. The attractor types, Lyapunov exponents, and Poincare section of the system under different parameters were analyzed. Additionally, a circuit was carried out, and a good similarity between the circuit experimental results and the theoretical analysis testifies the feasibility and practicality of the original system. Furthermore, a robust feedback controller was designed based on the finite-time stability theory, which guarantees the synchronization of 3D jerk master-slave system in finite time and asymptotically converges to the origin. Finally, we also give verification for the discussion in this paper by numerical simulation.


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## Body

## 1. Introduction

As Dr Passival says, “Chaos science is like a river, made up of many tributaries, it comes from all science” [1]. In 1963, Lorenz first constructed a 3D quadratic polynomial ODEs system, and he found the first chaotic attractor in this system [2]. In 1971, Ruelle and Takens first used the chaotic theory in dynamic systems and explained the nature of turbulence and discovered the existence of “strange attractors” [3]. In the following decades, many chaotic attractors were discovered, such as Logistic mapping [4], Rössler attractor [5], Chen’s attractor [6], and Lü attractor [7]. For many years, chaotic systems and hyperchaotic systems with self-excited attractors have been the focus of investigation [8], and these systems have unstable equilibria on the boundary of their basins of attraction. In 2011, Leonov pointed out that there are hidden attractors in addition to self-excited attractors in the dynamical systems [9]. The attraction basins of hidden attractors cannot intersect the neighborhood of any equilibria, which is a very important feature for chaotic systems with hidden attractors. The Shilnikov condition cannot verify the chaotic characteristics of the chaotic systems with hidden attractors because of the absence of homologous or heterozygous orbits. Sprott gave some simple chaotic systems with no or one equilibrium by computer numerical search [10–13]. Then, researchers proposed chaotic systems with hidden attractors with only one stable equilibrium [14–18], no equilibria [19–21], any number equilibria [22, 23], and a line equilibrium [24, 25]. Nowadays, chaotic systems with hidden attractors play an important role in nonlinear circuits [26], Van der Pol-Duffing oscillators [27], multilevel DC/DC converters [28], and relay systems with hysteresis [29]. In these dynamical systems, the existence of hidden oscillations can lead to the phenomenon of unstable states in life and industrial production. Hence, it is of great significance to understand the local and global dynamic behaviors of chaotic systems with hidden attractors.In general, the appearance of hidden attractors is not hoped when a system operates normally. If it happens, this will lead to sudden and large chaotic oscillations and the system may collapse by the perturbation of hidden attractors. Actually, like two sides of a coin, there have been potential application values in the fields of chaos-based secure communication [30], image encryption [31], robust embedded biometric authentication system based on chaos [32], and so on [33, 34]. It is now well established from a variety of studies that jerk oscillators can display chaos [35]. Kengne studied the finite-time synchronization of the jerk oscillators via theory and experiments [36]. Louodop examined the nonlinearity of Josephson junctions for design chaotic jerk oscillators [37]. Kingni has researched the dynamical behaviors of an autonomous Josephson jerk oscillator with two or no equilibrium points and its application to text encryption [38]. Panahi has studied the collective behavior of chaotic memristive jerk oscillator in dynamical networks [39]. However, in these studies, the dynamical behaviors and related engineering applications of the jerk system are researched, but the control methods of the jerk chaotic systems with hidden attractor are not involved. As regards chaos control, robust methods have always been the focus of research. In order to enhance the robustness of chaotic synchronization, Lenz proposed the robust control method of Lorenz system [40], and Mobayen also presented some robust control technologies [41, 42]. Shen proposed robust synchronization and parameter identification for uncertain chaotic systems with bounded unknown parameters [43]. Ji presented a robust adaptive backstepping synchronization approach for the estimation of uncertain chaotic systems with parameter uncertainties and external disturbances via a fuzzy disturbance observer [44]. Although most of the studies have been given the asymptotical stabilization of the synchronization errors, these studies cannot be synchronized in finite time or guarantee the robustness of the systems. Therefore, it is significant to design a robust finite-time feedback controller with parameter uncertainties for the master-slave synchronization of 3D jerk chaotic system with hidden attractor, which is helpful to solve the problem of sudden chaotic oscillation caused by hidden attractors, thus providing a good reference and inspiration for solving similar engineering oscillation problems. In this paper, we further designed the robust controller and the corresponding chaotic circuit for a 3D jerk system with hidden attractors [45]:(1)x˙1=x2,x˙2=x3,x˙3=−ax1−bx2−cx3+bx1x2+x22,with a,b, and c being real numbers and a≠0. The organizational structure of this paper is as follows: In Section 2, the chaotic dynamical behavior of 3D jerk system (1) is analyzed by numerical simulation, such as equilibrium stability, time series, bifurcations, Poincare mapping, Lyapunov exponents spectrum (LES), largest Lyapunov exponent (LLE), periodic orbit, and chaotic behavior. In Section 3, the finite-time synchronization problem of chaotic systems with uncertain parameters is discussed, and the finite-time robust feedback control of chaotic system is realized by combining the finite-time stability theorem. In Section 4, the 3D jerk system with hidden attractor circuit is implemented, and the reliability of the robust feedback controller is verified by numerical simulation. Finally, Section 5 gives a conclusive remark.


## 2. Dynamical Behaviors

### 2.1. Dissipativity

We can use the dissipativity formula:(2)DivV=∂x˙1t∂x1t+∂x˙2t∂x2t+∂x˙3t∂x3t=−c.SoV˙=−cV; V here is the volume of the moving space in the phase space motion. It can be easily verified that DivV=−c<0, so if t=0, its volume element is V0, the volume shrinks to V0e−c in time t, and each volume containing the system orbit shrinks to zero as t⟶∞ at an exponential rate −c. Thus, system (1) is dissipative under c>0, and its limit converges and is limited to a point in the form of exponential c, which is the attractor of a chaotic system.


### 2.2. Equilibria and Stability

Letx˙1=0,x˙2=0, and x˙3=0, At this time, the 3D jerk system (1) has only one equilibrium O0,0,0. Its Jacobian matrix is(3)J=010001−a−b−c.The characteristic polynomial equation isfλ=λ3+cλ2+bλ+a, according to Routh-Hurwitz criterion: When c>0, a>0, and bc−a>0, there are three negative real parts roots, and the 3D jerk system (1) has a stable node or stable node-foci. When c>0, a>0, and bc−a<0, the 3D jerk system (1) has only one saddle-focus.


### 2.3. Phase Trajectories and Time Series of the System

The phase orbit of the system with different parameters can directly reflect its states and motion behavior. When the parameters of the 3D jerk system (1) are set as a=3.4, b=1, and c=4 and the initial states x1,x2,x3=−2,0,2.4, the 3D jerk system (1) is in chaotic state. The 3D phase space attractor diagram and the projection of attractors of 3D jerk system (1) on three coordinate planes are shown in Figure 1.Figure 1
Chaotic attractor of the 3D jerk system (1) and projection in x1−x2 plane, x2−x3 plane, and x3−x1 plane. (a) x1−x2−x3. (b) x1−x2. (c) x2−x3. (d) x3−x1.
(a)(b)(c)(d)As shown in Figure1, attractors cannot be found by means of equilibria. Therefore, attractors of 3D jerk system (1) are hidden attractors. The time series diagram of x1, x2, and x3 can be obtained by integrating the 3D jerk system (1), as shown in Figure 2.Figure 2
The time series of 3D jerk system (1) in x1, x2, and x3 plane.
(a)(b)(c)Figure2 depicts the aperiodic characteristics of the system, and the oscillation curve is different from completely random noise; and each variable presents aperiodic changes in a continuous bounded range.


### 2.4. The Effect of Parameters on the System

The nonlinear behavior of the system is mainly influenced by the system parameters. In order to better study the complex dynamic behavior of the 3D jerk system (1), the numerical method is used to analyze the dynamic behavior of the system under different parameters, as shown in Table 1.Table 1
Various phase portraits of system (1) for some fixed parameters (PVs parameter values, PPs phase portraits, and Ps Poincare section).
PVsDynamicsLESPPsPsa=3.31,b=1,c=4Periodic 1[0, −0.073, −3.927]Figure3(a)Figure4(a)a=3.34,b=1,c=4Periodic 2[0, −0.056, −3.943]Figure3(b)Figure4(b)a=3.36,b=1,c=4Periodic 4[0, −0.038, −3.964]Figure3(c)Figure4(c)a=3.40,b=1,c=4Chaos[0.072, 0, −4.069]Figure1Figure4(d)Figure 3
Periodic 1, 2, or 4 orbit of 3D jerk system (1) and projection in x1−x2 plane, x2−x3 plane, and x3−x1 plane. (a) a = 3 : 31, b = 1, c = 4. (b) a = 3 : 34, b = 1, c = 4. (c) a = 3 : 36, b = 1, c = 4.
(a)(b)(c)Figure 4
Periodic 1, 2, and 4 and chaotic Poincare section of 3D jerk system (1) with different a. (a) a = 3 : 31, b = 1, c = 4. (b) a = 3 : 34, b = 1, c = 4. (c) a = 3 : 36, b = 1, c = 4. (d) a = 3 : 40, b = 1, c = 4.
(a)(b)(c)(d)


#### 2.4.1. Phase Space Orbit

The parameters are fixed asb=1 and c=4. When the parameter a is selected as 3.31, 3.34, and 3.36, respectively, the phase diagrams of attractors with periodic 1, periodic 2, and periodic 4 can be obtained, as shown in Figure 3.


#### 2.4.2. Poincare Section

A plane is cut out in the multidimensional phase space, and a pair of conjugate variablesx1 and x2 are fixed values in this section, which is the Poincare section. Like Section 2.4.1, the parameters are fixed as b=1 and c=4. When the parameter a is selected as 3.31, 3.34, 3.36, and 3.40, respectively, the Poincare section diagram in the state of periodic 1, periodic 2, and periodic 4 periods and chaos can be obtained, as shown in Figure 4.From Figure4, when the parameter a is selected as 3.31, 3.34, and 3.36, the Poincare section has only one moving point or a few discrete points, indicating that the behavior of the system is periodic, but when the parameters are selected as a=3.40, the Poincare section is a continuously dense point, and it has the characteristic of fractal structure, which indicates that chaotic behavior will occur.


#### 2.4.3. Lyapunov Exponent

Initial value sensitivity of chaotic systems means that when given a very close initial value, the phase space will diverge with an exponential rate. Lyapunov exponent is to identify whether the system is chaotic or not, according to the characteristics of whether the phase orbit has diffusion [46]. In addition, according to Lyapunov dimension,(4)Dλ=j+1λj+1∑i=1jλi.Through Table1, when λ1=0.072, λ2=0, and λ3=−4.069, substituting them into equation (4), the following equation can be obtained:(5)Dλ=j+1λj+1∑i=1jλi=2+λ1+λ2λ3=2.018.From equation (5), the Lyapunov dimension Dλ=2.018 for a=3.4,b=1, and c=4, so the 3D jerk system (1) has fractal Lyapunov dimension, which verifies that system (1) is in a chaotic state when a=3.4. In order to determine whether system (1) is chaotic or not, we should calculate the Lyapunov exponents spectrum of system (1) for fixed parameters b,c and let a vary. When a varies in the interval 3.2,3.4, the Lyapunov exponents spectrum of the 3D jerk system (1) is shown in Figure 5, and the largest Lyapunov exponent of the 3D jerk system (1) is shown in Figure 6.Figure 5
Lyapunov exponents spectrum of 3D jerk system (1) for b=1 and c=4.Figure 6
Largest Lyapunov exponent of 3D jerk system (1) for b=1 and c=4.According to Figure6, when a∈3.36,3.4, the largest Lyapunov exponent λ1>0 can be observed. This observation is also verified by Table 1.


#### 2.4.4. Bifurcation

The bifurcation diagram of a system can also be used to analyze the state of a system within the parameter range. It can describe the bifurcation state that the state variable changes with the parameter change. Fixed parametersb=1 and c=4. When parameter a∈3.3,3.4, the bifurcation diagram of the 3D jerk system (1) is shown in Figure 7.Figure 7
Bifurcation diagram of 3D jerk system (1) with parameters b=1 and c=4.Through the bifurcation state in Figure7, when a∈3.31,3.36, the 3D jerk system (1) exhibits periodic behaviors, and it presents typical states of periodic 1, periodic 2, and periodic 22; however, when a=3.40, chaotic behavior occurs. Therefore, the 3D jerk system (1) evolves from periodic to chaotic. Then, combined with Lyapunov exponential spectrum in Figure 5, it is clear that the state of system (1) changes when the parameters change, which implies that chaotic behavior can only occur within a certain parameter range.


## 2.1. Dissipativity

We can use the dissipativity formula:(2)DivV=∂x˙1t∂x1t+∂x˙2t∂x2t+∂x˙3t∂x3t=−c.SoV˙=−cV; V here is the volume of the moving space in the phase space motion. It can be easily verified that DivV=−c<0, so if t=0, its volume element is V0, the volume shrinks to V0e−c in time t, and each volume containing the system orbit shrinks to zero as t⟶∞ at an exponential rate −c. Thus, system (1) is dissipative under c>0, and its limit converges and is limited to a point in the form of exponential c, which is the attractor of a chaotic system.


## 2.2. Equilibria and Stability

Letx˙1=0,x˙2=0, and x˙3=0, At this time, the 3D jerk system (1) has only one equilibrium O0,0,0. Its Jacobian matrix is(3)J=010001−a−b−c.The characteristic polynomial equation isfλ=λ3+cλ2+bλ+a, according to Routh-Hurwitz criterion: When c>0, a>0, and bc−a>0, there are three negative real parts roots, and the 3D jerk system (1) has a stable node or stable node-foci. When c>0, a>0, and bc−a<0, the 3D jerk system (1) has only one saddle-focus.


## 2.3. Phase Trajectories and Time Series of the System

The phase orbit of the system with different parameters can directly reflect its states and motion behavior. When the parameters of the 3D jerk system (1) are set as a=3.4, b=1, and c=4 and the initial states x1,x2,x3=−2,0,2.4, the 3D jerk system (1) is in chaotic state. The 3D phase space attractor diagram and the projection of attractors of 3D jerk system (1) on three coordinate planes are shown in Figure 1.Figure 1
Chaotic attractor of the 3D jerk system (1) and projection in x1−x2 plane, x2−x3 plane, and x3−x1 plane. (a) x1−x2−x3. (b) x1−x2. (c) x2−x3. (d) x3−x1.
(a)(b)(c)(d)As shown in Figure1, attractors cannot be found by means of equilibria. Therefore, attractors of 3D jerk system (1) are hidden attractors. The time series diagram of x1, x2, and x3 can be obtained by integrating the 3D jerk system (1), as shown in Figure 2.Figure 2
The time series of 3D jerk system (1) in x1, x2, and x3 plane.
(a)(b)(c)Figure2 depicts the aperiodic characteristics of the system, and the oscillation curve is different from completely random noise; and each variable presents aperiodic changes in a continuous bounded range.


## 2.4. The Effect of Parameters on the System

The nonlinear behavior of the system is mainly influenced by the system parameters. In order to better study the complex dynamic behavior of the 3D jerk system (1), the numerical method is used to analyze the dynamic behavior of the system under different parameters, as shown in Table 1.Table 1
Various phase portraits of system (1) for some fixed parameters (PVs parameter values, PPs phase portraits, and Ps Poincare section).
PVsDynamicsLESPPsPsa=3.31,b=1,c=4Periodic 1[0, −0.073, −3.927]Figure3(a)Figure4(a)a=3.34,b=1,c=4Periodic 2[0, −0.056, −3.943]Figure3(b)Figure4(b)a=3.36,b=1,c=4Periodic 4[0, −0.038, −3.964]Figure3(c)Figure4(c)a=3.40,b=1,c=4Chaos[0.072, 0, −4.069]Figure1Figure4(d)Figure 3
Periodic 1, 2, or 4 orbit of 3D jerk system (1) and projection in x1−x2 plane, x2−x3 plane, and x3−x1 plane. (a) a = 3 : 31, b = 1, c = 4. (b) a = 3 : 34, b = 1, c = 4. (c) a = 3 : 36, b = 1, c = 4.
(a)(b)(c)Figure 4
Periodic 1, 2, and 4 and chaotic Poincare section of 3D jerk system (1) with different a. (a) a = 3 : 31, b = 1, c = 4. (b) a = 3 : 34, b = 1, c = 4. (c) a = 3 : 36, b = 1, c = 4. (d) a = 3 : 40, b = 1, c = 4.
(a)(b)(c)(d)


### 2.4.1. Phase Space Orbit

The parameters are fixed asb=1 and c=4. When the parameter a is selected as 3.31, 3.34, and 3.36, respectively, the phase diagrams of attractors with periodic 1, periodic 2, and periodic 4 can be obtained, as shown in Figure 3.


### 2.4.2. Poincare Section

A plane is cut out in the multidimensional phase space, and a pair of conjugate variablesx1 and x2 are fixed values in this section, which is the Poincare section. Like Section 2.4.1, the parameters are fixed as b=1 and c=4. When the parameter a is selected as 3.31, 3.34, 3.36, and 3.40, respectively, the Poincare section diagram in the state of periodic 1, periodic 2, and periodic 4 periods and chaos can be obtained, as shown in Figure 4.From Figure4, when the parameter a is selected as 3.31, 3.34, and 3.36, the Poincare section has only one moving point or a few discrete points, indicating that the behavior of the system is periodic, but when the parameters are selected as a=3.40, the Poincare section is a continuously dense point, and it has the characteristic of fractal structure, which indicates that chaotic behavior will occur.


### 2.4.3. Lyapunov Exponent

Initial value sensitivity of chaotic systems means that when given a very close initial value, the phase space will diverge with an exponential rate. Lyapunov exponent is to identify whether the system is chaotic or not, according to the characteristics of whether the phase orbit has diffusion [46]. In addition, according to Lyapunov dimension,(4)Dλ=j+1λj+1∑i=1jλi.Through Table1, when λ1=0.072, λ2=0, and λ3=−4.069, substituting them into equation (4), the following equation can be obtained:(5)Dλ=j+1λj+1∑i=1jλi=2+λ1+λ2λ3=2.018.From equation (5), the Lyapunov dimension Dλ=2.018 for a=3.4,b=1, and c=4, so the 3D jerk system (1) has fractal Lyapunov dimension, which verifies that system (1) is in a chaotic state when a=3.4. In order to determine whether system (1) is chaotic or not, we should calculate the Lyapunov exponents spectrum of system (1) for fixed parameters b,c and let a vary. When a varies in the interval 3.2,3.4, the Lyapunov exponents spectrum of the 3D jerk system (1) is shown in Figure 5, and the largest Lyapunov exponent of the 3D jerk system (1) is shown in Figure 6.Figure 5
Lyapunov exponents spectrum of 3D jerk system (1) for b=1 and c=4.Figure 6
Largest Lyapunov exponent of 3D jerk system (1) for b=1 and c=4.According to Figure6, when a∈3.36,3.4, the largest Lyapunov exponent λ1>0 can be observed. This observation is also verified by Table 1.


### 2.4.4. Bifurcation

The bifurcation diagram of a system can also be used to analyze the state of a system within the parameter range. It can describe the bifurcation state that the state variable changes with the parameter change. Fixed parametersb=1 and c=4. When parameter a∈3.3,3.4, the bifurcation diagram of the 3D jerk system (1) is shown in Figure 7.Figure 7
Bifurcation diagram of 3D jerk system (1) with parameters b=1 and c=4.Through the bifurcation state in Figure7, when a∈3.31,3.36, the 3D jerk system (1) exhibits periodic behaviors, and it presents typical states of periodic 1, periodic 2, and periodic 22; however, when a=3.40, chaotic behavior occurs. Therefore, the 3D jerk system (1) evolves from periodic to chaotic. Then, combined with Lyapunov exponential spectrum in Figure 5, it is clear that the state of system (1) changes when the parameters change, which implies that chaotic behavior can only occur within a certain parameter range.


## 2.4.1. Phase Space Orbit

The parameters are fixed asb=1 and c=4. When the parameter a is selected as 3.31, 3.34, and 3.36, respectively, the phase diagrams of attractors with periodic 1, periodic 2, and periodic 4 can be obtained, as shown in Figure 3.


## 2.4.2. Poincare Section

A plane is cut out in the multidimensional phase space, and a pair of conjugate variablesx1 and x2 are fixed values in this section, which is the Poincare section. Like Section 2.4.1, the parameters are fixed as b=1 and c=4. When the parameter a is selected as 3.31, 3.34, 3.36, and 3.40, respectively, the Poincare section diagram in the state of periodic 1, periodic 2, and periodic 4 periods and chaos can be obtained, as shown in Figure 4.From Figure4, when the parameter a is selected as 3.31, 3.34, and 3.36, the Poincare section has only one moving point or a few discrete points, indicating that the behavior of the system is periodic, but when the parameters are selected as a=3.40, the Poincare section is a continuously dense point, and it has the characteristic of fractal structure, which indicates that chaotic behavior will occur.


## 2.4.3. Lyapunov Exponent

Initial value sensitivity of chaotic systems means that when given a very close initial value, the phase space will diverge with an exponential rate. Lyapunov exponent is to identify whether the system is chaotic or not, according to the characteristics of whether the phase orbit has diffusion [46]. In addition, according to Lyapunov dimension,(4)Dλ=j+1λj+1∑i=1jλi.Through Table1, when λ1=0.072, λ2=0, and λ3=−4.069, substituting them into equation (4), the following equation can be obtained:(5)Dλ=j+1λj+1∑i=1jλi=2+λ1+λ2λ3=2.018.From equation (5), the Lyapunov dimension Dλ=2.018 for a=3.4,b=1, and c=4, so the 3D jerk system (1) has fractal Lyapunov dimension, which verifies that system (1) is in a chaotic state when a=3.4. In order to determine whether system (1) is chaotic or not, we should calculate the Lyapunov exponents spectrum of system (1) for fixed parameters b,c and let a vary. When a varies in the interval 3.2,3.4, the Lyapunov exponents spectrum of the 3D jerk system (1) is shown in Figure 5, and the largest Lyapunov exponent of the 3D jerk system (1) is shown in Figure 6.Figure 5
Lyapunov exponents spectrum of 3D jerk system (1) for b=1 and c=4.Figure 6
Largest Lyapunov exponent of 3D jerk system (1) for b=1 and c=4.According to Figure6, when a∈3.36,3.4, the largest Lyapunov exponent λ1>0 can be observed. This observation is also verified by Table 1.


## 2.4.4. Bifurcation

The bifurcation diagram of a system can also be used to analyze the state of a system within the parameter range. It can describe the bifurcation state that the state variable changes with the parameter change. Fixed parametersb=1 and c=4. When parameter a∈3.3,3.4, the bifurcation diagram of the 3D jerk system (1) is shown in Figure 7.Figure 7
Bifurcation diagram of 3D jerk system (1) with parameters b=1 and c=4.Through the bifurcation state in Figure7, when a∈3.31,3.36, the 3D jerk system (1) exhibits periodic behaviors, and it presents typical states of periodic 1, periodic 2, and periodic 22; however, when a=3.40, chaotic behavior occurs. Therefore, the 3D jerk system (1) evolves from periodic to chaotic. Then, combined with Lyapunov exponential spectrum in Figure 5, it is clear that the state of system (1) changes when the parameters change, which implies that chaotic behavior can only occur within a certain parameter range.


## 3. Design of Robust Feedback Controller

### 3.1. Provision of Robust Controller

Following the 3D jerk system (1), a class of master system with hidden attractors and uncertain parameters is given as(6)x˙1=x2,x˙2=x3,x˙3=−a+Δ1x1−b+Δ2x2−c+Δ3x3+b+Δ2x1x2,where x=x1,x2,x3T is the state variable of the master system, and the slave system can be described as follows:(7)y˙1=y2+u1t,y˙2=y3+u2t,y˙3=−a+Δ4y1−b+Δ5y2−c+Δ6y3+y22+b+Δ5y1y2+u3t,where y=y1,y2,y3T is the state variable of the slave system; uiti=1,2,3 is the external input control; Δii=1,2,3,4 is the parameter uncertain term of master and slave chaotic systems. The synchronization error is defined as ei=yi−xii=1,2,3. Therefore, its error system can be expressed as(8)e˙1=e2+u1t,e˙2=e3+u2t,e˙3=−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+v1x1+Δ2x2+Δ3x3+u3t.Before designing a finite-time robust feedback controller for the synchronization error system (1), the following Lemma 3.1, which plays an important role in the subsequent analysis, is recalled here for convenience.Lemma 1 (see [47]).
Provided thatVt is a differentiable and nonnegative scalar function and Vt satisfies the differential inequality V˙t≤−εVαt, where ε and α are positive constants, ε>1 and 0<α<1, and then we have(9)Vt=0ift≥tM,where the finite time tM≤V1−α0/ε1−α. V0 is the initial value and V0 is bounded.Based on the above Lemma 3.1, to drive the synchronization error in error system (1) to zero in the finite time, a robust feedback controller is proposed as follows:(10)u1t=−e1γ1+ae3−e2,u2t=−e1γ2+be3−e3,u3t=−e1γ3+ce3−y1y2b−δ5y1y2signe3+x1x2b+δ2x1x2signe3−y22+x22+δ4y1signe3+δ5y2signe3+δ6y3signe3−δ1x1signe3−δ2x2signe3−δ3x3signe3where 0<γi<1i=1,2,3 and remark γ=min1≤i≤3γi.


### 3.2. Performance Proof of Robust Feedback Controller

To verify the finite-time stability of the error system with the proposed controller (10), the following proof is given.Assumption 1.
Assumes that positiveδii=1,2,3… makes the parameter uncertainties as Δi≤δii=1,2,3,….Theorem 1.
Under the action of robust feedback controller (10), the master system (6) and slave system (7) can achieve synchronization in finite time T=2σ−1Vσ0/σ, where σ=1−γ/2.Proof.
Construct the Lyapunov functionVt=1/2∑i=13ei2. Differentiating Vt gives V˙t=3∑i=13ei·e˙i. Substituting (8) into V˙t, we can get:(11)V˙t=e1e˙1+e2e˙2+e3e˙3=e1e2+u1t+e2e3+u2t+e3−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+Δ1x1+Δ2x2+Δ3x3+u3t.
Then, substituting the controller in (10) into equation (11), we have(12)V˙t=e1e2+ae3−e2−e1γ1+e2e3+be3−e3−e1γ2+e3−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+Δ1x1+Δ2x2+Δ3x3+ce3−y1y2b−δ5y1y2signe3+x1x2b+δ2x1x2signe3−y22+x22+δ4y1signe3+δ5y2signe3+δ6y3signe3−δ1x1signe3−δ2x2signe3−δ3x3signe3−e3γ3.
Furthermore, simplifying equation (12) can be obtained as follows:(13)V˙t=−e1γ1+1−e2γ2+1−e3γ3+1+e3y1y2Δ5−δ5e3y1y2−e3x1x2Δ2+δ2e3x1x2−e3y1Δ4+δ4y1e3−e3y2Δ5+δ5y2e3−e3y3Δ6+δ6y3e3+e3x1Δ1−δ1x1e3+e3x2Δ2−δ2x2e3+e3x3Δ3−δ3x3e3,where, based on Assumption 1, we can get(14)V˙t≤−e1γ1+1−e2γ2+1−e3γ3+1,and then we can know that(15)−e1γ1+1−e2γ2+1−e3γ3+1=−2γ+1/2e122γ+1/2+e222γ+1/2+e322γ+1/2≤−2γ+1/2e122+e222+e322γ+1/2=−2γ+1/2Vγ+1/2t.
Thus, we have(16)V˙t≤−2γ+1/2Vγ+1/2t.
According to Lemma1, we have(17)T=V1−γ+1/202γ+1/21−γ+1/2=21−γ/2V1−γ/201−γ.
Letσ=1−γ/2; there are T=2σ−1Vσ0/σ; when t≥T, ei⟶0i=1,2,3.


## 3.1. Provision of Robust Controller

Following the 3D jerk system (1), a class of master system with hidden attractors and uncertain parameters is given as(6)x˙1=x2,x˙2=x3,x˙3=−a+Δ1x1−b+Δ2x2−c+Δ3x3+b+Δ2x1x2,where x=x1,x2,x3T is the state variable of the master system, and the slave system can be described as follows:(7)y˙1=y2+u1t,y˙2=y3+u2t,y˙3=−a+Δ4y1−b+Δ5y2−c+Δ6y3+y22+b+Δ5y1y2+u3t,where y=y1,y2,y3T is the state variable of the slave system; uiti=1,2,3 is the external input control; Δii=1,2,3,4 is the parameter uncertain term of master and slave chaotic systems. The synchronization error is defined as ei=yi−xii=1,2,3. Therefore, its error system can be expressed as(8)e˙1=e2+u1t,e˙2=e3+u2t,e˙3=−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+v1x1+Δ2x2+Δ3x3+u3t.Before designing a finite-time robust feedback controller for the synchronization error system (1), the following Lemma 3.1, which plays an important role in the subsequent analysis, is recalled here for convenience.Lemma 1 (see [47]).
Provided thatVt is a differentiable and nonnegative scalar function and Vt satisfies the differential inequality V˙t≤−εVαt, where ε and α are positive constants, ε>1 and 0<α<1, and then we have(9)Vt=0ift≥tM,where the finite time tM≤V1−α0/ε1−α. V0 is the initial value and V0 is bounded.Based on the above Lemma 3.1, to drive the synchronization error in error system (1) to zero in the finite time, a robust feedback controller is proposed as follows:(10)u1t=−e1γ1+ae3−e2,u2t=−e1γ2+be3−e3,u3t=−e1γ3+ce3−y1y2b−δ5y1y2signe3+x1x2b+δ2x1x2signe3−y22+x22+δ4y1signe3+δ5y2signe3+δ6y3signe3−δ1x1signe3−δ2x2signe3−δ3x3signe3where 0<γi<1i=1,2,3 and remark γ=min1≤i≤3γi.


## 3.2. Performance Proof of Robust Feedback Controller

To verify the finite-time stability of the error system with the proposed controller (10), the following proof is given.Assumption 1.
Assumes that positiveδii=1,2,3… makes the parameter uncertainties as Δi≤δii=1,2,3,….Theorem 1.
Under the action of robust feedback controller (10), the master system (6) and slave system (7) can achieve synchronization in finite time T=2σ−1Vσ0/σ, where σ=1−γ/2.Proof.
Construct the Lyapunov functionVt=1/2∑i=13ei2. Differentiating Vt gives V˙t=3∑i=13ei·e˙i. Substituting (8) into V˙t, we can get:(11)V˙t=e1e˙1+e2e˙2+e3e˙3=e1e2+u1t+e2e3+u2t+e3−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+Δ1x1+Δ2x2+Δ3x3+u3t.
Then, substituting the controller in (10) into equation (11), we have(12)V˙t=e1e2+ae3−e2−e1γ1+e2e3+be3−e3−e1γ2+e3−ae1−be2−ce3+y1y2b+Δ5−x1x2b+Δ2+y22−x22−Δ4y1−Δ5y2−Δ6y3+Δ1x1+Δ2x2+Δ3x3+ce3−y1y2b−δ5y1y2signe3+x1x2b+δ2x1x2signe3−y22+x22+δ4y1signe3+δ5y2signe3+δ6y3signe3−δ1x1signe3−δ2x2signe3−δ3x3signe3−e3γ3.
Furthermore, simplifying equation (12) can be obtained as follows:(13)V˙t=−e1γ1+1−e2γ2+1−e3γ3+1+e3y1y2Δ5−δ5e3y1y2−e3x1x2Δ2+δ2e3x1x2−e3y1Δ4+δ4y1e3−e3y2Δ5+δ5y2e3−e3y3Δ6+δ6y3e3+e3x1Δ1−δ1x1e3+e3x2Δ2−δ2x2e3+e3x3Δ3−δ3x3e3,where, based on Assumption 1, we can get(14)V˙t≤−e1γ1+1−e2γ2+1−e3γ3+1,and then we can know that(15)−e1γ1+1−e2γ2+1−e3γ3+1=−2γ+1/2e122γ+1/2+e222γ+1/2+e322γ+1/2≤−2γ+1/2e122+e222+e322γ+1/2=−2γ+1/2Vγ+1/2t.
Thus, we have(16)V˙t≤−2γ+1/2Vγ+1/2t.
According to Lemma1, we have(17)T=V1−γ+1/202γ+1/21−γ+1/2=21−γ/2V1−γ/201−γ.
Letσ=1−γ/2; there are T=2σ−1Vσ0/σ; when t≥T, ei⟶0i=1,2,3.


## 4. Circuit Implementation and Numerical Simulation

In this section, in order to verify the correctness of the 3D jerk system (1), the analog circuit is designed, and to illustrate the control effect of the robust feedback controller, the numerical simulation is presented.


### 4.1. Circuit Implementation

We use LM741 operational amplifiers, AD633 analog multipliers, resistors, and capacitors to design analog circuit, where the gain of multiplier AD633 is 0.1, and the power voltage of operational amplifier LM741 isE=±15V, and its output saturated voltages Vsat≈±13.5V. According to the 3D jerk system (1), a chaotic circuit with hidden attractor is designed, as shown in Figure 8.Figure 8
The circuit principle diagram of the 3D jerk system (1).Due to the limitation of LM741 and AD633 working voltages, the output voltage of the system is reduced to 1/10 of the original. Compress them according to 10 : 1, and carry on time-scale transformation; letτ=τ0t, where τ0=100. Then the 3D jerk system (1) can be expressed as(18)x˙1=−100−x2,x˙2=−100−x3,x˙3=−340x1−100x2−400x3−100−x1x2−100−x2x2.Applying Kirchhoff’s law and from Figure8, the corresponding circuit equations are written as(19)dx1dt=−1R1C1−x2,dx2dt=−1R4C2−x3,dx3dt=−1R7C3x1−1R8C3x2−1R10C3x3−110R9C3−x1x2−110R11C3−x2x2.where x1, x2, and x3 are related to the voltages on capacitors C1, C2, and C3, respectively. The capacitors C1=C2=C3=100nF. Comparing equation (18) with equation (19) and making the corresponding coefficients be equal, one can obtain the values of resistances as R1=R4=R7=R9=R11=100kΩ, R8=29.41kΩ, and R10=25kΩ. The other resistances are R2=R3=R5=R6=R12=R13=10kΩ. Under the circuit parameters mentioned above, one can get a phase diagram on the digital oscilloscope as shown in Figure 9.Figure 9
The screenshots of the digital oscilloscope with the 3D jerk system (1) in (a) x1−x2 plane, (b) x2−x3 plane, and (c) x1−x3 plane.
(a)(b)(c)


### 4.2. Numerical Simulation of Robust Feedback Controller

For the master system (6) and slave system (7), the parameters are set as a=3.4, b=1, and c=4, and when Δi=0i=1,2,3,4,5,6, the master-slave systems (6) and (7) are in chaotic state. The parameter uncertainties are taken as Δ1=0.01sinx3, Δ2=0.02cosx3,Δ3=0.01sinx3, Δ4=0.02siny3, Δ5=0.01siny3, and Δ6=0.01siny3. So δ1=0.01,δ2=0.02, δ3=0.01, δ4=0.02, δ5=0.01, and δ6=0.01. The control parameters are chosen as γ1=0.6, γ2=0.8, and γ3=0.9. The system initial values are chosen as x1=−2, x2=0,x3=2.4 and y1=7, y2=−3, y3=7.4. Under the robust feedback controller (10), the synchronization result is shown in Figure 10.Figure 10
3D jerk master-slave system synchronous orbits. (a) Synchronization ofx1, y1. (b) Synchronization of x2, y2. (c) Synchronization of x3, y3.
(a)(b)(c)In Figure10, the states of master system xii=1,2,3 and slave system yii=1,2,3 tend to converge, and synchronization is realized in finite time, and the states e1, e2, and e3 of the error system (8) are shown in Figure 11.Figure 11
Finite-time errore1, e2, and e3 orbit of 3D jerk master-slave system.From the above simulation results that the error system (8) realizes synchronization in finite time, the phase orbit of the synchronization error system gradually converges to the origin, the reliability of the synchronization method is verified, and the design of the robust feedback controller (10) meets the requirements.


## 4.1. Circuit Implementation

We use LM741 operational amplifiers, AD633 analog multipliers, resistors, and capacitors to design analog circuit, where the gain of multiplier AD633 is 0.1, and the power voltage of operational amplifier LM741 isE=±15V, and its output saturated voltages Vsat≈±13.5V. According to the 3D jerk system (1), a chaotic circuit with hidden attractor is designed, as shown in Figure 8.Figure 8
The circuit principle diagram of the 3D jerk system (1).Due to the limitation of LM741 and AD633 working voltages, the output voltage of the system is reduced to 1/10 of the original. Compress them according to 10 : 1, and carry on time-scale transformation; letτ=τ0t, where τ0=100. Then the 3D jerk system (1) can be expressed as(18)x˙1=−100−x2,x˙2=−100−x3,x˙3=−340x1−100x2−400x3−100−x1x2−100−x2x2.Applying Kirchhoff’s law and from Figure8, the corresponding circuit equations are written as(19)dx1dt=−1R1C1−x2,dx2dt=−1R4C2−x3,dx3dt=−1R7C3x1−1R8C3x2−1R10C3x3−110R9C3−x1x2−110R11C3−x2x2.where x1, x2, and x3 are related to the voltages on capacitors C1, C2, and C3, respectively. The capacitors C1=C2=C3=100nF. Comparing equation (18) with equation (19) and making the corresponding coefficients be equal, one can obtain the values of resistances as R1=R4=R7=R9=R11=100kΩ, R8=29.41kΩ, and R10=25kΩ. The other resistances are R2=R3=R5=R6=R12=R13=10kΩ. Under the circuit parameters mentioned above, one can get a phase diagram on the digital oscilloscope as shown in Figure 9.Figure 9
The screenshots of the digital oscilloscope with the 3D jerk system (1) in (a) x1−x2 plane, (b) x2−x3 plane, and (c) x1−x3 plane.
(a)(b)(c)


## 4.2. Numerical Simulation of Robust Feedback Controller

For the master system (6) and slave system (7), the parameters are set as a=3.4, b=1, and c=4, and when Δi=0i=1,2,3,4,5,6, the master-slave systems (6) and (7) are in chaotic state. The parameter uncertainties are taken as Δ1=0.01sinx3, Δ2=0.02cosx3,Δ3=0.01sinx3, Δ4=0.02siny3, Δ5=0.01siny3, and Δ6=0.01siny3. So δ1=0.01,δ2=0.02, δ3=0.01, δ4=0.02, δ5=0.01, and δ6=0.01. The control parameters are chosen as γ1=0.6, γ2=0.8, and γ3=0.9. The system initial values are chosen as x1=−2, x2=0,x3=2.4 and y1=7, y2=−3, y3=7.4. Under the robust feedback controller (10), the synchronization result is shown in Figure 10.Figure 10
3D jerk master-slave system synchronous orbits. (a) Synchronization ofx1, y1. (b) Synchronization of x2, y2. (c) Synchronization of x3, y3.
(a)(b)(c)In Figure10, the states of master system xii=1,2,3 and slave system yii=1,2,3 tend to converge, and synchronization is realized in finite time, and the states e1, e2, and e3 of the error system (8) are shown in Figure 11.Figure 11
Finite-time errore1, e2, and e3 orbit of 3D jerk master-slave system.From the above simulation results that the error system (8) realizes synchronization in finite time, the phase orbit of the synchronization error system gradually converges to the origin, the reliability of the synchronization method is verified, and the design of the robust feedback controller (10) meets the requirements.


## 5. Conclusion

In this work, the dynamical behaviors of the 3D jerk system with hidden attractor are analyzed by numerical calculation. Furthermore, an analog circuit is designed for implementing and testing our system model. A very good qualitative consistency is shown between the circuit experimental results and the simulations of the theoretical model. Then, for 3D jerk system synchronization and control, a finite-time robust feedback controller was proposed, and the synchronization of 3D jerk chaotic system with hidden attractor was realized in finite time. Finally, mathematical simulation result demonstrated that the performances of the proposed controller are excellent. However, there are still complex dynamics and the topological structure of this system should be exploited, and better synchronization control methods should be found. In addition, the synchronization and control of the master-slave system can be realized through circuit implementation. These will be provided in future works.



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