A Three-Dimensional Autonomous System with a Parabolic Equilibrium: Dynamical Analysis, Adaptive Synchronization via Relay Coupling, and Applications to Steganography and Chaos Encryption

Ramadoss, Janarthanan; Kengne, Romanic; Tokoue Ngatcha, Dianorré; Kamdoum Tamba, Victor; Rajagopal, Karthikeyan; Motchongom Tingue, Marceline

doi:https://doi.org/10.1155/2022/8362836

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Chaos in Applied Sciences and Engineering 2021

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Volume 2022 | Article ID 8362836 | https://doi.org/10.1155/2022/8362836

A Three-Dimensional Autonomous System with a Parabolic Equilibrium: Dynamical Analysis, Adaptive Synchronization via Relay Coupling, and Applications to Steganography and Chaos Encryption

Janarthanan Ramadoss,¹Romanic Kengne,²Dianorré Tokoue Ngatcha,³Victor Kamdoum Tamba,⁴Karthikeyan Rajagopal,⁵and Marceline Motchongom Tingue⁶

Academic Editor: Mustafa Cagri Kutlu

Received18 Feb 2022

Revised16 Apr 2022

Accepted23 Apr 2022

Published16 May 2022

Abstract

This paper is reporting on electronic implementation of a three-dimensional autonomous system with infinite equilibrium point belonging to a parabola. Performance analysis of an adaptive synchronization via relay coupling and a hybrid steganography chaos encryption application are provided. Besides striking parabolic equilibrium, the proposed three-dimensional autonomous system also exhibits hidden chaotic oscillations as well as hidden chaotic bursting oscillations. Electronic implementation of the hidden chaotic behaviors is done to confirm their physical existence. A good qualitative agreement is shown between numerical simulations and OrCAD-PSpice results. Moreover, adaptive synchronization via relay coupling of three three-dimensional autonomous systems with a parabolic equilibrium is analysed by using time histories. Numerical results demonstrate that global synchronization is achieved between the three units. Finally, chaotic behavior found is exploited to provide a suitable text encryption scheme by hidden secret message inside an image using steganography and chaos encryption.

1. Introduction

It is widely recognized that mathematically simple systems of nonlinear differential equations can exhibit chaos. With the advent of fast computers, it is now possible to explore the entire parameter space of these systems with the goal of finding parameters that result in some desired characteristics [1].

Recent research has involved categorizing periodic and chaotic attractors as either self-excited or hidden [2–10]. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, whereas a hidden attractor has a basin of attraction that does not intersect with small neighbourhoods of any equilibrium points. The classical attractors of Lorenz, Rössler, Chua, Chen, and Sprott systems (cases B to S) and other widely known attractors are those excited from an unstable equilibrium. From a computational point of view, this allows one to use a numerical method in which a trajectory started from a point on the unstable manifold in the neighbourhood of an unstable equilibrium and reaches an attractor, to identify it [2]. Hidden attractors cannot be found by this method and are important in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in structures like a bridge or an airplane wing.

The chaotic attractors in dynamical systems without any equilibrium points, with only stable equilibria, or with infinite number of equilibria are hidden attractors. That is the reason why such systems are rarely found. However, recently, such systems have been reported in literatures [11–47]. Especially, systems with infinite number of equilibria are rare and challenging to find. There are three families of chaotic systems with infinite number of equilibria: systems with line equilibria [33–37], systems with closed-curve equilibria [38–41], and systems with open-curve equilibria [42, 43]. Recently, systems with infinite number of equilibria have been studied as an exciting research subject [26, 30, 44, 45]. However, there is still a necessity and challenge to discover new chaotic systems with different opened-curve equilibria [23, 46].

A three-dimensional autonomous system with hidden attractors and a parabolic curve of equilibria is introduced in this paper. The proposed chaotic system has one positive control parameter and six terms among which four are nonlinear. Its simplicity is remarkable while it is capable of displaying chaotic oscillations and chaotic bursting oscillations depending solely on the control parameter. It is worth noting that chaotic bursting oscillations are usually found in systems with self-excited attractors [47–52]. To the best of our knowledge, there is no three-dimensional system with hidden attractors exhibiting chaotic oscillations and chaotic bursting oscillations. However, some questions arise: Can the three-dimensional autonomous system with parabolic curve of equilibria synchronize? Can chaotic behavior found in the three-dimensional autonomous system with parabolic curve of equilibria be exploited to provide suitable steganography and chaos encryption?

The rest of this paper is organized as follows. In Section 2, fundamental properties of the proposed three-dimensional system are investigated by means of equilibrium points, eigenvalue structures, phase portrait, time series, basin of attraction, bifurcation diagram, and Lyapunov exponents. The physical existence of the chaotic behavior found in the proposed system is verified using electronic implementation in Section 3. An adaptive synchronization via relay coupling of three three-dimensional autonomous systems with parabolic curve of equilibria is investigated in Section 4. An application to steganography and chaos encryption is performed in Section 5. Finally, the paper is concluded in Section 6.

2. Analysis of the Three-Dimensional Autonomous System with a Parabolic Equilibrium

Inspired by the method and structure proposed in [41], a three-dimensional autonomous system with a parabolic equilibrium is introduced in this section:where are state variables, is the time, and is a positive parameter. System (1) has only one parameter . System (1) has a parabolic equilibrium given by

The characteristic equation of system (1) evaluated at the parabolic equilibrium is

The roots of equation (3) depend on the sign of . If , the roots of equation (3) are and . The characteristic equations have at least a positive real root; therefore, is unstable. For , the roots of equation (3) are and . Since , equation (3) has a pair of complex conjugate eigenvalues with positive real root, so is unstable. Thus, is always unstable for .

To investigate the dynamical behaviors of system (1), Lyapunov exponents (LEs) and bifurcation diagram depicting maxima of versus the parameter for the initial conditions (x(0), y(0), z(0)) = (0, 0.1, 0.2) are plotted in Figure 1.

(a)

(b)

By increasing the parameter from 3.0 to 3.5, system (1) exhibits chaotic behavior followed by a reverse period-doubling leading to period-1-oscillation as shown in Figure 1(c). Depending on the value of the amplitude of the output , the chaotic region can be divided into two regions: chaotic bursting oscillations for and chaotic oscillations for . Chaotic behavior is confirmed by the LE shown in Figure 1(b). Chaotic behaviors found in the bifurcation diagram of Figure 1(a) are further detailed in Figure 2 which presents the time series of , , and and the corresponding phase portraits for specific values of parameter .

In the left panel of Figure 2, we can see that the variables x(t) and z(t) show a fast changing processes, while the variable y(t) describes a relatively slowly changing quantity. The fast-slow variables are confirmed by the corresponding phase portraits in the right panel of Figure 2. The signals x(t) and z(t) alternate between a silent and an active phase. These latter are thus called chaotic bursting oscillations. Chaotic behaviors found in Figure 1 for are depicted in Figure 3.

In the time domain, Figure 3(a) shows chaotic oscillations, and in Figure 3(b), the phase portraits display chaotic attractors. The basin of attraction of system (1) in the plane for is shown in Figure 4.

In Figure 4, the initial conditions in the white region lead to unbounded orbits, those in the light blue region lead to the strange attractor, and those on the red curve are the parabolic equilibrium. From Figure 4, one can notice that the proposed three-dimensional autonomous system with a parabolic equilibrium belongs to chaotic systems with hidden attractors [2–6, 9, 47] since the basin of attraction of the strange attractor intersects only a limited portion of the curve of equilibria.

3. Electronic Circuit Simulation of the Three-Dimensional Autonomous System with a Parabolic Equilibrium

Three state variables x, y, and z of system (1) are rescaled to overcome the difficulties in realization [53–55]. Therefore, system (1) is rewritten aswhere X = 10x, Y = 10y, and Z = 10z. The circuit in Figure 5 is designed to realize system (4). This circuit has been designed using a multiplier’s inherent characteristics [56, 57].

The circuit of Figure 5 consists of resistors, operational amplifiers, analogue multipliers, and capacitors. From Figure 5, the circuital equations are derived aswhere V_X, V_Y, and V_Z are the output voltages of the operational amplifiers (see Figure 5). The fixed constant of the multipliers is denoted as k_m, and k_m = 10 V. Normalizing circuital equations (5) by using the dimensionless states variablesand inserting equations (6) in (5), the following system is obtained:

Obviously, system (7) is equivalent to system (4) with the following conditions:

The resistor R₅ is used to vary the value of the parameter a. As a result, the values of circuit components are selected as R₁ = R₂ = R₃ = R₄ = R₆ = R = 10 kΩ, C₁ = C₂ = C₃ = C = 10 nF, and R₅ = 3.333 kΩ (for a = 3) or R₅ = 3.165 kΩ (for a = 3.16). Phase portraits in Figure 6 are obtained by using the electronic simulation package OrCAD-PSpice.

(a)

(b)

There is a good agreement between the numerical simulations and OrCAD-PSpice results when comparing the phase portraits of Figures 2 and 3 with Figure 6.

4. Adaptive Synchronization between the Relay Coupling of Three Three-Dimensional Autonomous Systems with a Parabolic Equilibrium

The relay coupling topology of three three-dimensional autonomous systems with a parabolic equilibrium used in this paper is represented in Figure 7.

In Figure 7, outer 1 is defined by the state variables , outer 2 by the state variables , and relay by the state variables . The variables and are the control signals between outer 1 and relay whereas and are the control signals between outer 2 and relay. Based on Figure 7, the rate-equations describing the three units coupled in relay are given by the following.

Outer 1 oscillator:

Outer 2 oscillator:and relay oscillator:where are the controller law defined as follows:

The coupling strength is updated with the law as follows:

The global error states are defined as follows:

After some mathematical simplifications, the dynamical error system is given bywhere and .

Assumption. . Since systems (9) to (11) exhibit chaotic behavior, the nonlinear functions and are bounded, so we can set and where and are positive parameters.

Theorem. . Systems (9) to (11) achieve the global synchronization under the controller and the update law (13) of the coupling strength if the following condition is satisfied:

Proof. The candidate Lyapunov function is chosen:The derivation of this Lyapunov function along the trajectories of system (15) givesBy using the above assumption, equation (18) becomesFrom equation (19), one can conclude that systems (9) to (11) achieve global synchronization if and only if . This completes the proof.
Figure 8 depicts the global errors of synchronization between the three coupled oscillators (Figures 8(a)–8(c)) and the update law of coupling strength (Figure 8(d)). The initial condition of the adaptive law is . The initial conditions of the coupled systems are set at values other than the equilibrium points. The synchronization between the three coupled oscillators is shown in Figure 9 which plots the times series of outer1 oscillator (in red) and those of outer 2 oscillator (in blue).
Figure 8 shows that the global errors described by equation (15) converge at zero from ; at this time, the updated law stabilizes around a value . It is noted that this result is not enough to guarantee the synchronization between the outer 1 and outer 2 because does not necessarily lead to and , i.e., . The time series of the states variables of outer 1 and outer 2 is illustrated in Figure 9.
From Figure 9, when , the outer 1 oscillator (in red) and outer 2 oscillator (in blue) converge to the same dynamic. This last result confirms the global synchronization between the three coupled oscillators outer 1, outer 2, and relay. By comparison with results found in the litterature on relay coupled oscillators [58–64], this result is more interesting because the coupling strength is not manual but it is adapted as a function of the changes in the environment [65].

(a)

(b)

(c)

(d)

5. Application to Steganography and Chaos Encryption

The flowchart of steganography and chaos encryptions is depicted in Figure 10.

A cover image and text file secret image are used in Figure 10. The text file secret message is firstly encrypted by using an affine cipher based on adaptive synchronization between the relay coupled three three-dimensional autonomous systems with a parabolic equilibrium with the support of the date of birth (DOB) keys. The DOB keys enable to construct a key by using birth day, month, and year of the sender (S) and receiver (R), respectively. Then, the least significant bit (LSB) algorithm is applied to hide the encrypted text file secret message in the cover image file by embedding the encrypted text file in the LSB of pixel values of the cover image. The color image considered is decomposed into 3 subimages component (red, green, and blue). Each pixel of components assumes a value between [0, 255] and represented with 8 bit. The LSB of some pixels of components is replaced by each bits of the text file secret message.

The encryption E(.) and decryption D(.) processes of affine cipher for a given text file secret message M are expressed [66] as follows:where l and m are parameters of affine cypher key k (k=(l, m)) and l’ is inverse of l modulo p and p is a positive integer. To exchange the text file secret message M, S and R have to generate their own secret key pair by using the DOB and three-dimensional chaotic autonomous system with a parabolic equilibrium. The DOB of S is , and the DOB of R is where is a pair of the DOB of S and R.

5.1. S and R Key Generation

(i)S solves the outer 1 of system (9) at time t and generates equations (21) and (22): with and .(ii)R solves the outer 2 of system (10) at time t and generates equations (23) and (24):

5.2. Encryption and Decryption Message

(i)S sends the text file secret message M to R secretly, and it encrypts M using E(.) function as follows:(ii)When the R receives the text file secret message M from S and recovers an original message M, it uses the decryption function D(.):

Figure 11 presents the 2 covers images with size [512 × 512] used to hide the secret message encrypted and the stegano images. The secret message is chosen as M = MY NAME IS STEGANO. By applying the encryption E(.) to the message M with the following parameters: the encrypted message obtained is M’ = 刮儇⸊ܖ”ᘣੜ瑑.

(a)

(b)

From Figure 11, it is noted that the flower and Lena cover images have the same visual aspect with flower and Lena stegano images. Thanks to the Peak Signal to Noise Ratio (PSNR), the difference between cover and stegano images is expressed [67, 68] as follows:where MSE is the mean squared error. By using the chosen text file secret message M, the PSNR of Lena and flower is 59.28 dB and 62.01 dB, respectively. Figure 12 shows the evolution of PSNR between cover and stegano images versus the length of the text file secret message M.

In Figure 12, the PSNR increases with the increase in the message length and it reaches a maximum at the message length of 150. By further increasing the message length, the PSNR decreases slowly. These results unveil the practicability and superiority of the proposed steganography and chaos encryption algorithms. Note that, implementing such an encryption system may also be designed with a discrete chaotic/hyperchaotic card [69].

6. Conclusion

This paper was devoted to the dynamical analysis, adaptive synchronization via relay coupling, and applications to steganography and chaos encryption of a three-dimensional autonomous system with a parabolic equilibrium. The dynamical behaviors of the three-dimensional chaotic autonomous system with a parabolic equilibrium were analysed both analytically and numerically, and it was found that the proposed system can generate chaotic oscillations and chaotic bursting oscillations. Then, an analog circuit was designed to realize the differential equations of the chaotic system under study. The designed circuit was implemented and tested using the OrCAD-PSpice software to verify the numerical simulations results. Comparison of the results obtained from the analog circuit and numerical simulations showed good qualitative agreement. Furthermore, it was demonstrated analytically and numerically that it is possible to achieve a global synchronization between a relay coupled three three-dimensional chaotic autonomous systems with parabolic equilibrium through an adaptive synchronization. Finally, a text message hidden inside an image was successfully realized by using steganography and chaos encryption.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially funded by the Center for Nonlinear Systems, Chennai Institute of Technology, India, via funding number CIT/CNS/2021/RD/064.

References

J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific, Singapore, 2010.
G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Localization of hidden Chuaʼs attractors,” Physics Letters A, vol. 375, no. 23, pp. 2230–2233, 2011.
View at: Publisher Site | Google Scholar
G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Hidden attractor in smooth Chua systems,” Physica D: Nonlinear Phenomena, vol. 241, no. 18, pp. 1482–1486, 2012.
View at: Publisher Site | Google Scholar
G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems. From hidden oscillations in hilbert-Kolmogorov, aizerman, and kalman problems to hidden chaotic attractor in Chua circuits,” International Journal of Bifurcation and Chaos, vol. 23, no. 01, Article ID 1330002, 2013.
View at: Publisher Site | Google Scholar
G. A. Leonov, N. V. Kuznetsov, M. A. Kiseleva, E. P. Solovyeva, and A. M. Zaretskiy, “Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 277–288, 2014.
View at: Publisher Site | Google Scholar
G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Communications in Nonlinear Science and Numerical Simulation, vol. 28, no. 1-3, pp. 166–174, 2015.
View at: Publisher Site | Google Scholar
B. A. Mezatio, M. Motchongom Tingue, R. Kengne, A. Tchagna Kouanou, T. Fozin Fonzin, and R. Tchitnga, “Complex dynamics from a novel memristive 6D hyperchaotic autonomous system,” International Journal of Dynamics and Control, vol. 8, no. 1, pp. 70–90, 2020.
View at: Publisher Site | Google Scholar
B. A. Mezatio, M. T. Motchongom, B. R. Wafo Tekam, R. Kengne, R. Tchitnga, and A. Fomethe, “A novel memristive 6D hyperchaotic autonomous system with hidden extreme multistability,” Chaos, Solitons & Fractals, vol. 120, pp. 100–115, 2019.
View at: Publisher Site | Google Scholar
D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Physics Reports, vol. 637, pp. 1–50, 2016.
View at: Publisher Site | Google Scholar
C. Li, Y. Peng, Z. Tao, J. C. Sprott, and S. Jafari, “Coexisting infinite equilibria and chaos,” International Journal of Bifurcation and Chaos, vol. 31, no. 05, Article ID 2130014, 2021.
View at: Publisher Site | Google Scholar
M. Molaie, S. Jafari, J. C. Sprott, and S. M. R. H. Golpayegani, “Simple chaotic flows with one stable equilibrium,” International Journal of Bifurcation and Chaos, vol. 23, no. 11, Article ID 1350188, 2013.
View at: Publisher Site | Google Scholar
S. Jafari, J. C. Sprott, V.-T. Pham, S. M. R. H. Golpayegani, and A. H. Jafari, “A new cost function for parameter estimation of chaotic systems using return maps as fingerprints,” International Journal of Bifurcation and Chaos, vol. 24, no. 10, Article ID 1450134, 2014.
View at: Publisher Site | Google Scholar
S. T. Kingni, S. Jafari, H. Simo, and P. Woafo, “Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form,” The European Physical Journal Plus, vol. 129, pp. 1–16, 2014.
View at: Publisher Site | Google Scholar
S.-K. Lao, Y. Shekofteh, S. Jafari, and J. C. Sprott, “Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor,” International Journal of Bifurcation and Chaos, vol. 24, no. 01, Article ID 1450010, 2014.
View at: Publisher Site | Google Scholar
V.-T. Pham, S. Jafari, C. Volos, X. Wang, and S. M. R. H. Golpayegani, “Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria,” International Journal of Bifurcation and Chaos, vol. 24, no. 11, Article ID 1450146, 2014.
View at: Publisher Site | Google Scholar
V.-T. Pham, C. Volos, S. Jafari, and X. Wang, “Generating a novel hyperchaotic system out of equilibrium,” Optoelectronics and Advanced Materials-Rapid Communications, vol. 8, pp. 535–539, 2014.
View at: Google Scholar
S. Jafari, J. C. Sprott, and F. Nazarimehr, “Recent new examples of hidden attractors,” The European Physical Journal - Special Topics, vol. 224, no. 8, pp. 1469–1476, 2015.
View at: Publisher Site | Google Scholar
M. Shahzad, V.-T. Pham, M. A. Ahmad, S. Jafari, and F. Hadaeghi, “Synchronization and circuit design of a chaotic system with coexisting hidden attractors,” The European Physical Journal - Special Topics, vol. 224, no. 8, pp. 1637–1652, 2015.
View at: Publisher Site | Google Scholar
J. C. Sprott, S. Jafari, V.-T. Pham, and Z. S. Hosseini, “A chaotic system with a single unstable node,” Physics Letters A, vol. 379, no. 36, pp. 2030–2036, 2015.
View at: Publisher Site | Google Scholar
S. Jafari, V.-T. Pham, S. M. R. H. Golpayegani, M. Moghtadaei, and S. T. Kingni, “The relationship between chaotic maps and some chaotic systems with hidden attractors,” International Journal of Bifurcation and Chaos, vol. 26, no. 13, Article ID 1650211, 2016.
View at: Publisher Site | Google Scholar
S. Jafari, V.-T. Pham, and T. Kapitaniak, “Multiscroll chaotic sea obtained from a simple 3D system without equilibrium,” International Journal of Bifurcation and Chaos, vol. 26, no. 02, Article ID 1650031, 2016.
View at: Publisher Site | Google Scholar
S. Jafari, J. C. Sprott, and M. Molaie, “A simple chaotic flow with a plane of equilibria,” International Journal of Bifurcation and Chaos, vol. 26, no. 06, Article ID 1650098, 2016.
View at: Publisher Site | Google Scholar
V.-T. Pham, S. Jafari, X. Wang, and J. Ma, “A chaotic system with different shapes of equilibria,” International Journal of Bifurcation and Chaos, vol. 26, no. 04, Article ID 1650069, 2016.
View at: Publisher Site | Google Scholar
Z. Wei, “Dynamical behaviors of a chaotic system with no equilibria,” Physics Letters A, vol. 376, no. 2, pp. 102–108, 2011.
View at: Publisher Site | Google Scholar
X. Wang and G. Chen, “Constructing a chaotic system with any number of equilibria,” Nonlinear Dynamics, vol. 71, no. 3, pp. 429–436, 2013.
View at: Publisher Site | Google Scholar
S. T. Kingni, V.-T. Pham, S. Jafari, and P. Woafo, “A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form,” Chaos, Solitons & Fractals, vol. 99, pp. 209–218, 2017.
View at: Publisher Site | Google Scholar
V.-T. Pham, S. Jafari, and T. Kapitaniak, “Constructing a chaotic system with an infinite number of equilibrium points,” International Journal of Bifurcation and Chaos, vol. 26, no. 13, Article ID 1650225, 2016.
View at: Publisher Site | Google Scholar
V.-T. Pham, C. Volos, S. Jafari, and T. Kapitaniak, “A novel cubic-equilibrium chaotic system with coexisting hidden attractors: analysis, and circuit implementation,” Journal of Circuits, Systems, and Computers, vol. 27, no. 04, Article ID 1850066, 2018.
View at: Publisher Site | Google Scholar
V.-T. Pham, F. Rahma, M. Frasca, and L. Fortuna, “Dynamics and synchronization of a novel hyperchaotic system without equilibrium,” International Journal of Bifurcation and Chaos, vol. 24, no. 06, Article ID 1450087, 2014.
View at: Publisher Site | Google Scholar
V. T. Pham, S. Jafari, C. Volos, and T. Kapitaniak, “A gallery of chaotic systems with an infinite number of equilibrium points,” Chaos, Solitons & Fractals, vol. 93, pp. 58–63, 2016.
View at: Publisher Site | Google Scholar
X. Wang, A. Akgul, S. Cicek, V.-T. Pham, and D. V. Hoang, “A chaotic system with two stable equilibrium points: dynamics, circuit realization and communication application,” International Journal of Bifurcation and Chaos, vol. 27, no. 08, Article ID 1750130, 2017.
View at: Publisher Site | Google Scholar
A. T. Azar, “A novel chaotic system without equilibrium: dynamics, synchronization, and circuit realization,” Complexity, Hindawi Publishing Corporation, vol. 2017, Article ID 7871467, p. 11, 2017.
View at: Publisher Site | Google Scholar
J. P. Singh and B. K. Roy, “The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour,” Nonlinear Dynamics, vol. 89, no. 3, pp. 1845–1862, 2017.
View at: Publisher Site | Google Scholar
E. Chen, L. Min, and G. Chen, “Discrete chaotic systems with one-line equilibria and their application to image encryption,” International Journal of Bifurcation and Chaos, vol. 27, no. 03, Article ID 1750046, 2017.
View at: Publisher Site | Google Scholar
S. Jafari and J. C. Sprott, “Simple chaotic flows with a line equilibrium,” Chaos, Solitons & Fractals, vol. 57, pp. 79–84, 2013.
View at: Publisher Site | Google Scholar
C. Li, J. C. Sprott, and W. Thio, “Bistability in a hyperchaotic system with a line equilibrium,” Journal of Experimental and Theoretical Physics, vol. 118, no. 3, pp. 494–500, 2014.
View at: Publisher Site | Google Scholar
J. Ma, Z. Chen, Z. Wang, and Q. Zhang, “A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium,” Nonlinear Dynamics, vol. 81, no. 3, pp. 1275–1288, 2015.
View at: Publisher Site | Google Scholar
X. Wang, V.-T. Pham, and C. Volos, “Dynamics, circuit design, and synchronization of a new chaotic system with closed curve equilibrium,” Complexity, Hindawi Publishing Corporation, vol. 2017, Article ID 7138971, p. 9, 2017.
View at: Publisher Site | Google Scholar
T. Gotthans and J. Petržela, “New class of chaotic systems with circular equilibrium,” Nonlinear Dynamics, vol. 81, no. 3, pp. 1143–1149, 2015.
View at: Publisher Site | Google Scholar
T. Gotthans, J. C. Sprott, and J. Petrzela, “Simple chaotic flow with circle and square equilibrium,” International Journal of Bifurcation and Chaos, vol. 26, no. 08, Article ID 1650137, 2016.
View at: Publisher Site | Google Scholar
S. Mobayen, C. K. Volos, S. Kaçar, and Ü. Çavuşoğlu, “New class of chaotic systems with equilibrium points like a three-leaved clover,” Nonlinear Dynamics, vol. 91, no. 2, pp. 939–956, 2018.
View at: Publisher Site | Google Scholar
V.-T. Pham, S. Jafari, C. Volos, S. Vaidyanathan, and T. Kapitaniak, “A chaotic system with infinite equilibria located on a piecewise linear curve,” Optik, vol. 127, no. 20, pp. 9111–9117, 2016.
View at: Publisher Site | Google Scholar
Y. Chen and Q. Yang, “A new Lorenz-type hyperchaotic system with a curve of equilibria,” Mathematics and Computers in Simulation, vol. 112, pp. 40–55, 2015.
View at: Publisher Site | Google Scholar
E. Tlelo-Cuautle, L. G. de la Fraga, V.-T. Pham, C. Volos, S. Jafari, and A. d. J. Quintas-Valles, “Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points,” Nonlinear Dynamics, vol. 89, no. 2, pp. 1129–1139, 2017.
View at: Publisher Site | Google Scholar
V.-T. Pham, C. Volos, S. Vaidyanathan, and X. Wang, “A Chaotic system with an infinite number of equilibrium points: dynamics, horseshoe, and synchronization,” Adv. Math. Phys., vol. 2016, Article ID 4024836, p. 8, 2016.
View at: Publisher Site | Google Scholar
K. Barati, S. Jafari, J. C. Sprott, and V.-T. Pham, “Simple chaotic flows with a curve of equilibria,” International Journal of Bifurcation and Chaos, vol. 26, no. 12, Article ID 1630034, 2016.
View at: Publisher Site | Google Scholar
Y. Ji and Q. Bi, “Bursting behavior in a non-smooth electric circuit,” Physics Letters A, vol. 374, no. 13-14, pp. 1434–1439, 2010.
View at: Publisher Site | Google Scholar
X. Han, B. Jiang, and Q. Bi, “3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation,” Nonlinear Dynamics, vol. 61, no. 4, pp. 667–676, 2010.
View at: Publisher Site | Google Scholar
Q. Bi and Z. Zhang, “Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales,” Physics Letters A, vol. 375, no. 8, pp. 1183–1190, 2011.
View at: Publisher Site | Google Scholar
S. T. Kingni, B. Nana, G. S. Mbouna Ngueuteu, P. Woafo, and J. Danckaert, “Bursting oscillations in a 3D system with asymmetrically distributed equilibria: mechanism, electronic implementation and fractional derivation effect,” Chaos, Solitons & Fractals, vol. 71, pp. 29–40, 2015.
View at: Publisher Site | Google Scholar
S. T. Kingni, L. Keuninckx, P. Woafo, G. Van der Sande, and J. Danckaert, “Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 1111–1123, 2013.
View at: Publisher Site | Google Scholar
S. T. Kingni, G. S. Mbouna Ngueuteu, and P. Woafo, “Bursting generation mechanism in a three-dimensional autonomous system, chaos control and synchronization in its fractional-order form,” Nonlinear Dynamics, vol. 76, pp. 1169–1183, 2010.
View at: Google Scholar
J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos, vol. 16, pp. 775–858, 2006.
View at: Google Scholar
S. Yu, W. K. S. Tang, J. Lü, and G. Chen, “Design and implementation of multi-wing butterfly chaotic attractors via Lorenz-type systems,” International Journal of Bifurcation and Chaos, vol. 20, no. 01, pp. 29–41, 2010.
View at: Publisher Site | Google Scholar
S. Yu, J. Lu, X. Yu, and G. Chen, “Design and implementation of grid multiwing hyperchaotic Lorenz system family via switching control and constructing super-heteroclinic loops,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 59, no. 5, pp. 1015–1028, 2012.
View at: Publisher Site | Google Scholar
J. Wu, C. Li, X. Ma, T. Lei, and G. Chen, “Simplification of chaotic circuits with quadratic nonlinearity,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 69, 2021.
View at: Google Scholar
R. Kengne, M. Motchongom Tingue, A. Kammogne Souop Tewa, G. Djuidjé Kenmoé, and T. C. Kofane, “Optimal phase control in a Remoissenet–Peyrard substrate potential: numerical and analogical investigations,” Indian Journal of Physics, 2022.
View at: Google Scholar
I. Fischer, R. Vicente, J. M. Buldú et al., “Zero-lag long-range synchronization via dynamical relaying,” Physical Review Letters, vol. 97, no. 12, Article ID 123902, 2006.
View at: Publisher Site | Google Scholar
R. Gutiérrez, R. Sevilla-Escoboza, P. Piedrahita et al., “Generalized synchronization in relay systems with instantaneous coupling,” Physical review. E, Statistical, nonlinear, and soft matter physics, vol. 88, Article ID 052908, 2013.
View at: Publisher Site | Google Scholar
A. Sharma, M. D. Shrimali, A. Prasad, R. Ramaswamy, and U. Feudel, “Phase-flip transition in relay-coupled nonlinear oscillators,” Physical review. E, Statistical, nonlinear, and soft matter physics, vol. 84, Article ID 016226, 2011.
View at: Publisher Site | Google Scholar
I. G. Da Silva, J. M. Buldú, C. R. Mirasso, and J. García-Ojalvo, “Synchronization by dynamical relaying in electronic circuit arrays,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 16, no. 4, Article ID 043113, 2006.
View at: Publisher Site | Google Scholar
B. Nana and P. Woafo, “Chaotic masking of communication in an emitter-relay-receiver electronic setup,” Nonlinear Dynamics, vol. 82, no. 1-2, pp. 899–908, 2015.
View at: Publisher Site | Google Scholar
R. Kengne, R. Tchitnga, S. Mabekou, B. R. W. Tekam, G. B. Soh, and A. Fomethe, “On the relay coupling of three fractional-order oscillators with time-delay consideration: global and cluster synchronizations,” Chaos, Solitons & Fractals, vol. 111, pp. 6–17, 2018.
View at: Publisher Site | Google Scholar
M. Motchongom Tingue, H. L. Ndassi, A. R. Tchamda, E. R. Mache Kengne, R. Tchitnga, and M. Tchoffo, “Bursting mechanism in a memristive Lorenz based system and function projective synchronization in its-fractional-order form: digital implementation under ATmega328P microcontroller,” Physica Scripta, vol. 96, no. 12, Article ID 125229, 2021.
View at: Publisher Site | Google Scholar
R. Kengne, R. Tchitnga, A. Mezatio, A. Fomethe, and G. Litak, “Finite-time synchronization of fractional-order simplest two-component chaotic oscillators,” The European Physical Journal B, vol. 90, no. 5, p. 88, 2017.
View at: Publisher Site | Google Scholar
P. MuthukumarP. BalasubramaniamK. Ratnavelu, P. Balasubramaniam, and K. Ratnavelu, “Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB),” Nonlinear Dyn, Springer, vol. 80, no. 4, pp. 1883–1897, 2015.
View at: Publisher Site | Google Scholar
A. Saleema and T. Amarunnishad, “A new steganography algorithm using hybrid fuzzy neural networks,” Procedia Technology, vol. 24, pp. 1566–1574, 2016.
View at: Publisher Site | Google Scholar
N. H. Alombah, A. E. T. Tchendjeu, A. E. T. Tchendjeu, K. Romanic, F. C. Talla, and H. B. Fotsin, “FPGA implementation of a novel two-internal-state memristor and its two component chaotic circuit,” Indian Journal of Science and Technology, vol. 14, no. 27, pp. 2257–2271, 2021.
View at: Publisher Site | Google Scholar
Y. Li, C. Li, S. Zhang, G. R. Chen, and Z. Zeng, “A self-reproduction hyperchaotic map with compound lattice dynamics,” IEEE Transactions on Industrial Electronics, 2022.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Janarthanan Ramadoss et al. 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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