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

Authors: Janarthanan Ramadoss; Romanic Kengne; Dianorré Tokoue Ngatcha; Victor Kamdoum Tamba; Karthikeyan Rajagopal; Marceline Motchongom Tingue
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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 Section2, 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:(1)dxdt=−z,dydt=xz2,dzdt=x−y2+zay2−z2,where x,y,andz are state variables, t is the time, and a is a positive parameter. System (1) has only one parameter a. System (1) has a parabolic equilibrium given by(2)E=x,y,z∈R3x=y∗2,y=y∗,z=0.The characteristic equation of system (1) evaluated at the parabolic equilibrium is(3)λλ2−ay∗2λ+1=0.The roots of equation (3) depend on the sign of a2y∗4−4. If a2y∗4−4>0, the roots of equation (3) are λ1=0 and λ2,3=ay∗2±a2y∗4−4/2. The characteristic equations have at least a positive real root; therefore, E is unstable. For a2y∗4−4<0, the roots of equation (3) are λ1=0 and λ2,3=ay∗2±ia2y∗4−4/2. Since a>0, equation (3) has a pair of complex conjugate eigenvalues with positive real root, so E is unstable. Thus, E is always unstable for a>0.To investigate the dynamical behaviors of system (1), Lyapunov exponents (LEs) and bifurcation diagram depicting maxima of xt versus the parameter a for the initial conditions (x(0), y(0), z(0)) = (0, 0.1, 0.2) are plotted in Figure 1.Figure 1
(a) Local maxima of (x) showing a typical reverse period-doubling route to chaos and (b) LE.
(a)(b)By increasing the parametera 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 xt, the chaotic region can be divided into two regions: chaotic bursting oscillations for 3.0≤a≤3.127 and chaotic oscillations for 3.127<a<3.198. 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 xt, yt, and zt and the corresponding phase portraits for specific values of parameter a.Figure 2
(a) Time series ofxt, yt, and zt, while the (b) phase portraits in the different planes for a = 3.In the left panel of Figure2, 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 3.127<a<3.198 are depicted in Figure 3.Figure 3
(a) Time series ofxt, yt, and zt while the (b) phase portraits in the different planes for a = 3.16.In the time domain, Figure3(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 z=0 for a=3.0 is shown in Figure 4.Figure 4
Cross section of the basins of attraction of the two attractors in thexy-plane at z = 0. Initial conditions in the white region lead to unbounded orbits, those in the red region are exactly on the curve equilibrium (it can be seen that there is no basin for the curve of equilibria which proves that the equilibrium is unstable), and those in the light blue region lead to the strange attractor shown in cross section as black dots.In Figure4, 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 variablesx, y, and z of system (1) are rescaled to overcome the difficulties in realization [53–55]. Therefore, system (1) is rewritten as(4)dXdt=−Z,dYdt=XZ2102,dZdt=X−Y210+aY2Z102−Z3102,where 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].Figure 5
Schematic diagram of the designed circuit for system (4).The circuit of Figure5 consists of resistors, operational amplifiers, analogue multipliers, and capacitors. From Figure 5, the circuital equations are derived as(5)dVXdt=−1R1C1VZ,dVYdt=1R2C2km2VXVZ2,dVZdt=1R3C3VX−1R4C3kmVY2+1R5C3km2VY2VZ−1R6C3km2VZ3,where VX, VY, and VZ are the output voltages of the operational amplifiers (see Figure 5). The fixed constant of the multipliers is denoted as km, and km = 10 V. Normalizing circuital equations (5) by using the dimensionless states variables(6)X=VX1.0V,Y=VY1.0V,Z=VZ1.0V,t′=τt=100μs⋅t,and inserting equations (6) in (5), the following system is obtained:(7)dXdt′=τC1−1R1Z,dYdt′=τC21R2km2XZ2,dZdt′=τC31R3X−1R4kmY2+1R5km2Y2Z−1R6km2Z3.Obviously, system (7) is equivalent to system (4) with the following conditions:(8)τR1C1=τR2C2=τR3C3=τR4C3=τR6C3=1,τR5C3=a.The resistorR5 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.Figure 6
Phase portraits of the designed circuit obtained by using OrCAD-PSpice for (a) chaotic bursting oscillations and (b) chaotic oscillations.
(a)(b)There is a good agreement between the numerical simulations and OrCAD-PSpice results when comparing the phase portraits of Figures2 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 Figure7.Figure 7
Schematic diagram of the relay coupling of the three three-dimensional autonomous systems with a parabolic equilibrium.In Figure7, outer 1 is defined by the state variables xm,ym,zm, outer 2 by the state variables xs,ys,zs, and relay by the state variables xr,yr,zr. The variables u1 and u2 are the control signals between outer 1 and relay whereas u3 and u4 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:(9)x˙m=−zmy˙m=xmzm2+u1z˙m=xm−ym2+zmaym2−zm2+u2.Outer 2 oscillator:(10)x˙s=−zsy˙s=xszs2+u3z˙s=xs−ys2+zsays2−zs2+u4,and relay oscillator:(11)x˙r=−zr,y˙r=xrzr2+u5,z˙r=xr−yr2+zrayr2−zr2+u6,where uii=1,…,6 are the controller law defined as follows:(12)u1=−ktym−yru2=−ktzm−zru3=−ktys−yru4=−ktzs−zru5=−u1+u3=−kt2yr−ym−ys,u6=−u2+u4=−kt2zr−zm−zs.The coupling strengthkt is updated with the law as follows:(13)k˙t=γ2yr−ym−ys2+2zr−zm−zs2The global error states are defined as follows:(14)e1t=xm+xs−2xr,e2t=ym+ys−2yr,e3t=zm+zs−2zr.After some mathematical simplifications, the dynamical error system is given by(15)e˙1t=−e3,e˙2t=fx,y,z−3kte2,e˙3t=e1+gx,y,z−3kte3,k˙t=γe22+e32,where fx,y,z=xmzm2+xszs2−2xrzr2 and gx,y,z=−ym2+ys2−2yr2+zmaym2−zm2+zsays2−zs2−2zrayr2−zr2.Assumption.
. Since systems (9) to (11) exhibit chaotic behavior, the nonlinear functions fx,y,z and gx,y,z are bounded, so we can set fx,y,z≤η1e2 and gx,y,z≤η2e3 where η1 and η2 are positive parameters.Theorem.
. Systems (9) to (11) achieve the global synchronization under the controller uii=1,…,6 and the update law (13) of the coupling strength if the following condition is satisfied:(16)k¯≥maxη1,η23.Proof.
The candidate Lyapunov function is chosen:(17)V=0.5e12+e22+e32+32γkt−k¯2.
The derivation of this Lyapunov function along the trajectories of system (15) gives(18)V˙=e1e˙1+e2e˙2+e3e˙3+3k˙tγkt−k¯=fx,y,ze2+gx,y,ze3−3k¯e22+e32.
By using the above assumption, equation (18) becomes(19)V˙≤−3k¯−η1e22−3k¯−η2e32.
From equation (19), one can conclude that systems (9) to (11) achieve global synchronization if and only if k¯≥maxη1,η2/3. This completes the proof.
Figure8 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 k0=0. 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).
Figure8 shows that the global errors described by equation (15) converge at zero from t=450; at this time, the updated law stabilizes around a value k¯=0.097. It is noted that this result is not enough to guarantee the synchronization between the outer 1 and outer 2 because X+Y−2Z=0 does not necessarily lead to X=Z and Y=Z, i.e., X=Y. The time series of the states variables of outer 1 and outer 2 is illustrated in Figure 9.
From Figure9, when t=450, 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].Figure 8
Global synchronization errors (a) to (c) and adaptive coupling strength (d) forγ=0.035 and a=3. The initial conditions are 0,0.1,0.2, 0,0.15,0.25, and 0.3,0.2,0.4, respectively, for the outer 1, outer 2, and relay oscillators.
(a)(b)(c)(d)Figure 9
Time series of the state variables of outer 1 and outer 2 forγ=0.035 and a=3. The initial conditions are 0,0.1,0.2, 0,0.15,0.25, and 0.3,0.2,0.4, respectively, for the outer 1, outer 2, and relay oscillators.


## 5. Application to Steganography and Chaos Encryption

The flowchart of steganography and chaos encryptions is depicted in Figure10.Figure 10
Schematic diagram of the proposed steganography and chaos encryption process.A cover image and text file secret image are used in Figure10. 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 encryptionE(.) and decryption D(.) processes of affine cipher for a given text file secret message M are expressed [66] as follows:(20)EM=l×M+mmodp,DM=l′×EM−mmodp,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 D1=DD−MM−YY, and the DOB of R is D2=dd−mm−yy where D=D1,D2 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):(21)K1=DD×xmt+MM×ymt+YYi×zmtmodp,(22)K2=dd×xmt+mm×ymt+yyi×zmtmodp,withi=1,…,4 and j=3,…,6.(ii)
R solves the outer 2 of system (10) at time t and generates equations (23) and (24):(23)L1=DD×xst+MM×yst+YYi×zstmodp,(24)L2=dd×xst+mm×yst+yyi×zstmodp.


### 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:(25)EM=M×K1+K2modp.(ii)
When theR receives the text file secret message M from S and recovers an original message M, it uses the decryption function D(.):(26)DM=EM−L2×L1−1modp.Figure11 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: DD=10;MM=06;YY=1,9,9,0;dd=22;MM=10;YY=1,9,8,4;p=128;xm=−24.67;ym=−75.42;zm=−4.381;xs=−24.67;ys=−75.42;zs=−4.381, the encrypted message obtained is M’ = 刮儇⸊ܖ”ᘣੜ瑑.Figure 11
(a) Cover and (b) stegano images.
(a)(b)From Figure11, 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:(27)PSNR=10log102552MSE,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.Figure 12
Variation of the PSNR of Lena and flower versus the length of messageM.In Figure12, 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].


## 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):(21)K1=DD×xmt+MM×ymt+YYi×zmtmodp,(22)K2=dd×xmt+mm×ymt+yyi×zmtmodp,withi=1,…,4 and j=3,…,6.(ii)
R solves the outer 2 of system (10) at time t and generates equations (23) and (24):(23)L1=DD×xst+MM×yst+YYi×zstmodp,(24)L2=dd×xst+mm×yst+yyi×zstmodp.


## 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:(25)EM=M×K1+K2modp.(ii)
When theR receives the text file secret message M from S and recovers an original message M, it uses the decryption function D(.):(26)DM=EM−L2×L1−1modp.Figure11 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: DD=10;MM=06;YY=1,9,9,0;dd=22;MM=10;YY=1,9,8,4;p=128;xm=−24.67;ym=−75.42;zm=−4.381;xs=−24.67;ys=−75.42;zs=−4.381, the encrypted message obtained is M’ = 刮儇⸊ܖ”ᘣੜ瑑.Figure 11
(a) Cover and (b) stegano images.
(a)(b)From Figure11, 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:(27)PSNR=10log102552MSE,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.Figure 12
Variation of the PSNR of Lena and flower versus the length of messageM.In Figure12, 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.



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