Fractional-Order and Memristive Nonlinear Systems: Advances and ApplicationsView this Special Issue
Dynamics, Circuit Design, and Synchronization of a New Chaotic System with Closed Curve Equilibrium
After the report of chaotic flows with line equilibrium, there has been much attention to systems with uncountable equilibria in the past five years. This work proposes a new system with an infinite number of equilibrium points located on a closed curve. It is worth noting that the new system generates chaotic behavior as well as hidden attractors. Dynamics of the system with closed curve equilibrium have been investigated by using phase portraits, bifurcation diagram, maximal Lyapunov exponents, and Kaplan–York dimension. In addition, we introduce an electronic implementation of the theoretical system to verify its feasibility. Antisynchronization ability of the new system with infinite equilibria is studied by applying an adaptive control. This study suggests that there exist other chaotic systems with uncountable equilibria in need of further investigation.
Although chaos has been an object of research since the 1960s, investigating chaotic systems is still a continuing concern within nonlinear dynamics field [1–7]. A great amount of different chaotic systems has been reported in the literature such as modified Lorenz chaotic system , jerk–based chaotic oscillators , three-dimensional (3D) system , 3D autonomous chaotic system with a single cubic nonlinearity , chaotic system with butterfly attractor , systems with multiscroll chaotic attractors [13–15], or fractional-order chaotic systems [16, 17]. Moreover, the past decade has seen the rapid development of chaos-based applications in many areas from hardware pseudorandom number generators , autonomous mobile robots , MOS oscillators , encryption , chaotic masking communication , to information processing .
Recently, there has been renewed interest in chaotic systems with uncountable equilibria  (see the illustration in Figure 1). One of the most important discoveries is the finding of chaotic flows with line equilibrium [24–26]. The presence of parallel lines of equilibrium points  and perpendicular lines of equilibria  has also been studied. Recent trends in systems with uncountable equilibria have led to systems with open curve equilibrium [27, 28] and closed curve equilibrium . The issue of finding different shapes of closed curve equilibrium especially has received considerable critical attention [29–31]. Chaotic system with circular equilibrium has been obtained by using a search routine . Gotthans and Petržela have constructed a chaotic flow with square equilibrium . Chaotic system with uncountable equilibria located on a rounded square has been reported in . From the computation point of view, it is now well established that systems with infinite equilibrium can be classified as special systems with “hidden attractor” [33–36], which have not treated much detail [37–39]. Discovering new chaotic systems with closed curve equilibrium is still an attractive research direction.
The main aim of this study is to investigate a novel chaotic system with closed-curve equilibrium. In Section 2, the model and the dynamics of the system are presented. Circuital implementation of the new system is reported in Section 3 while the ability of antisynchronization of such systems is discussed in Section 4. Finally, Section 5 gives conclusion remarks of our study.
2. Description and Dynamics of the System with Closed Curve Equilibrium
Recently, Gotthans et al.  have introduced chaotic systems with circle equilibrium and square equilibrium. Gotthans et al. system with circle equilibrium is given bywhile the system with square equilibrium is described asin which and are constants. This investigation is significantly important because authors indicated the existence of chaotic systems with different shapes of equilibrium points . Moreover, by generalizing the systems of Gotthans et al. we may get other new systems with an infinite number of equilibrium.
Based on the systems of Gotthans et al., we concentrate on a general model given byin which three state variables are , , and . The functions , are two arbitrary nonlinear functions. By solving , , and , it is straightforward to obtain the equilibrium point of general model (3). It means that we calculateAs can be seen from (4), we have . By substituting into (5) and (6), it is easy to confirm that the equilibrium points of general model (3) are laid on the curve described by (6) in the plane . Therefore numerous systems with closed curve equilibrium can be constructed by selecting appropriate functions and .
In this work, we select two nonlinear functions and described by the following forms:in which and are two positive parameters. Substituting (7) and (8) into the general model (3), our new system is derived asCombining (6) and (8), it is simple to verify that system (9) has an infinite number of equilibrium points . Remarkably, such equilibrium points are described byIn other words, the proposed system (9) has cloud-shaped curve equilibrium as illustrated in Figure 2. It is interesting to note that the cloud-shaped curve is different from basic shapes like circle, square, or ellipse [29–31]. Furthermore, system (9) belongs to a rare class of systems with “hidden attractors” . Therefore the investigation of system (9) will enhance our understanding of systems with “hidden attractors” which is an increasingly important area in practical engineering [40–42].
It is worth noting that system (9) with infinite equilibria displays chaotic attractors (see Figure 3) for , and the initial conditions . Chaotic behavior of the system is confirmed by the Lyapunov exponents , , and . The well-known Wolf et al.’s method has been used to calculate the Lyapunov exponents  and the time of computation is 10,000. In this case, the corresponding Kaplan–York dimension of system (9) is . It is noted that unexpected jumps in the values of the local Lyapunov exponents and Lyapunov dimension may occur. Thus the infimum over time interval often gives better estimates. In addition, it is difficult to get the same values of the finite-time local Lyapunov exponents and Lyapunov dimension for different points. Therefore the maximum of the finite-time local Lyapunov dimensions on the grid of point has to be considered [44–46].
By changing the value of the parameter , the dynamical properties of the system with infinite equilibria (9) are able to be discovered. Figures 4 and 5 display the bifurcation diagram and the diagram of maximal Lyapunov exponents of system (9), respectively. As can be seen from Figures 4 and 5, there is the presence of the classical period doubling route to chaos when decreasing the value of the bifurcation parameter . The system with cloud-shaped equilibrium generate periodical states in the range of . For instance, system (9) can display the period–1 state for (see Figure 6(a)) and the period–2 state for (see Figure 6(b)). For , chaotic states are able to be observed in system (9).
3. Circuit Design of the New System with Closed Curve Equilibrium
In order to illustrate the feasibility of the theoretical system with a closed curve of equilibrium points (9), we present its circuital implementation in this section. For the sake of simplicity, we applied the design approach based on operational amplifiers [47–49]. It is noting that this design approach is not complex and requires common electronic elements only [13, 50]. Our designed circuit is shown in Figure 7, in which there are three integrators () with corresponding output voltages (). The absolute nonlinearity is realized by the circuitry including two operational amplifiers () and two diodes ().
It is simple to derive the circuital equation of the circuit in Figure 7:The dimensionless system (12) is obtained by normalizing the circuital equation with , that is,As can be seen in (12), the state variables () are equivalent to the voltages of capacitors (). The dimensionless system (12) corresponds to the proposed system (9) with and ).
We select the values of electronic components to realize the theoretical systems (9) for and as follows: kΩ, kΩ, kΩ, kΩ, kΩ, kΩ, kΩ, , and nF. The PSpice projections of chaotic attractors with infinite equilibria are presented in Figure 8. From Figure 8 we can see that the designed circuit displays chaotic signals. The PSpice results also indicate that the circuit can emulate the theoretical model (9). It is necessary to remark that results of circuit simulation depend on the discretization step. As a result, we should consider the discretization step seriously when simulating electronic circuits, especially in the case of hidden oscillations [51, 52]. Moreover, it is worth noting that realizing the circuit with real analog devices is better than realizing the circuit in PSpice. Therefore, realizing the circuit with real analog devices will be our next work.
4. Antisynchronization of the Identical Systems with Infinite Equilibria
After the investigation of Pecora and Carrol related to synchronization in chaotic systems , various synchronization techniques were studied extensively, for example, global chaos synchronization , hybrid synchronization , ragged synchronizability , and so on . Remarkably, the possibility of synchronization of two identical chaotic systems has received considerable attention due to the vital role in practical applications [57, 58]. In this section, we discover the antisynchronization of two new systems with cloud-shaped equilibrium, named the master system and the slave system.
We consider the master system with closed curve equilibriumin which three state variables are , , and while and are two unknown parameters.
The slave system is described byin which the adaptive control is .
To indicate the difference between the slave system (14) and the master system (13), the state errors are defined asSimilarly, we calculate the parameter estimation errorsin which , are the estimations of two unknown parameters , , respectively. It is trivial to get the dynamics of the parameter estimation errors:
The aim of our work is to get the antisynchronization between the slave system and the master system; therefore, the adaptive control is proposed byIn (18), three positive gain constants are , , and while the parameter update law is defined by
By substituting (17) and (22) into (21), the differentiation of is rewritten in the formObviously, the differentiation of the Lyapunov function is a negative semidefinite function. As a result, it is simple to verify , , and exponentially as based on Barbalat’s lemma . The complete antisynchronization between the master system and the slave system is proved.
We take an example to illustrate the calculation of the antisynchronization scheme. The parameter values of the master system and slave system are fixed asWe assume that the initial states of the master system (13) arewhile the initial states of the slave system (14) are taken asThe positive gain constants are chosen as , , and . For numerical simulations, we take the initial condition of the parameter estimate asIt is apparent from Figure 9 that there is antisynchronization of the respective states of the systems with cloud-shaped equilibrium (13) and (14). The time-history of the synchronization errors , , and is shown in Figure 10. It is straightforward to verify that Figure 10 depicts the antisynchronization of the master and slave systems.
The main goal of the current study is to propose a novel unusual system with infinite number of equilibrium points which lay on a closed curve. Dynamics of the new system have been investigated via different tools such as phase portrait, bifurcation diagram, Kaplan–York dimension, and maximal Lyapunov exponents. The feasibility of the theoretical model is clearly verified by the circuit implementation. In addition, the research also shows that antisynchronization of systems with closed curve equilibrium is obtained by introducing an adaptive control. The findings in this study provide a new understanding of system with infinite equilibria. In terms of directions for future research, further work could explore chaos-based applications of such new system. For example, chaotic behaviors of other systems which are similar to the new one are useful for generating hardware pseudorandom number , controlling motions of autonomous mobile robots , or using in secure communications .
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge Professor GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting many helpful references. The author Xiong Wang was supported by the National Natural Science Foundation of China (nos. 11547117 and 61601306) and Shenzhen Overseas High Level Talent Peacock Project Fund (no. 20150215145C).
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