Microcontroller Implementation, Controls, and Synchronization of Three-Dimensional Autonomous System with a Parabolic Equilibrium Point

Te microcontroller implementation, controls, and synchronization of a three-dimensional (3D) autonomous system with a parabolic equilibrium point are investigated in this paper. Te system in question displays a reverse period doubling route hidden chaotic attractors with two diferent shapes. Ten, the partial and total amplitude controls of the system are achieved by inserting two parameters. A microcontroller implementation is performed in order to confrm the results obtained from the numerical simulations. It is found that the results from the numerical simulations and microcontroller implementation qualitatively agree with each other. Te sliding mode controllers are designed to control chaos in the system under study. With the sliding mode control method, the numerical simulations confrm that chaos can be controlled in the 3D autonomous system with a parabolic equilibrium point. In addition, two chaotic 3D autonomous systems with a parabolic equilibrium point and the same parameters are synchronized by the use of a unidirectional linear error feedback coupling scheme. Finally, an active control technique is applied to bring about chaos synchronization between two chaotic 3D autonomous systems with a parabolic equilibrium and diferent parameters.


Introduction
Nowadays, chaos theory is one of the biggest paradigm shifts in the nonlinear dynamics of complex systems. Wide varieties of dynamical systems with chaos have been reported in many modelling applications thus in many felds such as mechanics, biology, chemistry, and electronics. Te strong use of chaos has motivated researchers to come up with many new chaotic systems [1][2][3][4][5][6][7][8]. It is now evident from various studies that there exist uncommon chaotic systems with hidden attractors. Now, it is well established that no standard computational procedure can permit us to have a straightforward way of localizing hidden attractors. In chaos theory, hidden attractors are a very fascinating topic. Te basin of attraction of hidden attractors does not intersect with small neighborhoods of any equilibrium points. Tis plays a very signifcant role in the felds of nonlinear circuits.
According to Leonov and Kuznetsov, periodic and chaotic attractors can either be categorized as self-excited or hidden [1]. While the basin of attraction of a self-excited attractor is associated with an unstable equilibrium, that of a hidden attractor does not intersect with small neighborhoods of any equilibrium points. Because it is important to enable a chaotic system to maintain a desirable dynamical behavior, techniques such as static or dynamic feedback control or an open-loop control method have been proposed to achieve this goal. In some recent cases, the major task is to investigate the state feedback control law in order to stabilize the system around its unstable equilibrium point. Tis has so far been done using the active control method [9,10], the sliding mode control method [11,12], the adaptive control method [13,14], and the backstepping method [15,16].
Te foundation of science and engineering has been, in the past decades, impacted by the study of chaos with the discovery of chaos synchronization being one of the most exciting developments in recent times. Terefore, two coupled chaotic oscillators can be synchronized using the same methods as those used for controlling nonlinear systems. Pecora and Carroll [17] discovered that for synchronization of chaotic systems to be possible, an appropriate coupling design must be found, such that for instance, the chaotic time evolutions of the two systems become identical. Te use of state observers or the use of control laws according to the Lyapunov stability theory is another technique by which chaotic systems can be synchronized. Synchronization can either be the phase synchronization [18], the complete synchronization [18,19], the generalized synchronization [20], the antisynchronization [21], the generalized projective synchronization [22], the hybrid synchronization [23], the diference synchronization [15], or the diference-diference synchronization [16].
Inspired by these recent works [1,24] and motivated by the great interest in hidden attractors and some discoveries about coexisting asymmetric attractors, we proposed a three-dimensional autonomous system with a parabolic equilibrium point. Tis chaotic system is then analyzed by phase portraits, time histories, and the Lyapunov exponent. We will use the sliding mode control method to control chaos in the proposed 3D autonomous system with parabolic equilibrium point. A study of synchronization of two identic proposed 3D autonomous systems with parabolic equilibrium point will be investigated by using unidirectional feedback linear and nonlinear coupling. By using a microcontroller, a circuit is designed and built to plot the diferent phase portraits.
Te structure of the rest of this paper is as follows: the amplitude controls and microcontroller implementation of the proposed three-dimensional system with a parabolic equilibrium point are investigated in Section 2. Section 3 deals with the chaos control of a 3D system with a parabolic equilibrium point by using the sliding mode control method. In Section 4, a unidirectional linear error feedback coupling scheme is used to synchronize two chaotic autonomous 3D systems with a parabolic equilibrium point and the same parameters. In addition, an active control technique is employed to achieve chaos synchronization between chaotic 3D autonomous systems with a parabolic equilibrium point and diferent parameters. Finally, the paper is concluded in Section 5.

Amplitude Controls and Microcontroller
Implementation of 3D Autonomous System with a Parabolic Equilibrium A 3D autonomous system is proposed: With x, y, z being state variables, t the time, and a a positive parameter, the equilibrium point of system is parabolic with expression E � (x, y, z) ∈ R 3 /x � (y * ) 2 , y � y * , z � 0}. It is important to note that there is a difference between hyperbolic equilibrium point and parabolic equilibrium point. Te hyperbolic equilibrium can be defned, for example, as x � 1/(y * ) n where n is an odd integer number while the parabolic equilibrium point can be defned, for example, as x � (y * ) k where k is an even integer number. At the parabolic equilibrium point E, the characteristic equation of system (1a)-(1c) is given by λ λ 2 − a y * 2 λ + 1 � 0. (2) Te Routh-Hurwitz condition for the stability of this system at the parabolic equilibrium point E is − a(y * ) 2 > 0. Since a > 0, the parabolic equilibrium point E is always unstable. Figure 1 shows the plots of the dynamical behaviors of the 3D autonomous system with a parabolic equilibrium point.

(4)
By making use of the Routh-Hurwitz criteria, the roots of equation (4) are all real and negative if and only if − a(y * ) 2 > 0. Since a > 0, the parabolic equilibrium point E 1 is always unstable. So, the stability of the parabolic equilibrium point E 1 does not depend on the boosting controller β. In order to check the partial amplitude control of system (3a)-(3c), a plot of the average values of the state variables x, y, and z against the boosting controller β is done in Figure 2. Figure 2 shows that an increase in the boosting controller β leads to a decrease in the average of the state variable x while the two other state variables y and z remain almost constant. Te phase portraits of system (3a)-(3c) and time evolutions are depicted in Figure 3 for diferent values of the boosting controller β. Figure 3(a) shows that by increasing the boosting controller β, the amplitudes of the state variable x are shifted in the left in the plane (x, y) while Figure 3(b) shows that the amplitude of the state variable x is boosted from a bipolar signal to a unipolar signal by increasing the boosting controller β. Te amplitudes of system (1a)-(1c) can be totally controlled by replacing x ⟶ x/σ, y ⟶ y/σ, z ⟶ z/σ into system (1a)-(1c) where σ being the scaling amplitude.
By applying the Routh-Hurwitz criteria, all the roots of equation (6) have negative real parts if and only if − a(y * ) 2 /σ 2 > 0. Since a > 0, the parabolic equilibrium point E 2 is always unstable. So the stability of the parabolic equilibrium point E 2 is independent of the scaling amplitude σ. Te phase planes of system (5a)-(5c) are depicted in Figure 4 for specifc values of the scaling amplitude σ.
Te amplitudes of the state variables x, y, z are increased alongside the scaling amplitude σ as shown in Figure 4. Te microcontroller implementation of 3D autonomous system with a parabolic equilibrium point is depicted in Figure 5. Figure 5 is made of the integrated circuit ATmega328P microcontroller, the FTD232 module, and the TFT display of dimensions 800 × 480 connected to the microcontroller via the driver RA8875. Te Runge-Kutta method order four is used in this microcontroller. Te phase planes obtained from the circuit in Figure 5 point are depicted in Figure 6.
Te microcontroller results of Figure 6 qualitatively agree with the numerical simulations of Figure 1

Chaos Control of 3D Autonomous System with a Parabolic Equilibrium Point
Te sliding mode control method is used to control chaos in 3D autonomous system with a parabolic equilibrium point in this section. Te controlled system under study is described by With u 1 � x − y − xz 2 and u 2 � − [x − y 2 + z(ay 2 − z 2 )] + ] 1 being the controllers designed based on sliding mode control method, the variable ] 1 is the new input. Tus, system (7a)-(7c) becomes Tence, y ⟶ 0 if x, z ⟶ 0 as t ⟶ ∞, the second order of the above system is stabilized as Te sliding surface of system (9a) and (9b) is defned as By choosing ] 1 � − z + kσ 1 , k > 0, equation (10) System (8a)-(8c) has a single equilibrium point O 1 � (0, 0, 0) with the characteristic equation: Te Routh-Hurwitz conditions for the stability of system (8a)-(8c) at the only equilibrium point O 1 � (0, 0, 0) are Te solution of the set of inequalities (13a)-(13c) is k > 0. So, system (8a)-(8c) is asymptotically stable for k 1 > 1. System (9a) and (9b) has a single equilibrium point O 2 � (0, 0), and the characteristic equation at this equilibrium point is Te Routh-Hurwitz conditions for the stability of system (9a) and (9b) at the only equilibrium point O 2 � (0, 0) are Te solution of the set of inequalities (15a) and (15b) is k > 0. So, system (9a) and (9b) is asymptotically stable for k > 0. As a result, systems (8a)-(8c) and (9a) and (9b) are asymptotically stable for k > 0. Te state variables of controlled system (8a)-(8c) are depicted in Figure 7. Chaos control of system (1a)-(1c) by using the sliding mode control method is efcient as revealed by the numerical simulation results depicted in Figure 7.

Synchronization of Chaotic 3D Autonomous Systems by Using the Same or Different Parameters
In this section, the synchronization of the 3D autonomous systems with a parabolic equilibrium point is studied by using unidirectional feedback linear and nonlinear coupling terms. Te drive 3D autonomous system with a parabolic equilibrium point is given by Figure 6: Projections of phase planes obtained from the microcontroller implementation of 3D autonomous system with a parabolic equilibrium point. 6 Mathematical Problems in Engineering where a 1 is a value that assures chaotic behavior and the response 3D autonomous system with a parabolic equilibrium given by With a 2 being the parameter whose value assures chaotic behavior, the vector U � (u 1 , u 2 , u 3 ) T is a controller vector. Since the 3D autonomous system with a parabolic equilibrium point has two diferent shapes of chaotic attractors, it is interesting to study the synchronization between identical unidirectionally coupled 3D autonomous systems with a parabolic equilibrium point by using each shape of chaotic attractors and between two identical unidirectionally coupled 3D autonomous systems with a parabolic equilibrium by using diferent shapes of chaotic attractors. Te subsection that follows presents the achievement of the synchronization between two 3D proposed systems with the same parameters using the unidirectional linear feedback coupling scheme.

Synchronization of Proposed Chaotic Systems with the Same Parameters by Using Unidirectional Linear Feedback
Coupling. Here, the drive and response systems (16a)-(16c) and (17a)-(17c) are coupled by using the unidirectional linear feedback coupling: with the constant parameters k 1 , k 2 , and k 3 being the coupling strengths. For easy analysis, only the linear feedback coupling u 2 is used for the coupling of the two identical chaotic systems.
Te maximum synchronization error )and the transverse Lyapunov exponents are to estimate the quality of synchronization between the trajectories of the response and drive systems. Te plots of the largest transverse Lyapunov exponents and the maximum synchronization error e max as a function of the coupling strength k 2 are depicted in Figure 8.
Te response and drive systems are synchronized, if the maximum synchronization error becomes approximately equal to zero. From the left panel of Figure 8(a), one can see that the response and drive systems are synchronized in the chaotic regime (with a 1 � a 2 � 3.05) for 0.12 ≤ k 2 ≤ 0.4. Te boundary and the stability of synchronization are attested by the largest transverse Lyapunov exponent shown in the left panel of Figure 8(b). Te response and drive systems are synchronized in the chaotic regime (with a 1 � a 2 � 3.18) for 0.17 ≤ k 2 ≤ 0.35 (see the right panel of Figure 8(a)). Te boundary and the stability of synchronization are confrmed by the largest transverse Lyapunov exponent shown in the right panel of Figure 8(b). Te projection of the phase space trajectories of the response and drive systems for a given value of the coupling strength k 2 and the parameter values a 1 , a 2 are plotted in Figure 9.
Te drive and response systems (16a)-(16c) and (17a)-(17c) are synchronized in the two diferent shapes of chaotic attractors as shown in Figures 9(a) and 9(b), respectively. Worth noting is the fact that the synchronization between the drive and response systems (16a)-(16c) and (17a)-(17c) with diferent parameters (a 1 � 3.05 and a 2 � 3.18 or a 1 � 3.18 and a 2 � 3.05) is not achieved by using the unidirectional linear feedback coupling. In the next subsection, it is possible to achieve synchronization of identical proposed chaotic systems with diferent parameters by using nonlinear error feedback coupling.

Synchronization of Chaotic Systems with Diferent Parameters through Nonlinear Feedback Coupling.
Here, the drive and response systems (16a)-(16c) and (17a)-(17c) are considered where the controller U is an active control function to be designed by using a Lyapunov exponent function. Hereafter, the controller U for which the chaos synchronization of two proposed chaotic systems with diferent parameters can be achieved has to be designed. For this reason, the error dynamical variable between drive system (16a)-(16c) and response system (17a)-(17c) can be written as follows:   Figure 9: Projection of the phase space trajectories of the drive and response systems for a specifc value of the coupling strength k 2 , a 1 and Mathematical Problems in Engineering With e x � x 2 − x 1 , e y � y 2 − y 1 , and e z � z 2 − z 1 , the Lyapunov function has the derivative dL dt � e x − e z + u 1 + e y e x z 2 2 + e z x 1 z 1 + z 2 + u 2 + e z e x − e y y 1 + y 2 − e z z 2 A trivial choice of U � (u 1 , u 2 , u 3 ) T is u 1 � 0, u 2 � − e x z 2 2 − e z x 1 z 1 + z 2 − p 2 1 e y , u 3 � e y y 1 + y 2 + e z z 2 1 + z 1 z 2 + z 2 2 + a 1 z 1 y 2 1 − a 2 z 2 y 2 2 − p 2 2 e z , where p 1 and p 2 are real parameters. Ten, Equation (21) becomes By inserting equation (22) into equations (19a)-(19c), the error system (19a)-(19c) becomes System (24a)-(24c) has a single equilibrium point O 3 (0, 0, 0) whose characteristic equation is Te Routh-Hurwitz conditions for stability reveal that the zero solution of system (24a)-(24c) is locally asymptotically stable for any values of p 1 and p 2 . By choosing p 1 � 0.5 and p 2 � 1 with a 1 � 3.0 and a 2 � 3.16, the synchronization errors between the drive and response systems (16a)-(16c) and (17a)-(17c) are plotted in Figure 10. Figure 10 reveals that the drive-response systems (16a)-(16c) and (17a)-(17c) are asymptotically synchronized after a certain period of time.

Conclusion
Tis paper dealt with the microcontroller implementation, controls, and synchronization of a three-dimensional chaotic autonomous system with a parabolic equilibrium point. It was found that the system exhibited periodic oscillations and two diferent shapes of hidden chaotic attractors. Te addition of new parameters in the rate equations of this system made the partial and total amplitude controls possible. Ten, the microcontroller implementation of a chaotic autonomous system with a parabolic equilibrium was realized, and the comparison between the microcontroller implementation and numerical simulations results showed a good qualitative agreement. Moreover, the sliding mode controllers were designed in order to control chaos in the three-dimensional autonomous system with a parabolic equilibrium point. Te efectiveness of these designed controllers was confrmed by the numerical simulation results. Finally, the chaos synchronization of two chaotic autonomous systems with the same parameters and diferent initial conditions was achieved by using a unidirectional linear feedback coupling scheme. By using unidirectional nonlinear feedback coupling, it was demonstrated analytically and numerically that two chaotic autonomous systems with diferent parameters can synchronize.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.