#### Abstract

The microcontroller implementation, controls, and synchronization of a three-dimensional (3D) autonomous system with a parabolic equilibrium point are investigated in this paper. The system in question displays a reverse period doubling route hidden chaotic attractors with two different shapes. Then, the partial and total amplitude controls of the system are achieved by inserting two parameters. A microcontroller implementation is performed in order to confirm 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. The sliding mode controllers are designed to control chaos in the system under study. With the sliding mode control method, the numerical simulations confirm 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 different parameters.

#### 1. 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 fields such as mechanics, biology, chemistry, and electronics. The strong use of chaos has motivated researchers to come up with many new chaotic systems [1–8]. 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. The basin of attraction of hidden attractors does not intersect with small neighborhoods of any equilibrium points. This plays a very significant role in the fields 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. This 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].

The 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. Therefore, 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. The 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 difference synchronization [15], or the difference-difference 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. This 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 different phase portraits.

The 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 different parameters. Finally, the paper is concluded in Section 5.

#### 2. Amplitude Controls and Microcontroller Implementation of 3D Autonomous System with a Parabolic Equilibrium

A 3D autonomous system is proposed:

With being state variables, the time, and a positive parameter, the equilibrium point of system is parabolic with expression . It is important to note that there is a difference between hyperbolic equilibrium point and parabolic equilibrium point. The hyperbolic equilibrium can be defined, for example, as where *n* is an odd integer number while the parabolic equilibrium point can be defined, for example, as where *k* is an even integer number. At the parabolic equilibrium point , the characteristic equation of system (1a)–(1c) is given by

The Routh–Hurwitz condition for the stability of this system at the parabolic equilibrium point is . Since , the parabolic equilibrium point is always unstable. Figure 1 shows the plots of the dynamical behaviors of the 3D autonomous system with a parabolic equilibrium point.

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Two different shapes of chaotic hidden attractors (see Figures 1(a) and 1(b)), period-4-oscillations (see Figure 1(c)), period-2-oscillations (see Figure 1(d)), and period-1-oscillations (see Figure 1(e)) are shown at *a* = 3.05, *a* = 3.18, *a* = 3.22, *a* = 3.35, and *a* = 3.45, respectively.

The amplitudes of the state variable in system (1a)–(1c) can be partially controlled by making the change of variable in system (1a)–(1c) where is the boosting controller. So, system (1a)–(1c) becomes:

In this case, is the parabolic equilibrium point of system (3a)–(3c). Its characteristic equation is

By making use of the Routh–Hurwitz criteria, the roots of equation (4) are all real and negative if and only if . Since , the parabolic equilibrium point is always unstable. So, the stability of the parabolic equilibrium point does not depend on the boosting controller . In order to check the partial amplitude control of system (3a)–(3c), a plot of the average values of the state variables , , and against the boosting controller is done in Figure 2.

Figure 2 shows that an increase in the boosting controller leads to a decrease in the average of the state variable while the two other state variables and remain almost constant. The phase portraits of system (3a)–(3c) and time evolutions are depicted in Figure 3 for different values of the boosting controller .

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Figure 3(a) shows that by increasing the boosting controller , the amplitudes of the state variable are shifted in the left in the plane while Figure 3(b) shows that the amplitude of the state variable is boosted from a bipolar signal to a unipolar signal by increasing the boosting controller . The amplitudes of system (1a)–(1c) can be totally controlled by replacing into system (1a)–(1c) where being the scaling amplitude. So, system (1a)–(1c) becomes

System (5a)–(5c) has as the parabolic equilibrium point. The characteristic equation of the parabolic equilibrium point is

By applying the Routh–Hurwitz criteria, all the roots of equation (6) have negative real parts if and only if . Since , the parabolic equilibrium point is always unstable. So the stability of the parabolic equilibrium point is independent of the scaling amplitude . The phase planes of system (5a)–(5c) are depicted in Figure 4 for specific values of the scaling amplitude .

The amplitudes of the state variables are increased alongside the scaling amplitude as shown in Figure 4.

The 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. The Runge–Kutta method order four is used in this microcontroller. The phase planes obtained from the circuit in Figure 5 point are depicted in Figure 6.

The microcontroller results of Figure 6 qualitatively agree with the numerical simulations of Figure 1.

#### 3. Chaos Control of 3D Autonomous System with a Parabolic Equilibrium Point

The sliding mode control method is used to control chaos in 3D autonomous system with a parabolic equilibrium point in this section. The controlled system under study is described by

With and being the controllers designed based on sliding mode control method, the variable is the new input. Thus, system (7a)–(7c) becomes

Thence, if as , the second order of the above system is stabilized as

The sliding surface of system (9a) and (9b) is defined as

By choosing , , equation (10) becomes

System (8a)–(8c) has a single equilibrium point with the characteristic equation:

The Routh–Hurwitz conditions for the stability of system (8a)–(8c) at the only equilibrium point are

The solution of the set of inequalities (13a)–(13c) is . So, system (8a)–(8c) is asymptotically stable for . System (9a) and (9b) has a single equilibrium point , and the characteristic equation at this equilibrium point is

The Routh–Hurwitz conditions for the stability of system (9a) and (9b) at the only equilibrium point are

The solution of the set of inequalities (15a) and (15b) is . So, system (9a) and (9b) is asymptotically stable for . As a result, systems (8a)–(8c) and (9a) and (9b) are asymptotically stable for . The 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 efficient as revealed by the numerical simulation results depicted in Figure 7.

#### 4. 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. The drive 3D autonomous system with a parabolic equilibrium point is given bywhere is a value that assures chaotic behavior and the response 3D autonomous system with a parabolic equilibrium given by

With being the parameter whose value assures chaotic behavior, the vector is a controller vector. Since the 3D autonomous system with a parabolic equilibrium point has two different 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 different shapes of chaotic attractors. The 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.

##### 4.1. 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 , , and being the coupling strengths. For easy analysis, only the linear feedback coupling is used for the coupling of the two identical chaotic systems. The maximum synchronization error and the transverse Lyapunov exponents are to estimate the quality of synchronization between the trajectories of the response and drive systems. The plots of the largest transverse Lyapunov exponents and the maximum synchronization error as a function of the coupling strength are depicted in Figure 8.

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The 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 ) for . The boundary and the stability of synchronization are attested by the largest transverse Lyapunov exponent shown in the left panel of Figure 8(b). The response and drive systems are synchronized in the chaotic regime (with ) for (see the right panel of Figure 8(a)). The boundary and the stability of synchronization are confirmed by the largest transverse Lyapunov exponent shown in the right panel of Figure 8(b). The projection of the phase space trajectories of the response and drive systems for a given value of the coupling strength and the parameter values are plotted in Figure 9.

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The drive and response systems (16a)–(16c) and (17a)–(17c) are synchronized in the two different 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 different parameters ( and or and ) 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 different parameters by using nonlinear error feedback coupling.

##### 4.2. Synchronization of Chaotic Systems with Different Parameters through Nonlinear Feedback Coupling

Here, the drive and response systems (16a)–(16c) and (17a)–(17c) are considered where the controller is an active control function to be designed by using a Lyapunov exponent function. Hereafter, the controller for which the chaos synchronization of two proposed chaotic systems with different 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:

With , , and , the Lyapunov function

has the derivative

A trivial choice of iswhere and are real parameters. Then, 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 whose characteristic equation is

The Routh–Hurwitz conditions for stability reveal that the zero solution of system (24a)–(24c) is locally asymptotically stable for any values of and . By choosing and with and , 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.

#### 5. Conclusion

This 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 different shapes of hidden chaotic attractors. The addition of new parameters in the rate equations of this system made the partial and total amplitude controls possible. Then, 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. The effectiveness of these designed controllers was confirmed by the numerical simulation results. Finally, the chaos synchronization of two chaotic autonomous systems with the same parameters and different 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 different parameters can synchronize.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.