Abstract
In order to further improve the complexity of chaotic system, a new four-dimensional chaotic system is constructed based on Sprott B chaotic system. By analyzing the system’s phase diagrams, symmetry, equilibrium points, and Lyapunov exponents, it is found that the system can generate not only both two-wing and four-wing attractors but also the attractors with symmetrical coexistence, and the dynamic characteristics of the new system constructed are more abundant. In addition, the system is simulated by Multisim software, and the simulation results show that the results of circuit simulation and numerical simulation analysis are basically the same.
1. Introduction
Chaos is a complex, apparently random, and often surprising behavior in simple nonlinear dynamical systems [1]. Chaos, as a unique form of motion in nonlinear dynamic systems, is widely used in electronic engineering [2], information engineering [3], and other fields [4–6] because of its initial value sensitivity, boundedness, and inherent randomness [7]. In 1963, American meteorologist Lorenz put forward the first chaotic system model [8], which attracted wide attention of the scientific community, and then, new chaotic systems were constantly discovered. In 1976, Rössler proposed a new system named Rössler chaotic system [9], which had a different topology from Lorenz system. Chua proposed Chua’s circuit in 1986 [10, 11], which was one of the simplest chaotic oscillation circuits. In 1994, Sprott constructed several different simple chaotic systems [12]. In 1999, Chen and Ueta . discovered the Chen system while studying the anticontrol of chaos [13]. In 2002, Lü et al. proposed a kind of transition system named Lü system which connected Lorenz and Chen systems [14]. In 2003, Liu and Chen constructed the first four-wing butterfly chaotic attractor [15], which attracted many researchers’ attention. To improve the security of chaotic secure communication and chaotic information encryption, more and more researchers began to find chaotic systems with more complex dynamic behaviors [16–21].
In recent years, coexisting attractors had gradually become a research hotspot [22–24]. Compared with general chaotic attractors, the dynamic behaviors of coexisting attractors are more complex. In order to improve the security of information and reduce the possibility of information being decoded, coexisting attractors are more and more used in the field of encryption [25, 26]. In 2013, Li and Sprott proposed a multistable system with coexisting attractors [27] and found that the dynamic of the equilibrium points of the system depended on its stability and system structure. In 2014, Li and Sprott discovered a coexisting hidden attractor on a simple 4D Lorenz system [28], which had a large parameter region on a quasiperiodic torus. In 2017, Lai et al. proposed a unique 4D autonomous system with a signum function term [29], which can generate various types of coexisting attractors. In 2019, Zhou et al. proposed a chaotic system with multiple asymmetric coexisting attractors [30] and carried out circuit simulation and pulse synchronization research.
In this paper, a new 4D chaotic system based on Sprott B system is proposed. It includes the following elements: (i) It contains eight terms, including three nonlinear terms and one constant term. (ii) It is symmetric about the z-axis. (iii) It can produce two-wing and four-wing attractors at the same time. (iv) It can also produce symmetric coexisting attractors. (v) The realization of the system circuit in physics is verified by the circuit simulation software, which is favorable for future engineering applications. This paper is organized as follows: In Section 2, a new chaotic system is proposed, and the coexistence of two-wing and four-wing attractors is observed through phase diagrams. In Section 3, we analyze its dynamic behaviors by symmetry, equilibrium points, bifurcation diagrams, Lyapunov exponents, and trajectory diagrams and introduce its symmetric coexisting attractors. An electronic circuit is designed in Section 4, and the correctness of the theoretical analysis is verified by circuit simulation experiment. Finally, the conclusion of this paper is given in Section 5.
2. A New Four-Dimensional Chaotic System
In this section, we mainly design the new chaotic system, and the new system proposed in this paper is described as follows:where positive real numbers a, b, and c are system parameters and x, y, z, and are state variables. The new system (1) adds a state-feedback controller on the Sprott B chaotic system. Set , , and ; the chaotic system can be generated. By calculation, the Lyapunov exponents are , , , and . The corresponding Lyapunov exponential dimension is as follows:
Therefore, the attractor of the new system is a strange attractor with fractal dimension. Select the initial value . Through numerical simulation, we can get the chaotic attractors of system (1) as shown in Figure 1. As can be seen from Figure 1, system (1) presents two-wing butterfly chaotic attractors in the , , , and phase planes. The four-wing butterfly chaotic attractors appear in the and phase planes. This coexistence can be better observed in Figures 1(g)–1(h). It can be concluded that system (1) can generate chaotic butterfly attractors of two-wing and four-wing at the same time.
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3. Some Basic Properties of New System
3.1. Symmetric and Dissipative Properties
The Sprott B system is symmetric about the z-axis, and system (1) is also symmetric about the z-axis. A simple proof is shown in the following equation:
The state space of system (1) is four-dimensional, so the vector field of system (1) is defined as follows:
The divergence of system (1) is obtained from the vector field as follows:
According to equation (5), as long as , system (1) is dissipative, and the system converges in exponential form . As , all trajectories of the system will eventually be restricted to a set with a volume of zero, and the extreme motion will converge to an attractor, thus proving the existence of the attractors of the system.
3.2. Equilibria and Stability
In order to obtain the equilibrium points of system (1), let the right side of the equation be equal to zero. The system of equations is as follows:
By calculation, we get that the two equilibrium points of system (1) are and . System (1) has the same characteristic equation at :
According to the classical Routh–Hurwitz stability criterion, if , , and , the equilibrium points is unstable.
Let , , and , we can get the equilibrium points and . For the first equilibrium point , system (1) is linearized to Jacobian matrix as follows:
Let , the eigenvalues of Jacobian matrix are as follows:
It can be seen that and are negative real numbers, and are a pair of conjugate complex numbers, and the real part is positive, so the equilibrium point is a saddle-focus, and system (1) is unstable at .
For the second equilibrium point , system (1) is linearized to Jacobian matrix as follows:
In the same way, let , the eigenvalue obtained is shown in equation (9), so the equilibrium point is also a saddle-focus, and system (1) is unstable at . Obviously, two saddle-foci are the key to the chaotic motion of system (1).
3.3. Bifurcation Diagrams, Lyapunov Exponents, and Period-Doubling Bifurcation Process
The dynamic behaviors of system (1) can be further analyzed by bifurcation diagrams, Lyapunov exponents, and period-doubling bifurcation process.
Let , , and . We draw the bifurcation diagrams of the z peak of system (1) changing with b, as shown in Figure 2(a). In Figure 2(a), the red and blue branches represent the different attractors generated from different initial values and , and the overlapped parts represent the same attractors generated. As b increases in , the bifurcation diagrams clearly show the trajectory of system (1) from classical period-doubling bifurcation to chaos. Figures 2(b) and 2(c) show the Lyapunov exponents of system (1) which changes with the increase of parameter b, where . By comparing the three diagrams, it can be seen that the bifurcation diagrams are completely consistent with the dynamic behaviors described by Lyapunov exponents.
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Figures 3(a)–3(d) describe in detail the main orbital states through which system (1) operates. When , system (1) has a pair of period-1 attractors. When , system (1) has a pair of period-2 attractors. When , system (1) has a pair of strange attractors. When , system (1) has a strange attractor.
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Table 1 shows the comparison of the Lyapunov exponents of the new system with the literature [12, 28–30]. It can be seen that the maximum of the new system is larger. It indicates that the chaotic characteristics of the new system are more obvious, the chaotic degree is higher, and the dynamic characteristics of the system are more difficult to predict.
3.4. Coexisting Attractors
Let , , and , and we draw the bifurcation diagrams of the x peak of system (1) changing with parameter c. Similarly, the red and blue branches in Figure 4(a), respectively, represents the different attractors generated from different initial values of and , and the overlaps represent the same attractors generated. Figures 4(b) and 4(c) show the Lyapunov exponents of system (1) changing with the increase of parameter c. It is obvious that . Figure 4 shows that periodic attractors, chaotic attractors, and coexisting attractors exist in system (1).
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Figure 4(a) not only shows that system (1) has coexisting attractors but also shows that with the increase of c in , system (1) shows a trajectory from reverse period-doubling bifurcation to chaos. Figures 5(a)–5(d) describe the major orbital states of ergodic when symmetrically coexisting attractors appear in system (1). When , the system gets a pair of strange attractors. When , the system has a pair of period-2 attractors. When , the system has a pair of period-1 attractors. When , the system also has a pair of strange attractors. Figures 4(b)–4(c) Lyapunov exponents verify the above process and determine the property of the attractors.
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Figure 6 is a dynamical map, mainly depicting the influence of changing parameters b and c on the dynamical map characteristics of the system. Taking the relationship between the maximum Lyapunov exponent and 0 as the standard, the blue region represents the chaotic state of the system, and the maximum Lyapunov exponent is greater than 0; the yellow region represents the periodic state, and the maximum Lyapunov exponent is equal to 0; and the red region represents the system is stable under this parameter condition, and the maximum Lyapunov exponent is less than 0. It can be seen from the figure that as the parameter c increases, the system state alternates between chaotic state and periodic state and occasionally tends to be stable.
4. Circuit Implementation
In order to verify the dynamic behaviors of chaotic system, an actual circuit is designed to realize the chaotic system according to equation (1). The circuit is mainly realized by linear resistances of different resistance values, linear capacitances, operational amplifier TL082IP, and multiplier AD633. However, it should be noted that in the actual circuit, the allowable voltage range of the analog multiplier is and the allowable voltage range of the operational amplifier is . The dynamic range of variables x, y, z, and are approximately , , , and , respectively. It is beyond the allowable voltage range of analog multipliers and operational amplifiers, so it is necessary to make appropriate variable proportional compression transformation to the system state variables, so as to facilitate the implementation of the circuit. System (1) is transformed by proportional compression of uniform variables, so that x, y, z, and are compressed to the original , which is . The chaotic system equation after transformation is as follows:
In this way, the amplitude of the output chaotic signal can be reduced to 1/5 of the original system.
Make time-scale transformation of equation (11), and transform t into in the equation, where , and the results are as follows:
The modular circuit is designed according to the above formula, as shown in Figure 7.
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According to the circuit schematic diagrams, the corresponding self-excited oscillation circuit equation is obtained as follows:
By comparing equation (12) with equation (13), we can get
In this paper, Multisim software is used for circuit simulation, in which the output scaling factor of the analog multiplier AD633 is set as and the power supply voltage is . The integral time constant of the four circuit channels can be changed by adjusting the capacitance; set , , and . When , , and , it can be obtained from equation (14) that , , , , and . Circuit simulation results are shown in Figure 8. These trajectories are consistent with numerical simulation results.
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Similarly, we can use circuit simulations to observe the orbital states of system (1) as it moves towards chaos. Keep other parameters unchanged, and control the value of parameter b by changing . Figures 9(a)–9(d) describe the trajectories starting from the initial value , where corresponds to parameter , corresponds to parameter , corresponds to parameter , and corresponds to . By comparing with Figure 3, it can be seen that the results of circuit simulation and numerical simulation are consistent.
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The following circuit simulation is used to verify the symmetric coexisting attractors in system (1), mainly verifying the first three orbital states. Because the parameters have changed, the value of the corresponding resistances in the circuit should also be changed. Equation (14) can be used to calculate the corresponding resistances and when and . The parameter c is controlled by the resistance . , , and . The left graph of Figure 10 shows the attractors starting from initial value , and the right graph of Figure 10 shows the attractors starting from initial value . By comparing the circuit simulation diagrams with the numerical simulation diagrams in the previous section, we can see that the experimental results of the two are in good agreement.
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5. Conclusion
In this paper, a new four-dimensional chaotic system is designed by adding a state-feedback controller to the Sprott B system. It contains three nonlinear terms and one constant term and is symmetric about z-axis. Through the analysis of system diagrams, bifurcation diagrams, and Lyapunov exponents, it is found that the new system can generate not only two-wing and four-wing attractors but also symmetrical coexisting attractors, and the complexity of the system is further improved. We also design the electronic circuit, and the results of the circuit simulation experiment are consistent with those of the numerical simulation experiment, which proves the correctness of the theoretical analysis and the realizability of the system. The dynamic characteristics of the new system are more abundant, and it has great prospects in the fields of image encryption and secure communication.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (Nos. 61203004 and 61306142) and Natural Science Foundation of Heilongjiang Province (Grant no. F201220).