On a New Three-Dimensional Chaotic System with Adaptive Control and Chaos Synchronization

Saeed, Nasser A.; Saleh, Hend A.; El-Ganaini, Wedad A.; Kamel, Magdi; Mohamed, Mohamed S.

doi:https://doi.org/10.1155/2023/1969500

Shock and Vibration

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1969500 | https://doi.org/10.1155/2023/1969500

On a New Three-Dimensional Chaotic System with Adaptive Control and Chaos Synchronization

Nasser A. Saeed,^1,2,3Hend A. Saleh,⁴Wedad A. El-Ganaini,¹Magdi Kamel,¹and Mohamed S. Mohamed⁵

Academic Editor: Antonio Batista

Received29 Dec 2022

Revised19 Mar 2023

Accepted01 Apr 2023

Published14 Apr 2023

Abstract

A three-dimensional autonomous deterministic chaotic system having six parameters is explored within this article. The dynamical characteristics of the proposed system are investigated through eigenvalues structure, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponents, time response, and phase plane trajectories. For the suitable design of the system parameters, it is found that the system can exhibit periodic, period-n, or chaotic oscillations. Accordingly, the system’s dynamical behavior to the variation of its coefficients has been explored. The obtained results revealed that the proposed dynamical system does not lose its chaotic oscillations for the small fluctuations of one or more of the values of its parameters. In addition, chaos control and chaos synchronization have been studied by means of the adaptive control strategy relying on Lyapunov’s second method of stability. The numerical simulation revealed that superior chaos control and master-slave synchronization have been achieved by the applied control laws. Finally, the obtained results have been simulated via a nonlinear electronic circuit that demonstrated the feasibility of the purposed chaotic system for different engineering applications such as secure communications, cryptosystems, image encryption, and image processing.

1. Introduction

The chaos theory is defined as a branch of computer science and mathematics that studies the dynamical properties of nonlinear systems which are extremely sensitive to the initial conditions [1, 2]. Lyapunov exponents and the compactness property of the phase space are the most two important and necessary measures that may be used to investigate the chaotic behavior of such systems. Some of the initial paradigms of the three-dimensional chaotic oscillators in the literature are the Lorenz system [3], Rössler oscillator [4], Arneodo et al. system [5], Sprott system [6], Chen and Ueta oscillator [7], Lü and Chen system [8], Liu et al. system [9], Cai and Tan system [10], Chen and Lee system [11], and Tigan and Opris [12]. Recently, chaotic dynamics has found many applications in different areas such as mechanical systems [13, 14], microelectromechanical systems [15], radar systems [16], vehicle models [17], random number generators [18], robotic systems [19–22], memristive devices [23–25], maglev systems [26], rotor active magnetic bearings systems [27–29], biodynamics [30], biological systems [31–34], ecological systems [35, 36], cardiology [37, 38], chemical reactions [39–42], lasers [43, 44], unmanned aerial vehicles [45], rotating machinery [46–50], secure communications [51–53], image encryption [54–56], cryptosystems [57, 58], financial system [59–61], DC motor systems [62], and electronic circuits [63–65]. Any chaotic attractor has an infinite number of unstable periodic orbits. Therefore, chaotic motion arises when the system states move in the neighborhood of one of these unstable periodic orbits for a short period and then fall close to another unstable periodic orbit for a limited time and so forth. This mechanism results in unpredictable motion of the system state for a long time, where this motion is called chaotic oscillation. Chaos control aims to stabilize the chaotic wandering of the system states about its equilibrium points. Many control techniques have been applied for this purpose such as the optimal control system [66, 67], state-feedback control [68], sliding mode and integral sliding mode control [69–74], backstepping control method [75], adaptive control [76–78], and time-delayed feedback control [79].

Chaotic synchronization is a phenomenon that happens when two or more oscillators are coupled or when an oscillator drives another one. The common meaning of synchronization is that the phases of two or more systems change according to a specific pattern. The synchronization phenomenon is abundant in nature, science, social life, and engineering. Well-known systems such as clocks, firing neurons, applauding audiences, singing crickets, and cardiac pacemakers tend to operate in synchrony [80]. There are five synchronization techniques, which are phase synchronization, generalized synchronization, lag synchronization, amplitude envelope synchronization, anticipated synchronization, and complete synchronization. A complete synchronization regime is the common one that is used with a pair of chaotic systems known as the master system and the slave one. The main target of this synchronization technique is to force the slave system output according to a specific control law to track the output of the master system. Pecora and Carroll are the first people that found chaos synchronization in their experiments on circuits [81, 82], where the authors utilized two Lorenz systems where one of them has been used as the master system and the other as the slave. They found that the synchronization among the master and slave systems occurred when the first state variable of the slave system is replaced with the first state variable of the master system. This synchronization scheme is called the P-C scheme. In fact, many control techniques that are used for chaos control can also be utilized in chaos synchronization, where the active control method has been applied in chaos synchronization when all the system parameters are measurable [83–87]. The adaptive control strategy is also employed in chaotic system synchronization when some or all the system parameters are not measurable or when the estimation for some uncertain parameters is required [88–91]. In addition, sampled data feedback control strategies [92–94], time-delayed feedback control techniques [95, 96], and backstepping control methods [97, 98] are used in the chaos synchronization. Moreover, the sliding mode control has been applied extensively in chaos synchronization [99, 100].

In general, 3D-chaotic systems can be categorized depending on the number of terms, parameters, and equilibrium points as shown in Table 1 [101–108]. Also, the chaotic system with dimensions higher than 3D and having at least two positive Lyapunov exponents is called a hyperchaotic system [109, 110].

Within this article, a new 3D-chaotic system with four linear terms, two quadratic nonlinear terms, and six parameters is presented. Detailed bifurcation analysis for the considered system has been conducted through the time response, phase plane trajectories, Lyapunov exponents, bifurcation diagrams, and Kaplan–Yorke dimension. The fluctuation of the different system parameters on the system’s dynamics is explored. The obtained results illustrated that the considered system may respond with periodic, period-n, or chaotic oscillations depending on the values of its parameters. Accordingly, the optimal values of the system parameters are designed in such a way that makes the considered system oscillate chaotically, where both the Lyapunov exponents and the corresponding Kaplan–Yorke dimension are obtained. In addition, chaos control has been achieved by designing an adaptive controller based on Lyapunov’s second method of stability. Moreover, the chaos synchronization of the introduced chaotic system with itself as a master-slave system has been investigated by designing a globally stable adaptive control system. Finally, we have built an electronic circuit using MultiSim () to simulate the chaotic dynamics of the considered system.

2. The Novel 3D Chaotic System

The study of chaos arose from the discovery of the well-known Lorenz system in 1963. A chaotic system is a nonlinear dynamical system that is very sensitive to the initial conditions and produces aperiodic bounded signals that resemble noise despite not being generated from stochastic systems. The breakthrough of the huge applications of chaos (as in mathematics, computer science, engineering, population dynamics, robotics, biology, and so on) has prompted chaos generation to be a vital research subject. Therefore, this article introduces a novel 3D-chaotic autonomous system with four linear terms and two quadratic terms as follows.where and denote the state variables and and are positive constant parameters that form the coefficients of the considered system. At the system parameters and , system (1) exhibits complex dynamics. The chaotic motion of the suggested dynamical system will be proved in the flowing subsections.

2.1. Dissipativity, Attractor Existence, and Equilibrium Points

The nonlinear autonomous system given by equation (1) can be expressed in the state-space form as follows:

According to equation (2), the divergence of the vector field on can be calculated simply as follows:

Therefore, the necessary and sufficient conditions for the nonlinear autonomous system given by equation (1) to be a dissipative one are that the divergence should be a negative value. So, based on equation (3), system (1) is a dissipative system if and only if .

Suppose is a region in with smooth boundaries and let , where represents the flow of the vector field . Let represent the volume of . According to the Liouville theorem [111], we have

The solution of equation (4) can be written as . Therefore, any volume element in the space of system (1) will be contracted by the flow into the volume element at the time . This means that each volume containing the system trajectories will be shrunk to zero when at an exponential decay rate . Accordingly, all the orbits of the system given by equation (1) will be confined into a subset of zero volume, and the asymptotic motion of the system will settle into an attractor regardless of the initial conditions. So, for the designed parameters ( and ), the exponential contraction rate of the considered system is . In addition, the equilibrium points of the considered system are and . These equilibrium points are unstable when the system parameters are designed such that . Thus, the trajectories of the considered dynamical system (1) will diverge from the equilibrium points as long as the initial conditions do not satisfy one of these points.

2.2. Kaplan–Yorke Dimension

Within this section, Lyapunov exponents () of the considered dynamical system are obtained numerically as shown in Figure 1 at the system parameters: and . It is clear from the figure that the steady-state Lyapunov exponents of the considered system are and , where . Accordingly, one can find the Kaplan–Yorke dimension (i.e., ) as follows:

It is clear from equation (5) that the considered system has a fractional dimension, which confirms that the considered autonomous dissipative system has a nonperiodic solution.

2.3. Chaotic Response and Phase Trajectories

Based on the obtained Lyapunov exponents ( and ) as shown in Figure 1, the autonomous dynamical system (1) will respond with a chaotic bounded motion for any initial conditions in the three-dimensional space except the three equilibrium points and . The time response of the considered chaotic system is illustrated in Figure 2 at the initial conditions , while Figure 3 shows the corresponding phase plane trajectories. Figures 2 and 3 are obtained via solving equation (1) numerically using the ODE45 MATLAB solver when and . It is clear from Figure 2 that the system motion is chaotic. In addition, Figure 3 shows that the suggested chaotic system has chaotic attractors with shapes that are different from those Lorenz-like systems.

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3. Bifurcation Analysis

For the practical realization of the considered chaotic system, it should have high immunity to the slight fluctuations of one or more of its parameters. Therefore, this section aims to investigate the system dynamics when changing each one of the system parameters. Figure 4 shows the bifurcation diagram and the corresponding Lyapunov exponents utilizing as a bifurcation parameter along the range , with fixing the other parameters constant. Figure 4(a) illustrates that the system has dissipativity behaviors, where the contractions exponent on the interval . However, Figures 4(a) and 4(b) demonstrate that system motion may be either periodic, period-n, or chaotic depending on the value of the parameter . Also, one can notice from Figure 4(b) that the system can perform chaotic oscillation at a wide range of about (i.e., the system oscillates chaotically as long as ), which guarantees the immunity of the proposed chaotic system to the fluctuation of without losing its chaotic dynamics.

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Figure 5 illustrates the bifurcation of the system motion and the corresponding Lyapunov exponents when utilizing as a bifurcation parameter on the interval . Figure 5(a) demonstrates that the system may perform periodic oscillations as long as , but increasing beyond may result in a periodic-doubling bifurcation, that is, the route to a chaotic motion. In addition, the figure shows that the system may lose its chaotic oscillation via periodic-halving bifurcation if is increased beyond . However, Figure 5(b) confirms that the proposed system can exhibit chaotic motion as long as , which guarantees the system immunity for the small fluctuation of about the designed value .

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Figure 6 depicts the motion bifurcation and the corresponding Lyapunov exponents when utilizing as the main bifurcation parameters along the interval . Figure 6(a) shows that the system can perform periodic motion as long as . But, increasing beyond results in periodic-doubling bifurcation, which ultimately leads to chaotic oscillations as shown in Figure 6(b) when . Accordingly, one can confirm the system’s immunity to the small fluctuation of about the designed value . Figures 7–9 are a repetition of Figures 4–6 but concerning the rest of the system parameters and , respectively. By examining Figures 4–6, one can demonstrate that the motion bifurcation and the corresponding Lyapunov exponent of the considered chaotic system are insensitive to the variation of the parameters and on the intervals and . Based on the abovementioned investigations, it is clear that the introduced system (1) with the designed parameter values (i.e., and ) has high immunity for the slight fluctuations of the values of its parameters.

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4. Chaos Control

4.1. Adaptive Controller Design

Chaos control of the proposed dynamical system has been investigated within this section utilizing an adaptive control strategy. Thus, the suggested controlled chaotic system (1) is modified to becomewhere and are the suggested adaptive control signals to stabilize the chaotic motion of system (1). The main strategy of the adaptive controller is to generate control signals and in order to cancel the nonlinearity of the considered chaotic system (6) and force it to respond as a dissipative linear system with () stable equilibrium point. Accordingly, and are designed such thatwhere and are positive constants that form the linear feedback gains, while and denote the estimated parameters of the system coefficients and . Now, by substituting equation (7) into equation (6), we have the following controlled system:

Notice that when the estimated parameters (i.e., , and ) reach the same values of the system parameters (i.e., and ), equation (8) becomes and . Accordingly, let us denote the error estimation of the parameters as follows:

Based on equation (9), the derivatives of the parameter estimation errors can be expressed as follows:

Substituting equation (9) into equation (8) yields

To obtain the control law that will adjust the parameter estimations, let us build up the Lyapunov positive definite function for the controlled chaotic system given by equation (11) as follows:

Differentiating the constructed Lyapunov function , we have

Eliminating and from equation (13) utilizing equations (10) and (11) yields

According to equation (14), the parameters estimation law can be chosen as follows:where are positive constants. Based on the designed estimation law given by equation (15), the derivative of the Lyapunov function is a negative definite function that can be written as follows:

Theorem 1. The controlled system that is given by equation (6) with the unknown coefficients and is globally stabilized regardless of the initial conditions by both the designed control law given by equation (7) and the parameters estimation law given by equation (15), where and are positive constants.

Proof. The above theorem is a simple consequence of Lyapunov’s second method for stability [112]. We showed that the Lyapunov function that is given by equations (12) is a positive definite function on . In addition, we illustrated that the first derivative given by equations (15) is a negative definite on . Hence, according to Lyapunov’s second method for stability, it follows that the system states and tend to zero exponentially as the time tends to infinity, which completes the proof that the system given by equation (6) is globally stable.

4.2. Numerical Simulation of Chaos Adaptive Control

Numerical simulations for the introduced adaptive control system given in Section 4.1 are illustrated within this section. The time response of the controlled chaotic system is simulated numerically via solving equations (6), (7), and (15) using the ODE45 MATLAB solver as in Figure 10 when , and at the initial conditions . Figures 10(a)–10(c) show the system’s chaotic motion before control on the time interval and after turning on the introduced adaptive control law at up to . In addition, Figure 10(d) shows the evolution of the estimated parameters () after turning on the controller at . It is clear from Figures 10(a)–10(c) that the chaotic states and on the interval have been forced to enter the equilibrium point as soon as turning on the controller at (i.e., and as soon as the controller is activated). Moreover, Figure 10(d) demonstrates the exponential convergence of the estimated parameters to the system parameters (i.e., ).

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5. Chaos Synchronization

5.1. Adaptive Controller Design

Based on the investigation given in Section 3, the parameters and have negligible influence on the system’s dynamical behaviors. Accordingly, these parameters are treated as fixed values such that , and within this section. Therefore, the modified novel chaotic system (i.e., equation (1)) that represents the master system is given as follows:

In addition, let the slave system be given by the following dynamical equation:where and denote the slave system states, and and are the control signals to be designed in order to achieve the global synchronization between the master system (17) and the slave one (18). Accordingly, the master-slave state errors can be defined as follows:

Based on equations (17)–(19), the error dynamics can be defined as follows:

According to equation (20), let us design the control law as follows:where and are the estimated parameters of the unknown system parameters and . In addition, and are positive constants that represent the control gains. Now, substituting equation (21) into equation (20) yieldswhere and are the parameter estimation errors, and . To obtain the control laws that will adjust the states’ synchronization and the parameter estimations, let us build up the Lyapunov positive definite function for the chaotic system given by equation (22) as follows:

Accordingly, the derivative of equation (23) can be expressed as follows:

Eliminating and from equation (24) utilizing equation (22) yields

According to equation (25), the parameters estimation law can be designed as follows:where are positive constants. Based on the designed estimation law given by equation (26), the derivative of the Lyapunov function is a negative definite function that can be expressed as follows:

Theorem 2. The coupled synchronized master-slave chaotic system that is given by equations (17) and (18) with the unknown coefficients and is globally stabilized via both the designed control law given by equation (21) and the updated parameters law that is given by equation (26), where and are positive constants.

Proof. The abovementioned theorem is a simple consequence of Lyapunov’s second method for stability [112]. We designed the Lyapunov function given by equation (23) to be a positive definite function on . In addition, its first-order derivative (i.e., ) given by equation (27) is a negative definite on . Therefore, based on Lyapunov’s second method for stability, it follows that and tend to zero exponentially as the time tends to infinity, which completes the proof that the master-slave system given by equations (17) and (18) is globally stable.

5.2. Numerical Simulation of Chaos Synchronization

Based on equations (17)–(19), (21), and (26), the block diagram describing the sequential execution of the synchronized master-slave system is depicted in Figure 11. The figure shows that the chaotic states of both the master and slave systems (i.e., and ) are firstly fed into both the adaptive control law (i.e., equation (21)) and parameters estimation law (i.e., equation (26)) using an appropriate sensors network simultaneously. Then, the parameter estimation law estimates the system parameters and according to the predefined estimation rule (i.e., equation (26)) to feed them into the adaptive control law. After that, the adaptive control law computes the control signals and based on the input signals and to apply them to the slave system in order to follow typically the same chaotic motion of the master one.

Accordingly, the master-slave synchronized motion has been simulated within this section via solving equations (17)–(19), (21), and (26) numerically using MATLAB ODE45 solver as shown in Figures 12 and 13 when the system parameters , control gains , and initial conditions . Figures 12(a)–12(c) illustrate the instantaneous oscillations of both the master and slave systems before synchronization (i.e., ) on the interval and after turning on the synchronization controller at up to . In addition, Figure 12(d) demonstrates the evolution of the estimated parameters ( and ) after turning on the controller at . It is clear from Figures 12(a)–12(c) that the unsynchronized states of the slave system (i.e., and ) on the time interval have been forced to follow the master system states when the controller is turned on at (i.e., and when ). Also, Figure 12(d) demonstrates the exponential convergence of the estimated parameters to the system parameters with time (i.e., when ).

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On the other hand, Figures 13(a)–13(c) show the temporal oscillations of both the master and slave systems before synchronization (i.e., ) on the time interval and after turning on the synchronization controller at up to . In addition, Figure 13(d) depicts the evolution of the estimated parameters ( and ) along the time interval . It is clear from Figures 13(a)–13(c) that the unsynchronized states of the slave system (i.e., and ) on the time interval have been forced to follow the master system states as soon as the controller is turned on at faster than that in Figures 12(a)–12(c) (i.e., and when ). Moreover, Figure 13(d) demonstrates the abrupt convergence of the estimated parameters to the system parameters when (i.e., when ).

By comparing Figures 12 and 13, one can notice that in Figure 12, both the control signals and the estimated parameters have been set to zero on the time interval (i.e., both the adaptive control law and the parameters estimation law have been stopped from up to ), but in Figure 13, the control signals only have been set zero on the time interval (i.e., the adaptive control law has been stopped from up to but the parameters estimation law is activated from up to ). It is clear from Figure 13(d) that the estimated parameters and are very close to the actual system parameters and on the time interval even though the adaptive controller has stopped. So, as soon as the controller is turned on at , the correct control signals have been applied to the slave system, which makes it abruptly follows the master system from up to . Accordingly, to make the slave follows the master system exactly at a short transient time, it is recommended to activate firstly the parameters estimation law before turning on the adaptive control law to avoid the transient time required by the adaptive controller to compute the correct control signals, which was of 4 time steps approximately as shown in Figure 12.

6. Nonlinear Circuit Design

Using MultiSim software (), the proposed chaotic system (1) has been simulated within this section using an electronic circuit consisting of five operational amplifiers (i.e., ) as shown in Figure 14, where three of these amplifiers serve as analog integrators (i.e., ) while the other two are employed as inverting amplifiers (i.e., ). According to this electronic circuit, the straight-forward equations of motion that govern the states and shown in Figure 14 can be expressed as follows:

Based on equation (28), the circuit components are designed such that and . Relying on these designed parameters, the circuit output has been visualized using the Multisim oscilloscope as in Figure 15. Figure 15(a) shows the chaotic attractor when the circuit outputs and are the input channels to the oscilloscope. In addition, Figure 15(b) shows the phase plane trajectory when the circuit outputs and are the input channels to the oscilloscope. Moreover, Figure 15(c) illustrates the phase plane when the circuit outputs and are the input channels to the oscilloscope. By comparing Figure 3 with the simulation results in Figure 15, one can notice the typical correspondence between the numerical solutions in Figure 3 and the output of the electronic circuit shown in Figure 15. Accordingly, one can confirm the possible implementation of the introduced chaotic system for different engineering applications such as secure communications, cryptosystems, image processing, and image encryption.

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7. Conclusion

A novel three-dimensional autonomous chaotic oscillator having both four linear terms and two quadratic nonlinear terms with six parameters has been studied in this work. The system’s dynamics are explored utilizing Lyapunov exponents, bifurcation diagrams, Kaplan–Yorke dimension, time response, and phase plane trajectories. The obtained results demonstrated that the proposed dynamical system may perform periodic, period-n, or chaotic oscillations depending on the designed values of its parameters. In addition, the obtained bifurcation diagrams illustrated that the considered system does not lose its chaotic oscillations for the small fluctuations of one or more of the values of its parameters. Moreover, adaptive control strategies based on Lyapunov’s second method of stability have been applied for the purposes of chaos control and chaos synchronization. The numerical simulation proved that the designed adaptive control laws can achieve superior chaos control and master-slave synchronization. Finally, a simple electronic circuit that simulates the system dynamics demonstrated the feasibility of the designed chaotic system for different engineering applications.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The authors would like to thank Taif University, where this work was supported and funded by the Taif University Researchers Supporting Project number TURSP-2020/160, Taif, Saudi Arabia. This work was supported by the Polish National Science Centre, Poland, under the grant OPUS 18 No. 2019/35/B/ST8/00980.

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