Dynamical Analysis and Offset Boosting in a 4-Dimensional Quintic Chaotic Oscillator with Circulant Connection of Space Variables

Sandrine, Nzoulewa Dountsop; Jacques, Kengne

doi:https://doi.org/10.1155/2023/7735838

Complexity

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Research Article | Open Access

Volume 2023 | Article ID 7735838 | https://doi.org/10.1155/2023/7735838

Dynamical Analysis and Offset Boosting in a 4-Dimensional Quintic Chaotic Oscillator with Circulant Connection of Space Variables

Nzoulewa Dountsop Sandrine^1,2and Kengne Jacques^1,2,3

Academic Editor: Atila Bueno

Received22 Jun 2022

Revised23 Nov 2022

Accepted08 Mar 2023

Published08 May 2023

Abstract

In recent years, much energy has been devoted to the study of chaotic models with specific features particularly those with cyclic connection of the variables. Previous ones provide multistability, amplitude control, and so on. Concerning the first phenomenon, models with ring connection of variables presented a coexistence of up to twelve disconnected attractors. In order to emphasize the complexity of circulant chaotic oscillators and their use in the engineering domain, a quintic chaotic model with cyclic connection of variables is considered and studied, which has complex equilibria located on the line . Therefore, it experiences, amongst other, the phenomenon of offset boosting obtained by introducing four constants into the equations of the model, which has not be done in the past. Multistability is also revealed and the coexistence of eight and sixteen attractors is demonstrated using phase portraits. The system’s dynamics has been investigated considering its two parameters. Nonlinear dynamical tools such as bifurcation diagrams, phase portraits, time evolutions, two-parameter diagram, and Lyapunov exponents help to highlight the important phenomena encountered. The numerical results are confirmed using PSpice and particularly show the double-band chaotic attractor. Moreover, total amplitude control (TAC) is shown, proving that our oscillator can be used as an attenuator or amplifier in the engineering domain. The method of adaptive synchronization has been applied to the considered oscillator to emphasize the possible implication into the secure of communication systems.

1. Introduction

The designing and implementation of chaotic oscillators with new performances has interested a good number of scholars nowadays. This is because of its various applications to secure communication [1, 2], random bits generation [3], image encryption [4], and neural network. Such systems provide relevant features such as transient chaos [5–7], multistability [8–10], antimonotonicity [11, 12], and many others applicable in the previously mentioned field of engineering. In addition, the symmetry property of chaotic systems has captivated many researchers in their study. Some based their analysis on inversion symmetry because a system that provides it brings in the notion of bistability [13, 14]; therefore, increasing the number of coexisting attractors in the concerned systems. The complexity of such chaotic oscillators lets a good number of researchers to break the inversion symmetry which leads to new phenomena like the coexistence of asymmetric type of oscillations [15–19]. In fact, it can be found in any of these references the simultaneous existence of nonsymmetrical attractors for the same rank of parameters. In reference [16], the coexistence of two different attractors is revealed (two different chaotic states or a periodic state with chaotic state). The cyclic chaotic system of [17] provides the coexistence of four attractors (a periodic state with four different chaotic attractors) that are not symmetric either with inversion or cyclic symmetry. Additionally, the authors of reference [18] show the coexistence of two different periodic states and two different chaotic attractors. The authors of reference [19] emphasized the appearance of five asymmetric attractors for the same parameter’s rank using different initial states. Moreover, reference [20] revealed the coexistence of a periodic state with a chaotic state. Apart from inversion symmetry, the cyclic one interested many persons such as Thomas R. who introduced the first chaotic model with that connection of variables [20]. He used the cubic and sine nonlinearities and showed the pathway to chaos using phase portraits with multistability. Later on, many others were introduced [21–25]. In most of these systems, studies were focused on the stability of equilibria, the different routes to chaos, and multistability. The later phenomenon was characterized either by hidden attractors [26–29] or self-exited ones [8, 30]. This year, Veeman et al. proposed a “new chaotic system with coexisting attractors” with spherical symmetry having three unstable equilibrium points and the coexistence of three hidden attractors [31]. Another type of multistability is megastability defined as a property of a chaotic/hyperchaotic system with an infinite number of coexisting nested attractors which form a layer structure [32]. In this reference, authors presented for odd numbers of integers, this important feature. In this regard, Kengne et al. [16] introduced in 2021 a new cyclic chaotic attractor with piecewise quadratic nonlinearity derived from Thomas one in 1999 [20]. With , the inversion symmetry of the system was perturbed and three unstable fix points where obtained on the line x = y = z providing self-exited attractors. Then, using the Letellier and Gilmore [33] transformation formulae, the cyclic connection of variables was clearly demonstrated. The entire dynamics of the 3-D system was studied in the symmetric and asymmetric case of oscillations. Through deep investigation, transient chaos, TAC (total amplitude control), and multistability were put on. In the case of mutistability, the coexistence of symmetry and asymmetry attractors was revealed. Moreover, the case of asymmetry oscillations showed coexistence of bubble branches. Then, the highest number of attracting sets was twelve disconnected attractors. To emphasize the feasibility of the model, PSpice simulations were provided and in accordance with the numerical ones. According to their wide application in large space circular antenna, blade disk, and magnetic store devices [34], it is relevant to propose, analyze, and validate new cyclic chaotic models with new features.

It is in this regard that we propose a new quintic cyclic chaotic system derived from the one studied by Rajagopal et al. [34] in 2019, in which cubic nonlinearity was used in a 4-D cyclic system. Here, the novelty of the system is its capacity of experienced offset boosting which is very rare in cyclic chaotic system’s literature, its presentation of five complex equilibria, and the coexistence of up to sixteen attractors for the same rank of parameters. To inquiry the dynamics of the model, bifurcation diagrams, Lyapunov Exponents, phase portraits, time evolutions, and others are employed. PSpice environment helps us to show the practical implementation of the model. Furthermore, the adaptive synchronization method helps to emphasize the use of such oscillator to secure communication, an asset in the engineering domain. The rest of the document is organized as follows.

Section 2 provides circuit and mathematical model of the quintic circulant system with relevant properties like dissipation and symmetry. The study of the stability of equilibria is also performed. Section 3 deals with numerical analysis from which different phenomena are revealed like TAC known as total amplitude control, offset boosting thus multistability to name few. The general overview of the system is shown in Section 4 through one- and two-dimension plots. Section 6 emphasizes the easy implementation of the model into engineering domain through its easy adaptive synchronization, and finally, we end with concluding remarks.

2. Preliminary Insights of the Quintic Oscillator

This section consists of the description of the model with the use of differential equations. Its property of dissipation is verified to conclude on its aptitude to develop attracting sets. Also, the two types of symmetry, including inversion and cyclic, sources of the high number of coexisting attractors in dynamical systems are studied. The investigation on fixed points, a key stage in the study of paramount complex dynamical behaviors of this chaotic system, is performed toward the end of the section.

2.1. Circuit Implementation and Mathematical Model of the Quintic Cyclic Chaotic System

Derived from the work of Rajagopal et al. 2019 [35], the implemented circuit diagram of the model is that of Figure 1, which consists of twelve multipliers, six operational amplifiers, twenty resistors, and three capacitors. Amongst the six amplifiers integrated in the circuit, three work as integrators while the rest are inverters. The four quintic nonlinearities in the model are represented by a set of three multipliers in each block. The multipliers and operational amplifiers are all supplied by a symmetrical . X, Y, Z, and W represent the different voltages, respectively, across the capacitors C₁, C₂, C₃, and C₄. The determination of the mathematical model of the system lets us to use some assumptions for the off–the-shelf components. Linear resistors and capacitors are employed; then, the operational amplifiers work in their linear zone and are ideal. Therefore, with the help of Kirchhoff’s law, the following system is derived.

(a)

(b)

(c)

(d)

Using the following change of variables and parameters,

We obtain the following dimensionless system of equations:where a and b are the control parameters and the state variables of the system. This system is derived from the work done by Rajagopal et al. in 2019 [34] in which he used the cubic nonlinearity in a cyclic system but only showed the zones of multistability for a normal system and that of its fractional order. He also showed the different routes to chaos in his work. Here, we are going to reveal the order complex phenomena developed in the new system.

2.2. Dissipation and Symmetry of the 4-Dimensional Chaotic Model

At a given point of a phase space , the total volume of expansion/contraction rate of this oscillator is given by the following expression:

Thus, the system is dissipative and can develop attractors. Additionally, equation (3) is an invariant under the transformation, , so the chaotic model is symmetric about the origin. Apart from this property, each derivation of (3) depends on itself and the next nearest variable; then, the functions are same except the rotation of variables. Therefore, the system presents the cyclic connection of space variables. These two symmetric properties are the source of the great number of coexisting attractors found in the model. They are also used to simplify the algorithm employed numerically to solve system (3).

2.3. Equilibria and Study of Their Stability

2.3.1. Equilibria

The characterization of the dynamics of chaotic systems lies on fixed points because it provides its general overview [36]. Due to the nature of equilibria, nonlinear systems can be classified as self-excited and hidden attractor oscillators [29]. For the first category, fixed points exist and are unstable while the second case concerns those without equilibria or with stable ones [37]. To have the equilibria of this system, the right hand side of (3) is equalized to zero. Using the same method, Rajagopal et al. [34] studied a fourth-order cyclic chaotic system with cubic nonlinearity possessing 81 equilibrium points, emphasizing therefore the complexity of chaotic models with ring connection of space variables. Then, this new one is obtained by replacing the cubic nonlinearity with the quintic one. Setting the right hand side of (3) to zero, we obtain as follows:

Those situated on the line are: ,

2.3.2. Study of Stability of Equilibria

The best understanding of the dynamics of chaotic systems passes by the study of the nature of fixed points. To do it, using the Ruth–Hurwith criterion of stability, we use Jacobian’s matrix which, at a stationary point , is given by the following expression:

At the points , the characteristic equation is given by the following expression:

Throughout this work, the couple satisfies the condition ; the characteristic equation has coefficients with different signs, and thus, all equilibria are unstable using the previous criterion. In this case, the system develops self-excited attractors as mentioned by many authors before [8, 29, 38].

3. Numerical Results

Nonlinear dynamic systems are captivating the research community’s attention based on the different key behaviors that they reveal. Here, the numerical study of the quintic system considered here provides very fascinating states of oscillations.

3.1. Linear Transformation Property

Linear transformation concerns with offset boosting and amplitude control. Amplitude control is relevant in engineering applications, for it helps in optimizing and achieving amplitude [39, 40]. It can be partial or total depending on the variables implicated [41, 42].

Offset-boosting refers to a partial amplitude control as defined in reference [43] as the method of repositioning any attractor as well as its basin of attraction in the solution space of a dynamical system in the absence of solutions destroyed.

Theorem 1. An attractor in a smooth dynamical system can always obtain offset boosting by introducing an appropriate constant into any dimension of the system.

Proof. Consider a smooth dynamical system .
Introducing a vector into each variable will give the system offset boosted within an interval of a. More precisely, let ; then, and . This shows that the same governing equation will produce the same solution except a constant offset a.
It is in this line that many authors have based this condition of offset-boostable systems on the fact that one of the variables should appear once, in the mathematical equation of the system [44–47]. However, in a case of all variables appearing at least twice in the equations, can such model experience offset boosting? This question is that of cyclic systems in which each of the variables appears at least twice in the equations. Then, a particular approach was used by Li et al. in 2018 [48] to produce an infinite lattice from the offset boosting phenomenon. He used rather than one variable, all in a 4-D system to produce offset boosting and hatching with a periodic function. Here, we apply the same approach to experience an offset boosting process in this cyclic system with quintic nonlinearity. Then, the new system obtained is given by (5).The constants can have the same value or not. Therefore, this action on phase portraits, time series, bifurcation diagrams, and Lyapunov exponents are represented in Figure 2. Any color is associated to a common value of the five constants: . In Figure 2(a)(A), we have bifurcation diagrams that show the maxima of the variable x as the parameter b evolves, which is obtained using five different values of the constants. Figure 2(a)(B) shows the Lyapunov spectrum indicating the evolution of the four exponents of the system in terms of the control parameter b when it undergoes offset boosting phenomenon. In Figure 2(b), we have the chaotic attractor offset boosted for the same values in Figure 2(a)(A) used therefore to emphasize this state of functioning. For the same values of the control parameters, time series are provided in Figure 2(b)(B). This particular offset boosting phenomenon is rare in the literature of cyclic chaotic systems and deserves dissemination.
Total amplitude control is also exhibited by the chaotic model, which is obtained by multiplying the quintic nonlinearity with a positive parameter m used here as a rescaling factor to tune the amplitude of the four system variables x, y, z, and . It can be noticed that the parameter m acts only on the nonlinearity and therefore is up to fulfill the amplitude control tasks.
This relevant feature of the system emphasizes the aptitude of the analog circuit of Figure 1 to be used as a possible amplifier or attenuator circuits in a secure transmission system as a signal source. To tune the four variables of system (3), the following replacement is performed only with the quintic nonlinearity: and the new system is obtained.
It can be noticed that the controller m acts only on the quintic function, and therefore, the amplitude control strategy changes the attractors according to the four axes. The recapitulated results of Figure 3 expose from 3(a) to 3(d) the bifurcation diagrams of different variables in terms of the rescaling factor m, which really indicate the total control of all the space variables. Here, as the parameter m changes, the amplitude decreases while the entire dynamics of the system is not modified. The representation of Figure 3(e) shows the largest Lyapunov exponent of the system in function of m which remains approximately constant all through reinforcing none distruction of the system dynamics when the amplitude control parameter evolves. In Figure 3(f), we have a typical example of the double-band chaotic attractor which has undergone the amplitude control process successfully. Therefore, the magenta, green, and black colours correspond, respectively, to (m = 1), (m = 2.5), and (m = 5). This system presents multistability which is the appearance of two or more attractors for the same rank of parameters, with different initial states.

(a)

(b)

(a)

(b)

(c)

(d)

(e)

(f)

3.2. Multistability

In 2018, Rajagopal and collaborators presented some chaotic systems with cyclic connection of variables and they studied deeply the one with polynomial function which exhibits a zone of multistability. In that domain, he showed a coexistence of three different attractors [23]. In the same line in 2019, they studied a fourth-order cyclic chaotic system with cubic nonlinearity and its fractional order [34]. Here, he highlights the routes to chaos and multistability windows. For these two works, they used only one control parameter. Due to its results, we are presenting a new 4-D chaotic system with the dynamics based on two parameters, which presents windows of multistability and a coexistence of eight and sixteen attractors. Here, with the help of downward and upward continuation of bifurcation, the windows of irregularities are uncovered and their exploitation leads to the simultaneous appearance of attractors indicated before. In Figure 4, the coexistence of eight attractors is shown and is revealed using the two parameters a and b. In fact, Figures 4(a) and 4(b) provide the first coexistence of eight attractors for a = 6 then b = 10.73, while 4(c) shows the second case when a = 4.167 with b = 8. Apart from these states of oscillations, a set of sixteen coexisting oscillations are represented in Figure 5 with the table of initial conditions in Table1. Concerning this case, (a) and (b) represent periodic states while (c) and (d) are chaotic states. This particular simultaneous appearance of sixteen states of functioning has not yet been presented in the cyclic chaotic systems and deserves dissemination.

(a)

(b)

(c)

(a)

(b)

(c)

(d)

4. Two-Parameter and One-Parameter Bifurcation Diagrams

In this section, we introduce the general dynamics of the quintic cyclic chaotic system with the help of the evolution of variables maxima in terms of the two control parameters known as bifurctaion diagrams. Else, the evolution of parameters is represented indicating the general evolution aspect of the model in respect of the control parameters.

4.1. Two-Parameter Bifurcation Diagram

From the engineering application point of view, the simultaneous change of the control parameters shown in a diagram provides interesting information on the general overview of the dynamics of a chaotic system. Thus, in Figure 6, we point up the two-parameter diagram of system (3) which gives the space of variation of the two control parameters. This diagram was obtained by varying a(resp.b) in the range (resp. ) and recording the corresponding values of the maximum Lyapunov exponent for the initial state . System (3) is solved numerically using the Runge–Kutta algorithm with the grid 300 × 300 values of the concerned parameters from Free Pascal and plot by the use of MATLAB. The colorful diagram of Figure 6 shows the zone of periodicity through the blue, cyan, and yellow colors. Then, the rest of the colors enlighten the chaotic domains. This quantification is performed with the help of the color bar representing the maximum Lyapunov exponent (MLE). The contrast zones indicate multistability.

4.2. One-Parameter Bifurcation Diagram

In this subsection, we evaluate the influence of the two control parameters a and b on the dynamics of the quintic cyclic chaotic system. Downward and forward continuation techniques are used to plot the bifurcation diagrams to identify the zones of multistability. Here, at each iteration, the final state serves as an initial one to the next iteration. First, we fix b = 8 and represent the local maxima of the variable x in function of the control parameter a. The scenario to chaos is the period de-doubling route to chaos.

The forward (red) and downward (black) bifurcation diagrams are represented in Figure 7(a) with different windows of multistability exploited previously. In Figure 7(b), we have the corresponding spectrum of Lyapunov exponents which clearly indicate the way to chaos. From these two subfigures, the dynamics starts periodically within the zone , then continues chaotically with the symmetrical merging crisis in the domain . This captivating domain is followed by Lest quantified chaotic region in after which follows periodic oscillations in . Another chaotic zone follows in ; then, the dynamics continues periodically within the zone . Therefore, a chaotic zone follows in and the dynamics ends periodically. Second, the influence of the dynamics of the system due to the evolution of the parameter b is represented in Figure 8, and the spectrum of the Lyapunov exponents is attached then, as well, matched with the bifurcation diagrams. The route to chaos is the period-doubling bifurcation phenomenon. The techniques used previously permit to reveal the different windows of the multistability as indicated on the diagrams. Here, it can be noticed that the dynamics is the reverse of the previous on.

(a)

(b)

(a)

(b)

5. PSPICE Verifications

Basing on the numerical results obtained previously, it can be seen that the quintic cyclic system can experience relevant dynamical phenomena when the appropriate parameters are used. Therefore, it is pertinent to verify the feasibility of such diverse oscillations types [16, 19, 49]. Here, we focus on the PSpice simulations of the dynamics of the system represented in Figure 1 and particularly on the double-band chaotic attractor. The PSpice implementation of Figure 1 is performed using the operational amplifier TL084 and the multiplier AD633/AD, all supplied by a symmetrical to achieve the provided goal. Then, using capacitors and resistors, the concerned figure is connected. Each block is therefore realized representing each of the four variables of the system. The quintic nonlinearity is built using three multipliers and for each, the two multiplicand inverting inputs with the summing input are connected to the ground. For the first multiplier, the two multiplicand noninverting inputs are connected to the same variable to do the first multiplication. Then, the output of the first multiplier is used as the two multiplicand noninverting inputs of the second. For the third one, the output of the second one is used as one of the noninverting input while the other one is connected to the input variable of the first one. Then, the output of the third multiplier is the quintic nonlinearity. With all the connections performed, when varying the control resistors R_a (ie., a) or R_b (ie., b), the dynamics represented either in Figures 7 or 8 is observed and particularly, for , and using the initial states , the double-band chaotic attractor is obtained and can be observed on the right hand side of Figure 9. Moreover, the transient analyses are configured as follows: Print Step (0 s), Final Time (500 ms), No-Print Delay (60 ms), and Step Celling (1 s). On the left hand side, the numerical double-band chaotic attractor is observed and a good similarity between the two figures is well noticed.

(a)

(b)

The numerical attractor is obtained using the values of the parameters and initial states in the caption of the concerned figure.

6. Adaptive Synchronization

Increasing attention is directed to the study of chaotic systems due to their different applications in the engineering domain, amongst which appears synchronization and its particular use on secure communication [1, 2, 50, 51]. From reference [1], synchronization is defined as the phenomenon that accomplishes when two or more oscillators combine their oscillations in time such that their movements become correlated permanently. There exist different types of synchronization and accordingly different techniques used [52–54], such as adaptive synchronization, a common method [10].

Here, the use of this last method of synchronization lets us to propose the master and slave sub-systems given successively by the following expressions:where and are the state variables of, respectively, master and slave subsystems. Furthermore, is the nonlinear controller having a goal to be appropriately chosen for the master and slave to synchronize. The dynamics of the error between the master and slave is given in equation (12).

With

For the reasons of simplicity, the controllers are considered as follows:where and are the estimated parameters of a and b, respectively. is the positive feedback gain. By substituting (15) into (13) and (12), we obtain the following expressions for the slave subsystem (16) and (17), the dynamics of error.where

According to the principle of synchronization, the two identical quintic cyclic models experience the phenomenon whenever as , the error [55] or asymptotically stable. For this to be achieved, let us consider the following positive definite function as Lyapunov candidate:

The differential of the Lyapunov function along the trajectory of error system (13) is as follows:

Substituting equation (17) into equation (20), we will have the following expression:

For system (11) and (12) to synchronize, equation (22) has to be satisfied.

This lead us to the following expressions for and :

We have simulated equations (11), (16), and (23) numerically, then Figures 10 and 11 are recorded. The starting points used are the following: and for the two subsystems. The different gains are choosen as with the initial estimated parameters . The system’s parameters are a = 3.7 and b = 8.

(a)

(b)

(c)

(d)

Figure 10 shows the time evolution of the different error components and it is noticed that they converge asymptotically to zero as time evolves. In addition, Figure 11 represents the time evolution of the two estimated parameters which converge asymptotically to 3.61 for a and 11.52 for b. We can therefore say that the quintic cyclic system well synchronizes using the adaptive method.

7. Conclusion

In this paper, the dynamics of a quintic cyclic chaotic model is investigated through various dynamical tools. The accomplishment of this task starts with the determination of equilibria and their stability conditions. The complexity of the system lets us to determine fixed points located on the line which is complex. The use of the Routh–Hurwitz criterion of stability permits us to show that the five equilibria are always unstable. One- and two-dimension diagrams are used to investigate the overall dynamics and predict the zones of multistability. In this case, bifurcation diagrams showing routes to chaos are given in terms two control parameters a and b. Also, time series, Lyapunov exponents, and phase portraits help to accentuate the types of oscillations provided. The different scenarios to chaos are well traced by the spectrum of Lyapunov exponents. While exploring the quintic cyclic chaotic system, diverse types of oscillations are discovered. The key one is offset boosting which is a great input in such models. This relevant phenomenon is obtained by introducing four constants to the space variables which has not been done before. To determine the zones of multistability, we use the forward and backward continuation methods of simulation which allow to discover the coexistence of eight and sixteen attractors at different points of functioning. To the best of the author’s knowledge, the second one has not yet been mentioned and therefore emphasizes the importance of the quintic chaotic system. In the engineering domain, Total Amplitude Control (TAC) is demonstrated, proving therefore the appropriate integration of the model in a case of attenuation or amplification of the signal for modulation. Additionally, the synchronization of the system enlightens its easy application in the domain of communication secure. PSpice simulation of the double-band chaotic attractor confirms the numerical one and as such ensures the validity of the proposed model. For future outcomes, inversion and cyclic symmetries will be broken to uncover more fascinated types of oscillations. Also, the numbers of variables will be added to sort out many other complicated phenomena.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Nzoulewa Dountsop Sandrine and Kengne Jacques. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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