# A New Hyperchaotic Two-Scroll System: Bifurcation Study, Multistability, Circuit Simulation, and FPGA Realization

Authors: Sundarapandian Vaidyanathan; Aceng Sambas; Esteban Tlelo-Cuautle; Ciro Fabian Bermudez-Marquez; Khaled Benkouider; Samy Abdelwahab Safaan
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

With the swift advancement of chaos theory, the modeling, chaotic oscillations, and engineering applications of chaotic and hyperchaotic systems are important topics in research. In this research paper, we elucidate our findings of a new four-dimensional two-scroll hyperchaotic system having only two quadratic nonlinearities and carry out a detailed bifurcation study of the proposed dynamical model. Also, an electronic circuit has been constructed for the new system using MultiSim (Version 14). The implementation of the new 4-D hyperchaotic system in a field-programmable gate array (FPGA) is performed herein by applying two numerical methods,viz. Forward Euler Method and Trapezoidal Method. The experimental results show a good match with the simulated hyperchaotic attractors. We also provide details of the hardware resources used for an FPGA Basys 3 Xilinx Artix-7 XC7A35T-ICPG236C.


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## Body

## 1. Introduction

Due to the rapid advancement of chaos theory, chaotic and hyperchaotic systems are applicable in several engineering branches such as memristive circuits [1–4], chemical oscillators [5–7], information sensing systems [8–10], lasers [11, 12], and communication systems [13–16]. Xiu et al. [17] designed a 5-D hyperchaotic system with hardware circuit design developed from a memristive cellular neural network (CNN) and applied it to build a secure communication system. Nazir et al. [18] proposed a secure communication system for encryption based on a 4-D hyperchaotic system and genetic codes. Bian and Yu [19] showed the use of a 6-D hyperchaotic system to enhance the security of a communication system. Boumaraf and Merazka [20] demonstrated the use of a new 4-D hyperchaotic cryptosystem for encryption applications.Modeling of two-scroll hyperchaotic systems has attracted good research in the literature [21–23]. This research work reports the findings of a new four-dimensional nonlinear dynamical system having two quadratic nonlinear terms and depicting a two-scroll hyperchaotic attractor. Since the maximal Lyapunov exponent (MLE) of the two-scroll system is τmax=2.6174, we deduce that the two-scroll system (1) has high complexity. By a rigorous mathematical analysis, it is shown that the proposed mathematical model has a unique balance point at the origin. This is followed by an extensive bifurcation analysis of the new 4-D two-scroll system reported in this work. We study the changes in the qualitative behavior of the proposed 4-D two-scroll system with respect to changes in the values of the system parameters by means of a bifurcation analysis. Bifurcation analysis for systems exhibiting chaos or hyperchaos is useful to get illuminated on the various qualitative properties like oscillations, quasi-periodicity, chaoticity, and hyperchaoticity when the parameters take various values in specified intervals [24–26]. Our research study also shows that the new hyperchaotic system exhibits multistability, which is a special property of coexistence of attractors for a selected set of values for the parameters but differing sets of values for the initial data of the trajectories [27, 28].Circuit implementations of chaotic and hyperchaotic systems are of great usability in practice due to direct applications in many engineering disciplines [3, 29, 30]. An electronic model of the proposed hyperchaotic two-scroll system has been envisioned using MultiSim. Embedded and nonembedded implementations of chaotic and hyperchaotic systems are of paramount importance to increase their applications in many scientific and engineering fields since those FPGA designs can be linked directly to the digital world [31–35]. FPGA implementation of the proposed hyperchaotic two-scroll system has also been carried out at the end of this work by applying two numerical methods, viz. Forward Euler Method and Trapezoidal Method. Experimental implementation using FPGA facilitates practical applications with the new hyperchaotic model.


## 2. A New Hyperchaotic System with Two-Scroll Attractor

A two-scroll attractor of a new hyperchaotic system is the main contribution of the modelling part of this research paper. Our new system is the following 4-D dynamics given by(1)z˙=fz,wherefz=az2−z1−z3+z4cz1−z2−z1z3+z4−bz3+z1z2−dz2.In the 4-D dynamics (1),z=z1,z2,z3,z4 and a,b,c,d are positive system constants. We note that the vector field fz has nine linear terms and two quadratic nonlinear terms.When we take the constants asa,b,c,d=25,8,90,6 and the initial data as z0=0.8,0.2,0.4,0.6, the Lyapunov characteristic exponents of the model (1) can be estimated using MATLAB for T=1E4 seconds as follows:(2)τ1=2.6174,τ2=0.2743,τ3=0,τ4=−36.8194.We note that the sum of the Lyapunov characteristic exponents of the model (1) is a negative quantity,viz.(3)τ1+τ2+τ3+τ4=−33.9277.This calculation demonstrates that the 4-D system stated in the equation (1) has a dissipative hyperchaotic attractor with two positive Lyapunov characteristic exponents.IfVt denotes a volume element of the flow of the 4-D hyperchaotic system (1), then the divergence of the flow of V is calculated as follows:(4)∇⋅V=∂f1∂z1+∂f2∂z2+∂f3∂z3+∂f4∂z4=−a−b−1.This also shows that 4-D hyperchaotic system (1) is dissipative for all positive values of the system constantsa and b. Since the maximal Lyapunov exponent (MLE) of the two-scroll system is τmax=2.6174, we deduce that the two-scroll system (1) has high complexity.Figure1 shows various MATLAB signal portraits of the 4-D hyperchaotic two-scroll dynamics (1) for the initial data z0=0.8,0.2,0.4,0.6 and parameter data a,b,c,d=25,8,90,6.Figure 1
Various MATLAB signal portraits of the 4-D hyperchaotic two-scroll dynamics (1).
(a)(b)(c)(d)The Kaplan–Yorke dimension of the 4-D hyperchaotic two-scroll dynamics (1) can be found as(5)DKY=3+τ1+τ2+τ3τ4=3.0785.The balance points (or equilibrium points) of system (1) are the roots of the system:(6)fz=0.Thus, we proceed to find the roots of the following nonlinear system:(7a)az2−z1−z3+z4=0,(7b)cz1−z2−z1z3+z4=0,(7c)−bz3+z1z2=0,(7d)−dz2=0.Let us assume thata>0, b>0,c>0, and d>0.From (7d), −dz2=0 or z2=0.When we substitutez2=0 in (7c), we get −bz3=0 or z3=0.Furthermore, substitution ofz2=z3=0 into equations (7a) and (7b) lead to the following:(8a)−az1+z4=0,(8b)cz1+z4=0.Subtracting (8a) from (8b), we get(9)c+az1=0.Sincec and a are positive constants, c+a>0.From (9), we get z1=0.Substitutingz1=0 into (8a) or (8b), we get z4=0.As a result of this calculation, we conclude thatZ0=0 is the only balance point for the 4-D two-scroll hyperchaotic system (1). The stability nature of Z0=0 is found by calculating the eigenvalues of the Jacobian matrix A=Df0.We take the constants for the hyperchaos case asa,b,c,d=25,8,90,6.For these values, we calculate the system Jacobian matrix as(10)A=Df0=−2525−1190−10100−800−600.With the use of MATLAB, we estimate the eigenvalues of the matrixA as(11)λ1=0.3121,λ2=35.6688,λ3=−8.0000,λ4=−61.9809.This pinpoints that the unique balance pointZ0=0 of the two-scroll system (1) is a saddle-point, which is unstable. Hence, the model (1) has a self-excited two-scroll hyperchaotic attractor.


## 3. A Bifurcation Study of the Hyperchaotic Two-Scroll Attractor

In this section of the paper, the dynamical behavior of the novel, extremely complex two-scroll hyperchaotic system (1) is examined withZ0=0.8,0.2,0.4,0.6.


### 3.1. Behavior Evolution When a Changes

Here, with the values of the constantsb,c,d set to b=8, c=90 and d=6, the dynamic responses of the 4-D dynamical system (1) are displayed when the value of the constant a increases in the interval of values [5, 25]. The outcomes of the simulation show that, depending on the value of parameter a, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 2 displays the bifurcation diagram for system (1). Figure 3 depicts the Lyapunov exponents (LE) spectrum τ1,τ2,τ3,τ4 for system (1).Figure 2
Bifurcation diagram for the model (1) whena varies in 5,25 and b,c,d=8,90,6.Figure 3
Lyapunov exponents for the model (1) whena varies in 5,25 and b,c,d=8,90,6.From Figures2 and 3, it can be seen that there are no positive LE value for system (1) in the range of 5<a<9.5. Figure 4 shows a periodic response for system (1) when a=6. The corresponding LE values of (1) are estimated using MATLAB as follows:(12)τ1=0,τ2=−0.959,τ3=−2.141,τ4=−11.901.Figure 4
A periodic response of the model (1) fora=6 and b,c,d=8,90,6.When9.5<a<11.4, τ1 increases from zero to a positive value while τ2 reaches zero. These results indicate that system (1) has a chaotic attractor for this range of parameter a, as clearly seen in Figure 5, when a=11. The corresponding LE values of the new 4-D system are estimated using MATLAB as follows:(13)τ1=0.296,τ2=0,τ3=−0.593,τ4=−19.703.Figure 5
A chaotic attractor of the model (1) fora=11 and b,c,d=8,90,6.Also, whena=11, system (1) has a fractional Kaplan–York dimension:(14)DKY=2+τ1+τ2τ3=2.4992.When11.4<a<2.5, system (1) has two positive LE values, which leads into a hyperchaotic region. Figure 6 presents the hyperchaotic attractor generated by system (1) when a=12. The corresponding LE values of (1) are estimated using MATLAB as follows:(15)τ1=1.065,τ2=0.691,τ3=0,τ4=−22.758.Figure 6
A hyperchaotic attractor of model (1) fora=12 and b,c,d=8,90,6.Also, whena=12, system (1) has a fractional Kaplan–York dimension:(16)DKY=3+τ1+τ2+τ3τ4=3.0772.From Figure3, it can be seen that τ1 increases with the increase of parameter a, until it reaches over 2.6 when a=25. This causes more disordering in the attractor, as shown in Figure 7. Figures 4–7 portray a 2-D plot of the attractor of the 4-D system (1) in z1,z2 coordinate plane for different values of the system constant a, while b,c,d=8,90,6.Figure 7
A hyperchaotic attractor of the model (1) whena=25 and b,c,d=8,90,6.


### 3.2. Behavior Evolution Whenb Changes

Here, with the values of the constantsb,c,d set to a=25, c=90, and d=6, the dynamic responses of the 4-D dynamical system (1) are displayed when the value of the constant b increases in the interval of values 8,20. The outcomes of the simulation show that, depending on the value of parameter b, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 8 displays the bifurcation diagram for system (1). Figure 9 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 8
Bifurcation diagram for the model (1) whenb varies in 8,20 and a,c,d=25,90,6.Figure 9
Lyapunov exponents for the model (1) whenb varies in 8,20 and a,c,d=25,90,6.From Figures8 and 9, it can be seen that τ1 decreases with the increase of the constant b. When 8<b<14, system (1) possesses two positive LE values. Figure 10 shows the hyperchaotic attractor of system (1) when b=10. The corresponding LE values of (1) are estimated using MATLAB as follows:(17)τ1=2.666,τ2=0.273,τ3=0,τ4=−38.942.Figure 10
A hyperchaotic attractor of the model (1) forb=10 and a,c,d=25,90,6.Also, whenb=10, system (1) has a fractional Kaplan–York dimension:(18)DKY=3+τ1+τ2+τ3τ4=3.0755.Whenb=13.8, system (1) exhibits a less disordered hyperchaotic attractor shown in Figure 11, when compared with the one in Figure 10. The hyperchaotic attractor exhibited in Figure 11 has τ1=2.442 and DKY=3.064.Figure 11
A hyperchaotic attractor of the model (1) forb=13.8 and a,c,d=25,90,6.When14<b<15 the second Lyapunov exponent τ2 decreases from a positive value to zero while the first exponent τ1 stays still positive. This transition indicates that system (1) has a chaotic response for this range of parameter b. Figure 12 shows the chaotic attractor of system (1) when b=15. This strange attractor is characterized by the following Lyapunov exponents:(19)τ1=0.061,τ2=0,τ3=−0.772,τ4=−40.284.Figure 12
A chaotic attractor of the model (1) forb=15 and a,c,d=25,90,6.Also, whenb=10, system (1) has a fractional Kaplan–York dimension:(20)DKY=2+τ1+τ2τ3=2.0790.When15<b<20, there are no positive exponents which mean that system (1) has a periodic response in this region of parameters. The periodic orbit is shown in Figure 13 when b=20. The corresponding Lyapunov exponents are calculated and obtained as follows:(21)τ1=0,τ2=−1.794,τ3=−2.083,τ4=−42.124.Figure 13
A periodic attractor of the model (1) forb=20 and a,c,d=25,90,6.Figures10–12 and 1 portray a 2-D plot of the attractor of the 4-D system (1) in z1,z3 coordinate plane for different values of the system constant b, while a,c,d=25,90,6.


### 3.3. Behavior Evolution Whenc Changes

Here, with the values of the constantsa,b,d set to a=25, b=8, and d=6, the dynamic responses of system (1) are displayed when the value of the constant c increases in the interval of values 30,90. The outcomes of the simulation show that, depending on the value of parameter c, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 14 displays the bifurcation diagram for system (1). Figure 15 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 14
Bifurcation diagram for the model (1) whenc varies in 30,90 and a,b,d=25,8,6.Figure 15
Lyapunov exponents for the model (1) whenc varies in 30,90 and a,b,d=25,8,6.From Figures14 and 15, it can be seen that the first Lyapunov exponent τ1 increases with the increase of the constant c. We define A=30,30.9∪31.5,35.1∪37.5,39. When c∈A, system (1) does not have any positive LE value. Hence, system (1) has a periodic attractor for the values of c in this region. Figure 16 shows the periodic attractor of system (1) when c=30.5. The corresponding LE values of (1) are estimated using MATLAB as follows:(22)τ1=0,τ2=−0.222,τ3=−1.035,τ4=−32.744.Figure 16
A periodic attractor of the 4-D system (1) whenc=30.5 and a,b,d=25,8,6.We defineB=30.9,31.5∪35.1,37.5∪39,44.7. When c∈B,τ1 increases from zero to a positive value while the second Lyapunov exponent τ2 reaches zero. These results indicate that system (1) has a chaotic behavior for this range of the constant c∈B. Figure 17 shows the chaotic attractor of system (1) when c=40. The corresponding LE values of (1) are estimated using MATLAB as follows:(23)τ1=0.218,τ2=0,τ3=−0.601,τ4=−33.618.Figure 17
A chaotic attractor of the 4-D system (1) whenc=40 and a,b,d=25,8,6.Also, the Kaplan–Yorke dimension of (1) for c=40 is obtained as:(24)DKY=2+τ1+τ2τ3=2.3627.When44.7<c<90, system (1) has two positive LE values, which leads into a wide hyperchaotic region. Figure 18 presents the hyperchaotic attractor exhibited by (1) when c=50.Figure 18
A hyperchaotic attractor of the model (1) forc=50 and a,b,d=25,8,6.The corresponding LE values of (1) are estimated using MATLAB as follows:(25)τ1=1.130,τ2=0.497,τ3=0,τ4=−35.629.Also, the Kaplan–Yorke dimension of (1) for c=50 is obtained as:(26)DKY=3+τ1+τ2+τ3τ4=3.0457.From Figure15, it can be seen that τ1 increases with the increase of the constant c until it reaches over τ1=2.6 when c=90. This causes a hyperchaotic attractor in which the trajectory is highly disordered, as seen in Figure 19.Figure 19
A hyperchaotic attractor of the model (1) forc=90 and a,b,d=25,8,6.Figures16–19 portray a 2-D plot of the attractor of system (1) in z2,z3 coordinate plane for different values of the constant c, while a,b,d=25,8,6.


### 3.4. Behavior Evolution Whend Changes

Here, with the values of the constantsa,b,c set to a=25, b=8, and c=90, the dynamic responses of system (1) are displayed when the value of the constant d increases in the range of values 5,1100. The outcomes of the simulation show that, depending on the value of parameter d, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 20 displays the bifurcation diagram for system (1). Figure 21 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 20
Bifurcation diagram for the model (1) whend varies in 0,1100 and a,b,c=25,8,90.Figure 21
Lyapunov exponents for the model (1) whend varies in 0,1100 and a,b,c=25,8,90.When5<d<140, system (1) possesses two positive LE values. Figure 22 shows the hyperchaotic attractor of system (1) when d=40. The corresponding LE values of (1) are estimated using MATLAB as follows:(27)τ1=1.525,τ2=1.296,τ3=0,τ4=−36.825.Figure 22
A hyperchaotic attractor of the model (1) ford=40 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=40 is obtained as:(28)DKY=3+τ1+τ2+τ3τ4=3.0766.We defineC=140,875∪1035,1065. When d∈C,τ2 decreases from a positive value to zero while τ1 stays still positive. This transition indicates that system (1) has a chaotic response for this very wide region. When d=180,τ1 reaches its largest value of 3.392.Figure23 shows the chaotic attractor of system (1). The corresponding LE values of (1) are estimated using MATLAB as follows:(29)τ1=3.392,τ2=0,τ3=−0.316,τ4=−37.089.Figure 23
A chaotic attractor of the model (1) ford=180 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=180 is obtained as(30)DKY=3+τ1+τ2+τ3τ4=3.0829.Whend=1045, τ1 decreases to 0.544 providing less strength to the system’s chaotic dynamics. Figure 24 shows the chaotic attractor of system (1). The corresponding LE values of (1) are estimated using MATLAB as follows:(31)τ1=0.544,τ2=0,τ3=−2.811,τ4=−31.735.Figure 24
A chaotic attractor of the model (1) ford=1045 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=1045 is obtained as(32)DKY=2+τ1+τ2τ3=2.1935.We defineD=875,1035∪1065,1100. When d∈D, there are no positive LE values which means that system (1) has a periodic response in this region of constants. The periodic orbit is shown in Figure 25 when d=1020.Figure 25
A periodic attractor of the model (1) ford=1020 and a,b,c=25,8,90.The corresponding LE values of (1) are estimated using MATLAB as follows:(33)τ1=0,τ2=−0.550,τ3=−0.555,τ4=−32.901.Figures22–25 portray a 2-D plot of the attractor of the 4-D system (1) in z3,z4 coordinate plane for different values of the constant d while a,b,c=25,8,90.


## 3.1. Behavior Evolution When a Changes

Here, with the values of the constantsb,c,d set to b=8, c=90 and d=6, the dynamic responses of the 4-D dynamical system (1) are displayed when the value of the constant a increases in the interval of values [5, 25]. The outcomes of the simulation show that, depending on the value of parameter a, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 2 displays the bifurcation diagram for system (1). Figure 3 depicts the Lyapunov exponents (LE) spectrum τ1,τ2,τ3,τ4 for system (1).Figure 2
Bifurcation diagram for the model (1) whena varies in 5,25 and b,c,d=8,90,6.Figure 3
Lyapunov exponents for the model (1) whena varies in 5,25 and b,c,d=8,90,6.From Figures2 and 3, it can be seen that there are no positive LE value for system (1) in the range of 5<a<9.5. Figure 4 shows a periodic response for system (1) when a=6. The corresponding LE values of (1) are estimated using MATLAB as follows:(12)τ1=0,τ2=−0.959,τ3=−2.141,τ4=−11.901.Figure 4
A periodic response of the model (1) fora=6 and b,c,d=8,90,6.When9.5<a<11.4, τ1 increases from zero to a positive value while τ2 reaches zero. These results indicate that system (1) has a chaotic attractor for this range of parameter a, as clearly seen in Figure 5, when a=11. The corresponding LE values of the new 4-D system are estimated using MATLAB as follows:(13)τ1=0.296,τ2=0,τ3=−0.593,τ4=−19.703.Figure 5
A chaotic attractor of the model (1) fora=11 and b,c,d=8,90,6.Also, whena=11, system (1) has a fractional Kaplan–York dimension:(14)DKY=2+τ1+τ2τ3=2.4992.When11.4<a<2.5, system (1) has two positive LE values, which leads into a hyperchaotic region. Figure 6 presents the hyperchaotic attractor generated by system (1) when a=12. The corresponding LE values of (1) are estimated using MATLAB as follows:(15)τ1=1.065,τ2=0.691,τ3=0,τ4=−22.758.Figure 6
A hyperchaotic attractor of model (1) fora=12 and b,c,d=8,90,6.Also, whena=12, system (1) has a fractional Kaplan–York dimension:(16)DKY=3+τ1+τ2+τ3τ4=3.0772.From Figure3, it can be seen that τ1 increases with the increase of parameter a, until it reaches over 2.6 when a=25. This causes more disordering in the attractor, as shown in Figure 7. Figures 4–7 portray a 2-D plot of the attractor of the 4-D system (1) in z1,z2 coordinate plane for different values of the system constant a, while b,c,d=8,90,6.Figure 7
A hyperchaotic attractor of the model (1) whena=25 and b,c,d=8,90,6.


## 3.2. Behavior Evolution Whenb Changes

Here, with the values of the constantsb,c,d set to a=25, c=90, and d=6, the dynamic responses of the 4-D dynamical system (1) are displayed when the value of the constant b increases in the interval of values 8,20. The outcomes of the simulation show that, depending on the value of parameter b, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 8 displays the bifurcation diagram for system (1). Figure 9 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 8
Bifurcation diagram for the model (1) whenb varies in 8,20 and a,c,d=25,90,6.Figure 9
Lyapunov exponents for the model (1) whenb varies in 8,20 and a,c,d=25,90,6.From Figures8 and 9, it can be seen that τ1 decreases with the increase of the constant b. When 8<b<14, system (1) possesses two positive LE values. Figure 10 shows the hyperchaotic attractor of system (1) when b=10. The corresponding LE values of (1) are estimated using MATLAB as follows:(17)τ1=2.666,τ2=0.273,τ3=0,τ4=−38.942.Figure 10
A hyperchaotic attractor of the model (1) forb=10 and a,c,d=25,90,6.Also, whenb=10, system (1) has a fractional Kaplan–York dimension:(18)DKY=3+τ1+τ2+τ3τ4=3.0755.Whenb=13.8, system (1) exhibits a less disordered hyperchaotic attractor shown in Figure 11, when compared with the one in Figure 10. The hyperchaotic attractor exhibited in Figure 11 has τ1=2.442 and DKY=3.064.Figure 11
A hyperchaotic attractor of the model (1) forb=13.8 and a,c,d=25,90,6.When14<b<15 the second Lyapunov exponent τ2 decreases from a positive value to zero while the first exponent τ1 stays still positive. This transition indicates that system (1) has a chaotic response for this range of parameter b. Figure 12 shows the chaotic attractor of system (1) when b=15. This strange attractor is characterized by the following Lyapunov exponents:(19)τ1=0.061,τ2=0,τ3=−0.772,τ4=−40.284.Figure 12
A chaotic attractor of the model (1) forb=15 and a,c,d=25,90,6.Also, whenb=10, system (1) has a fractional Kaplan–York dimension:(20)DKY=2+τ1+τ2τ3=2.0790.When15<b<20, there are no positive exponents which mean that system (1) has a periodic response in this region of parameters. The periodic orbit is shown in Figure 13 when b=20. The corresponding Lyapunov exponents are calculated and obtained as follows:(21)τ1=0,τ2=−1.794,τ3=−2.083,τ4=−42.124.Figure 13
A periodic attractor of the model (1) forb=20 and a,c,d=25,90,6.Figures10–12 and 1 portray a 2-D plot of the attractor of the 4-D system (1) in z1,z3 coordinate plane for different values of the system constant b, while a,c,d=25,90,6.


## 3.3. Behavior Evolution Whenc Changes

Here, with the values of the constantsa,b,d set to a=25, b=8, and d=6, the dynamic responses of system (1) are displayed when the value of the constant c increases in the interval of values 30,90. The outcomes of the simulation show that, depending on the value of parameter c, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 14 displays the bifurcation diagram for system (1). Figure 15 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 14
Bifurcation diagram for the model (1) whenc varies in 30,90 and a,b,d=25,8,6.Figure 15
Lyapunov exponents for the model (1) whenc varies in 30,90 and a,b,d=25,8,6.From Figures14 and 15, it can be seen that the first Lyapunov exponent τ1 increases with the increase of the constant c. We define A=30,30.9∪31.5,35.1∪37.5,39. When c∈A, system (1) does not have any positive LE value. Hence, system (1) has a periodic attractor for the values of c in this region. Figure 16 shows the periodic attractor of system (1) when c=30.5. The corresponding LE values of (1) are estimated using MATLAB as follows:(22)τ1=0,τ2=−0.222,τ3=−1.035,τ4=−32.744.Figure 16
A periodic attractor of the 4-D system (1) whenc=30.5 and a,b,d=25,8,6.We defineB=30.9,31.5∪35.1,37.5∪39,44.7. When c∈B,τ1 increases from zero to a positive value while the second Lyapunov exponent τ2 reaches zero. These results indicate that system (1) has a chaotic behavior for this range of the constant c∈B. Figure 17 shows the chaotic attractor of system (1) when c=40. The corresponding LE values of (1) are estimated using MATLAB as follows:(23)τ1=0.218,τ2=0,τ3=−0.601,τ4=−33.618.Figure 17
A chaotic attractor of the 4-D system (1) whenc=40 and a,b,d=25,8,6.Also, the Kaplan–Yorke dimension of (1) for c=40 is obtained as:(24)DKY=2+τ1+τ2τ3=2.3627.When44.7<c<90, system (1) has two positive LE values, which leads into a wide hyperchaotic region. Figure 18 presents the hyperchaotic attractor exhibited by (1) when c=50.Figure 18
A hyperchaotic attractor of the model (1) forc=50 and a,b,d=25,8,6.The corresponding LE values of (1) are estimated using MATLAB as follows:(25)τ1=1.130,τ2=0.497,τ3=0,τ4=−35.629.Also, the Kaplan–Yorke dimension of (1) for c=50 is obtained as:(26)DKY=3+τ1+τ2+τ3τ4=3.0457.From Figure15, it can be seen that τ1 increases with the increase of the constant c until it reaches over τ1=2.6 when c=90. This causes a hyperchaotic attractor in which the trajectory is highly disordered, as seen in Figure 19.Figure 19
A hyperchaotic attractor of the model (1) forc=90 and a,b,d=25,8,6.Figures16–19 portray a 2-D plot of the attractor of system (1) in z2,z3 coordinate plane for different values of the constant c, while a,b,d=25,8,6.


## 3.4. Behavior Evolution Whend Changes

Here, with the values of the constantsa,b,c set to a=25, b=8, and c=90, the dynamic responses of system (1) are displayed when the value of the constant d increases in the range of values 5,1100. The outcomes of the simulation show that, depending on the value of parameter d, system (1) can display a variety of types of behavior such as periodic, chaotic, or hyperchaotic behavior with varying degrees of complexity. Figure 20 displays the bifurcation diagram for system (1). Figure 21 depicts the LE spectrum τ1,τ2,τ3,τ4 for system (1).Figure 20
Bifurcation diagram for the model (1) whend varies in 0,1100 and a,b,c=25,8,90.Figure 21
Lyapunov exponents for the model (1) whend varies in 0,1100 and a,b,c=25,8,90.When5<d<140, system (1) possesses two positive LE values. Figure 22 shows the hyperchaotic attractor of system (1) when d=40. The corresponding LE values of (1) are estimated using MATLAB as follows:(27)τ1=1.525,τ2=1.296,τ3=0,τ4=−36.825.Figure 22
A hyperchaotic attractor of the model (1) ford=40 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=40 is obtained as:(28)DKY=3+τ1+τ2+τ3τ4=3.0766.We defineC=140,875∪1035,1065. When d∈C,τ2 decreases from a positive value to zero while τ1 stays still positive. This transition indicates that system (1) has a chaotic response for this very wide region. When d=180,τ1 reaches its largest value of 3.392.Figure23 shows the chaotic attractor of system (1). The corresponding LE values of (1) are estimated using MATLAB as follows:(29)τ1=3.392,τ2=0,τ3=−0.316,τ4=−37.089.Figure 23
A chaotic attractor of the model (1) ford=180 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=180 is obtained as(30)DKY=3+τ1+τ2+τ3τ4=3.0829.Whend=1045, τ1 decreases to 0.544 providing less strength to the system’s chaotic dynamics. Figure 24 shows the chaotic attractor of system (1). The corresponding LE values of (1) are estimated using MATLAB as follows:(31)τ1=0.544,τ2=0,τ3=−2.811,τ4=−31.735.Figure 24
A chaotic attractor of the model (1) ford=1045 and a,b,c=25,8,90.Also, the Kaplan–Yorke dimension of (1) for d=1045 is obtained as(32)DKY=2+τ1+τ2τ3=2.1935.We defineD=875,1035∪1065,1100. When d∈D, there are no positive LE values which means that system (1) has a periodic response in this region of constants. The periodic orbit is shown in Figure 25 when d=1020.Figure 25
A periodic attractor of the model (1) ford=1020 and a,b,c=25,8,90.The corresponding LE values of (1) are estimated using MATLAB as follows:(33)τ1=0,τ2=−0.550,τ3=−0.555,τ4=−32.901.Figures22–25 portray a 2-D plot of the attractor of the 4-D system (1) in z3,z4 coordinate plane for different values of the constant d while a,b,c=25,8,90.


## 4. Coexisting Attractors and Multistability for the New System

Multistability, often referred to as coexisting attractors, is a special nonlinear phenomenon wherein two or more distinct attractors evolve simultaneously from various starting points.LetZ01 and Z02 represent two distinct starting points for the new 4-D system (1), where:Z0=0.5,0.5,0.5,0.5 (Blue color).W0=−0.5,−0.5,0.5,−0.5 (Red color).First, we fix the system constants asa=25, b=8, c=32, and d=6. As shown in Figure 26, system (1) exhibits two different coexisting attractors. The blue one is a periodic attractor that begins from Z0 and characterized by the following LE spectrum:(34)τ1=0,τ2=−0.312,τ3=−0.741,τ4=−32.950.Figure 26
The coexistence of one periodic and one chaotic attractor for system (1) projected in thez1,z4 coordinate plane for the choice of parameters a,b,c,d=25,8,32,6.The red one is a chaotic attractor that begins fromW0 and characterized by the following LE spectrum:(35)τ1=0.041,τ2=0,τ3=−1.111,τ4=−32.928.Next, we fix the system constants asa=25, b=14, c=90, and d=6 As shown in Figure 27, system (1) exhibits two different coexisting chaotic attractors. The blue chaotic attractor begins from Z0 and characterized by the following LE spectrum:(36)τ1=0.322,τ2=0,τ3=−0.393,τ4=−39.942.Figure 27
The coexistence of chaotic attractors for system (1) projected in thez2,z3 coordinate plane for the choice of parameters a,b,c,d=25,14,90,6.The red chaotic attractor begins fromW0 and characterized by the following LE spectrum:(37)τ1=0.236,τ2=0,τ3=−0.396,τ4=−39.842.Finally, we fix the system constants asa=25, b=8, c=65, and d=6. As shown in Figure 28, system (1) exhibits two different coexisting hyperchaotic attractors. The blue hyperchaotic attractor begins from Z0 and characterized by the following LE spectrum:(38)τ1=1.970,τ2=0.341,τ3=0,τ4=−36.313.Figure 28
The coexistence of hyperchaotic attractors for system (1) projected in thez1,z3 coordinate plane for the choice of parameters a,b,c,d=25,8,65,6.The red hyperchaotic attractor begins fromW0 and characterized by the following LE spectrum:(39)τ1=1.958,τ2=0.339,τ3=0,τ4=−36.300.


## 5. Electronic Circuit Design for the New Hyperchaotic Two-Scroll System

We detail the electronic circuit realization of the proposed two-scroll hyperchaotic model using MultiSim software (Version 14). Figure29 depicts an analog electronic circuit for the implementation of the proposed hyperchaotic circuit, which includes two analog multipliers (AD63JN), seven operational amplifiers (TL083CD), seventeen resistors, and four capacitors.Figure 29
The electronic circuit design for the 4-D hyperchaotic system (29).For the circuit design, we rescale the two-scroll model (1) using the following transformation of coordinates:(40)Zi=12zi,i=1,2,3,4.Replacing the old variables with the new variables on the new 4-D hyperchaotic system (1), we obtain the following system:(41)Z˙1=aZ2−Z1−Z3+Z4,Z˙2=cZ1−Z2−2Z1Z3+Z4,Z˙3=−bZ3+2Z1Z2,Z˙4=−dZ2.Using Kirchhoff’s electrical circuit laws, we derive the circuit model for the rescaled hyperchaotic model (41) as follows:(42)C1Z˙1=1R1Z2−1R2Z1−1R3Z3+1R4Z4,C2Z˙2=1R5Z1−1R6Z2−110R7Z1Z3+1R8Z4,C3Z˙3=−1R9Z3+110R10Z1Z2,C4Z˙4=−1R11Z2.Here,Z1, Z2, Z3, Z4 are the voltages across the capacitors, C1, C2, C3, C4, respectively. We choose the values of the circuital elements as follows:R1=R2=16kΩ, R3=R4=R6=R8=400kΩ, R5=4.44kΩ,R7=R10=20kΩ, R9=50kΩ, R11=66.67kΩ,R12=R13=R14=R15=R16=R17=100kΩ, C1=C2=C3=C4=3.2nF.With MultiSim 14.0, we obtain the experiment observations of system (42) in Figures30–33. It can be seen that the good qualitative agreement between the MATLAB simulations (Figures 1–4) and the MultiSim results (Figures 30–33) of the hyperchaotic two-scroll system is confirmed.Figure 30
Electronic circuit simulation using MultiSim 14.0 for the new hyperchaotic two-scroll circuit (42) in theZ1,Z2− coordinate plane.Figure 31
Electronic circuit simulation using MultiSim 14.0 for the new hyperchaotic two-scroll circuit (42) in theZ2,Z3− coordinate plane.Figure 32
Electronic circuit simulation using MultiSim 14.0 for the new hyperchaotic two-scroll circuit (42) in theZ3,Z4− coordinate plane.Figure 33
Electronic circuit simulation using MultiSim 14.0 for the new hyperchaotic two-scroll circuit (42) in theZ1,Z4− coordinate plane.


## 6. FPGA Implementation of the New Two-Scroll Hyperchaotic Model

The use of field-programmable gate arrays (FPGA) has shown advantages in the implementation of chaotic and hyperchaotic systems, as done in [31–34]. One can find guidelines on applying numerical methods that are directly synthesized into an FPGA [35]. In this work, we show the experimental results of the FPGA implementation of proposed model (1) by applying two common numerical methods. The first one is the forward Euler, whose iterative algorithm is given in equation (30), and the second one is the implicit algorithm known as trapezoidal method that is given in equation (31).(43)xn+1=xn+hfxn,tn,xn+1=xn+h2fxn,tn+fxn+1,tn+1.It is well-known that a system of ordinary differential equations, as the one associated for the proposed model (1), can be solved by discretizing the equations using a numerical method. In the case of applying the forward Euler, one gets the discrete equations given by equation (44), from which one can clearly see the requirement of arithmetic operations as adders, subtractors, and multipliers. The control of the iterations is performed by designing a state machine control that includes registers, as detailed in [35], where one can see the design of the arithmetic blocks, and a single-constant multiplier block to reduce hardware resources.(44)z1n+1=z1n+haz2n−z1n−z3n+z4n,z2n+1=z2n+hcz1n−z2n−z1nz3n+z4n,z3n+1=z3n+h−bz3n+z1nz2n,z4n+1=z4n+h−dz2n.The design of each digital block requires the definition of the number of bits to be processed. In this case, the computer arithmetic is performed by applying fixed-point representation with the format 15.49. The 64 bits are distributed in one bit for the sign, 14 for the integer part, and 49 bit for the fractional part. The bit distribution considers the maximum amplitude values that can appear for each state variable during the processing of the data. For example, from the simulation results of the new 4-D hyperchaotic system (1), the four state variables have amplitudes in the range−199.02,342.50, but in the discretization process by applying the numerical methods, an analysis of all the internal arithmetic operations, mainly those that result during the multiplications among state variables, resulted in an absolute range of values in the range −11299.60,11116.51, thus leading to use the format 15.49.Figure34 depicts a block diagram showing the discretization of the proposed model (1) using the forward Euler method given in equation (32). The registers are controlled by a state machine to perform the iteration process. The FPGA implementation is also done in a similar way by applying the trapezoidal method so that the hardware resources for the FPGA design using the FPGA Basys 3 Xilinx Artix-7 XC7A35T-ICPG236C are as summarized in Table 1. One can see the number of lookup tables (LUTs), flip-flops (FF), digital signal processors (DSP), multipliers, adders, and subtractors. We also give the number of clock cycles required by an iteration, and the latency in nanoseconds (ns).Figure 34
Block diagram of the discretization of the proposed model (1) using the forward Euler method.Table 1
Hardware resources for the FPGA design of (1) using the FPGA basys 3 xilinx artix-7 XC7A35T-ICPG236C using two numerical methods with h=0.001.
ResourcesForward EulerTrapezoidalAvailableLUTs46791057020,800FF567125341,600DSP458290Multipliers1024–Adders518–Subtractors612–Clock cycles by iteration23–Latency (ns)80120–The FPGA experimental setup for the proposed model (1) is shown in Figure35. Furthermore, the FPGA experimental outputs of (1) are depicted in Figures 36–39.Figure 35
Experimental setup to measure the attractors of the proposed model (1) using a FPGA basys 3 xilinx artix-7 XC7A35T-ICPG236, a 16 bit digital-analog converter, a linear power supply and a keysight oscilloscope to visualize the hyperchaotic attractor.Figure 36
Experimental hyperchaotic attractor of the proposed model (1) in thez1,z2− plane.Figure 37
Experimental hyperchaotic attractor of the proposed model (1) in thez2,z3− plane.Figure 38
Experimental hyperchaotic attractor of the proposed model (1) in thez3,z4− plane.Figure 39
Experimental hyperchaotic attractor of the proposed model (1) in thez1,z4− plane.


## 7. Conclusions

In this research work, we reported our findings of a new four-dimensional two-scroll hyperchaotic system having only two quadratic nonlinearities. We conducted a detailed dynamic study of the proposed model and noted the coexistence of attractors for fixed parameter values but different initial states. Since the maximal Lyapunov exponent (MLE) of the two-scroll system isτmax=2.6174, we deduced that the two-scroll system (1) has high complexity. Also, a design for an electronic circuit has been made for the new hyperchaotic system using MultiSim (Version 14). The experimental observation of the hyperchaotic attractors has been shown from the FPGA implementation by applying numerical methods, viz. forward Euler method and trapezoidal method. It was observed that the FPGA experimental results using the forward Euler discretization method are in good agreement with the MATLAB simulation results of the proposed hyperchaotic two-scroll system.



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