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Dynamic Analysis and Circuit Design of a Novel Hyperchaotic System with Fractional-Order Terms.

1. Introduction

Hyperchaos was discovered by Rossler in 1979 [1] and the first hyperchaotic circuit was implemented by Matsumoto in 1986 [2]. In these last years, hyperchaotic systems have gained the interest of the scientific community and new systems and circuits are proposed [3-8]. This great interest can be explained by the aptitude of hyperchaotic systems to generate complex dynamics characterized by more than one positive Lyapunov exponent and attractors deployed in multiple directions. In practical applications and particularly in secure communication, chaotic synchronization has been explored by using electronic circuits, namely, Duffing circuit [9], Chua circuit [10], and Rossler circuit [11]. However, for hyperchaotic circuits, many challenging problems are still pending due to their complex behaviors.

On the other hand, several researches have attempted to construct chaotic and hyperchaotic models with simple algebraic structures highly recommended for circuit design. The most famous chaotic one is the Jerk system proposed by sprott, in 1994 [12, 13], which contains simple nonlinear terms. However, it is well known that most systems contain conventional nonlinear terms like piecewise linear functions [14-17], integer order polynomials [8, 18], sine functions [19], time delayed functions [20], and switching functions [21]. In this framework, fractional-order polynomials could be used to build complex chaotic behaviors and, to the best of our knowledge, they have not been harnessed until now.

The purpose of this paper is to build a novel hyperchaotic system with more complex dynamics than those proposed by related works. Expecting that the PWNL function with FO terms gives us more complex chaotic proprieties than the piecewise linear one, this PWNL function is constructed from absolute functions and FO polynomials. To enhance the potential application of the proposed system, its related circuit is designed afterwards with MultiSIM Software.

The rest of this paper is structured as follows. In Section 2, the mathematical model of the hyperchaotic system is proposed and its basic properties are presented. In Section 3, the dynamic analysis of the novel system is investigated by pointing out its elementary characteristics such as the Lyapunov exponents, the attractor forms, and the equilibrium points. In Section 4, the oscillator circuit of the hyperchaotic system is designed afterwards.

2. Mathematical Model and Basic Properties

Let consider the mathematical model of the novel hyperchaotic system with FO terms expressed by the following differential equations:

[mathematical expression not reproducible], (1)

with G(x) being a nonlinear function defined as

G(x) = -c[x.sup.2] + d[absolute value of x]x + m[[absolute value of x].sup.r] [x.sup.-1], 1 < r < 2, (2)

where (a, b, c, d, h, k, m, r) are the system's parameters and (x, y, z, w) are the state variables. r is a fractional number satisfying 1 < r < 2. Since r [not equal to] 1, [[absolute value of x].sup.r][x.sup.-1] will never be an indeterminate form. The nonlinear function G(x) can be written as follows:

[mathematical expression not reproducible]. (3)

System (1) can exhibit chaotic behavior if the general condition of dissipativity is satisfied such as

[mathematical expression not reproducible]. (4)

As long as a + h > 0, system (1) is dissipative and it converges to an attractor. Thus, when the parameters (a, b, c, d, m, r, h, k) are equal to (0.93, 1.11, -0.11, -0.21, 6.26, 1.32, 0.001, 14) and the initial condition is equal to (1, 1, 1, 1),system(1) generates a strange attractor displayed in Figure 1. This attractor has an asymmetrical form with respect to all the principal axes characterized by two scrolls of different sizes.

The time series of the state variables x and w are described in Figures 2(a) and 2(b). These signals represent the chaotification rates of each variable. On the other hand, system (1) is sensitive to initial conditions as shown in Figures 2(c) and 2(d). Note that the variation range of the variable w is extended within [-250, 200], unlike the other variables. This point must be considered in practical applications.

3. Dynamic Analysis

3.1. Equilibrium and Stability. The equilibrium points of system (1) are obtained by solving these equations:

y = z = w = 0,

-c[x.sup.2] + d[absolute value of x]x + m[[absolute value of x].sup.r] [x.sup.-1] = 0. (5)

Proposition 1. (i) If x = 0, then w = 0 and the origin [H.sub.1] = (0, 0, 0, 0) is the first equilibrium of system (1).

(ii) If x > 0, then w = 0 and [H.sub.2] = ([((c - d)/m).sup.1/[alpha]], 0, 0, 0) is an equilibrium of system (2) where [alpha] = r - 3.

(iii) If x < 0, then w = 0 and [H.sub.3] = ([(-(-c - d)/m).sup.1/[alpha]], 0, 0, 0) is an equilibrium of system (2) where [alpha] = r - 3.

Proof. Case (i) is obvious.

For case (ii), we should solve the following equation: -c + d + m[x.sup.r-3] = 0 which admits the solution [x.sup.[alpha]] = (c - d)/m with [alpha] = r - 3. The roots of this equation are given by [22]

[mathematical expression not reproducible] (6)

with n [member of] N, [[theta].sub.1] being the phase of [x.sup.[alpha]], and [alpha] being a fractional number. Notice that the term (c - d)/m is positive when c = -0.11, d = -0.21, and m = 6.26. Then, [[theta].sub.1] is equal to zero and we have x = [((c - d)/m).sup.1/[alpha]].

For case (iii), we should solve the following equation: -c - d - m[(-x).sup.r-3] = 0 which admits the solution [(-x).sup.[alpha]] = [X.sup.[alpha]] = (-c - d)/m with [alpha] = r - 3. The roots of this equation are given by [22]

[mathematical expression not reproducible] (7)

with n [member of] N, [[theta].sub.2] being the phase of [x.sup.[alpha]], and a being a fractional number. Notice that the term (-c - d)/m is positive when c = -0.11, d = -0.21, and m = 6.26. Then, [[theta].sub.2] is equal to zero and we have x = [(-(-c - d)/m).sup.1/[alpha]].

When the parameters (a, b, c, d, m, r, h, k) are equal to (0.93, 1.11, -0.11, -0.21, 6.26, 1.32, 0.001, 14), system (1) admits three equilibrium points: [H.sub.1] = (0, 0, 0, 0), [H.sub.2] = (11.73, 0, 0, 0), and [H.sub.3] = (-5.87, 0, 0, 0). For the stability analysis, Table 1 gives the Jacobian matrix J and its corresponding eigenvalues calculated for each equilibrium point.

3.2. Lyapunov Exponents Analysis. System (1) exhibits four Lyapunov exponents (LEs). These LEs are esteemed using the Wolf algorithm [23], as shown in Figure 3 as

[LE.sub.1] = 0.232,

[LE.sub.2] = 0.020,

[LE.sub.3] = 0,

[LE.sub.4] = -1.169. (8)

Since the LE spectrum has two positive Lyapunov exponents; thus system (1) is hyperchaotic. [[lambda].sub.1] is the largest positive one. This exponent increases the expansion degree of the attractor in the phase space.

In addition, the corresponding Kaplan-Yorke dimension is

[D.sub.L] = 3 + [([[lambda].sub.1] + [[lambda].sub.2] + [[lambda].sub.3])/[absolute value of [[lambda].sub.4]]] = 3.19. (9)

3.3. Routes to Chaos. System (1) can display periodic orbits, chaos, and hyperchaos attractors under different conditions. In fact, when the parameter m varies and the parameters (a, b, c, d, r, h, k) are fixed, two Hopf bifurcations are detected as shown in Figure 4. These bifurcations are denoted H in the bifurcation diagram and they appear when m = 0.99 and m = 2.11, respectively. Each Hopf point is characterized by a first Lyapunov coefficient (FLC). A positive FLC indicates the existence of a supercritical Hopf bifurcation, whereas a negative one indicates a subcritical Hopf bifurcation. In system (1), the two points obtained are supercritical Hopf bifurcations. This type of bifurcation indicates that the evolution to chaotic behavior is possible.

In addition, as the parameter of bifurcation m increases, system (1) undergoes the following routes:

(i) If -1 [less than or equal to] m [less than or equal to] 0.3, then system (1) exhibits periodic orbit. Figure 5(a) shows this regular attractor with m = -0.5.

(ii) If 0.3 < m [less than or equal to] 2.1, then system (1) converges to a fixed point as shown in Figure 5(b).

(iii) If 2.1 < m [less than or equal to] 3.8, then another periodic orbit is obtained as shown in Figure 5(c) with m = 1.

(iv) If 3.8 < m [less than or equal to] 5.5, system (1) exhibits chaotic attractor. Figure 5(d) shows this strange attractor with m = 4.5.

(v) If 5.5 < m < 7, then system (1) exhibits hyperchaotic attractor. Figure 5(e) shows this strange attractor with m = 6.

Some typical attractors are tabulated in Table 2 according to the parameter m.

3.4. Comparative Analysis. Referring to the survey paper [24], the first Lyapunov exponent can be one of the comparative criteria between hyperchaotic systems. Table 3 presents a comparative analysis between system (1) and two related ones, recently proposed in literature. Such a choice is based on the fact that, identical to system (1), the first comparative example contains linear piecewise functions whereas the second one is based on the jerk equation. Based on Table 3, it is clear that system (1) exhibits more complex dynamics. Thus, this confirms the highlight potential applications of noninteger order terms with respect to classical nonlinear terms.

4. Circuit Design

It is obvious that hardware implementation of chaotic systems is an interesting task in engineering applications, namely, for secure communications and random bits generation. Therefore, the aim of this section is to design an analog circuit that can build hyperchaotic behaviors according to system (1).

4.1. Design of the Analog Circuit with MultiSIM. For the circuit implementation, we choose the particular case study when the system parameter r is fixed to 1.5. Thus, the proposed system will be defined by the following model:

[mathematical expression not reproducible], (10)

where the system parameters (a, b, c, d, m, h, k) are equal to (1, 1, -0.11, -0.21, 5, 0.01, 14). System (10) exhibits four LEs such as

[LE.sub.1] = 0.18,

[LE.sub.2] = 0.04,

[LE.sub.3] = 0,

[LE.sub.4] = -1.2. (11)

Despite the observation of the obtained phase portraits of system (10), we deduce that the maximum value of the signal w can reach the level of 250. Thus, 250 Visa sufficiently high voltage for the common components used in the proposed circuit. Therefore, a linear transformation for system (10) is necessary to decrease the amplitude of the state variables. Letting u = x/2, v = y/2, g = z/2, and f = w/160 and then setting the original state variables x, y, z, and w instead of the variable u, v, g, and f, the adjusted system becomes the following one:

[mathematical expression not reproducible]. (12)

The amplitude of the state variables of system (10) has decreased as shown in Figure 6. Moreover, the two systems (10) and (12) are equivalent since the linear transformation does not change the physical properties of nonlinear systems.

To design the hyperchaotic circuit of system (12), only common electronic components are used such as resistors, capacitors, diodes, multipliers, and operational amplifiers. In fact, the nonlinear terms of system (12) should be designed first, namely, the quadratic term, the absolute function, the sign function, and the square root function. The quadratic term is implemented with the analog multiplier. The square root element is designed with two operational amplifiers as only active elements [27]. The analog circuit of the square root element is provided in Figure 7.

For the theoretical study and based on [27], the second voltage source in Figure 7 should be fixed to 2.878 V. However, in experimentation applications, we have obtained the root square function by using a stabilized voltage equal to 2.9 V as shown in Figure 8. This figure describes two voltages; the first one is a positive source signal and the second one is the output signal of the square root circuit. Based on these results, the observed maximum voltages are equal to 720 mV and 900 mV ([equivalent] [square root of 0.72] = 0.88), respectively. Thus, the square root function is correctly obtained with 2.9 V. In addition, based on MultiSIM results and experimental simulations, if the source voltage is included in [2.7 V, 3 V] then system (12) generates strange attractors. To avoid making this paper more cumbersome, details on experiments and experimental results will be soon presented in future works, confirming the MultiSIM results.

The corresponding circuit equation of the hyperchaotic system can be described as

[mathematical expression not reproducible], (13)

According to system (12) and (13) and design considerations, we fixed the values of the resistances and the capacitors as

[mathematical expression not reproducible]. (14)

Finally, the obtained circuit diagram, designed with MultiSIM Software, is provided in Figure 9 where the multiplier is AD633 and the operator amplifier is UA741.

4.2. Simulation Results. For the oscillator circuit, all active devices (UA741 and AD633) are powered by [+ or -]15 V. Several design considerations were taken into account to prevent degrading the hyperchaotic behavior such as the adjustment of the resistors and the capacitors for the integration operations.

The oscilloscope traces of the proposed circuit are shown in Figure 10. Comparing the different hyperchaotic attractors shown in Figures 6 and 10, a good qualitative agreement between the numerical simulations with Matlab and the electrical simulations with MultiSIM Software is observed. In fact, for MultiSIM Software, we have obtained the same attractors forms as those obtained by Matlab simulations. However, in these last attractors, some saturation effects are detected due to the operational amplifiers responses. To avoid making this paper more cumbersome, details on experiments and experimental results will be presented in future works, where saturation effects of amplifiers will be deeply analyzed.

5. Conclusions

In this paper, a novel hyperchaotic system is proposed by considering fractional-order polynomials. Analytical and numerical results show that this system exhibits more complex behaviors than those proposed by related works. Moreover, its analog circuit is designed and simulated with MultiSIM Software. In future works, experimental realization of the hyperchaotic circuit will be proposed and the saturation effects induced by the operational amplifiers will be analyzed. Thereafter, the proposed circuit will be considered for secure image encryption and decryption applications.

https://doi.org/10.1155/2017/3273408

Conflicts of Interest

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

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[27] T. Kamsri, P. Julsereewong, and V. Riewruja, "Simple square-root extractor using op amps," in Proceedings of the 2008 International Conference on Control, Automation and Systems, ICCAS 2008, pp. 1812-1815, October 2008.

Abir Lassoued and Olfa Boubaker

National Institute of Applied Sciences and Technology (INSAT), Centre Urbain Nord, BP 676,1080 Tunis Cedex, Tunisia

Correspondence should be addressed to Abir Lassoued; lassoued.abir5@gmail.com

Received 28 June 2017; Revised 19 September 2017; Accepted 1 October 2017; Published 26 October 2017

Academic Editor: Dimitri Volchenkov

Caption: Figure 1: Projections of the attractor related to the hyperchaotic system (1) onto the spaces (x, y, z), (x, y), (x, z), (y, z), and (w, y).

Caption: Figure 2: Time series and sensitive dependence on initial conditions (1, 1, 1, 1) (blue) and (1.001, 1, 1, 1) (red): (a) and (d) variable x; (b) and (c) variable w.

Caption: Figure 3: The Lyapunov exponent spectrum of the hyperchaotic system (1).

Caption: Figure 4: Hopf bifurcations.

Caption: Figure 5: Different attractors forms exhibited by system (1) when parameter m varies.

Caption: Figure 6: Projections of the attractor related to the adjusted hyperchaotic system (12) onto the spaces (x, w), (x, z), and (y, z).

Caption: Figure 7: Circuit design of the square root function with MultiSIM.

Caption: Figure 8: Experimental results of the square root function.

Caption: Figure 9: Circuit design of the hyperchaotic system with MultiSIM.

Caption: Figure 10: Simulation results of the hyperchaotic system with MultiSIM Software.

Table 1: Stability analysis of system (1).

Equilibrium          Jacobian matrix
point

[H.sub.1]       [mathematical expression
                    not reproducible]

[H.sub.2]       [mathematical expression
                    not reproducible]

[[DELTA].sub.1] = 2dx - 2cx + (r - 1)m[x.sup.r-2]

[H.sub.3]       [mathematical expression
                    not reproducible]

[[DELTA].sub.2] = -2dx - 2cx - (r - 1)m[x.sup.r-2]

Equilibrium          Corresponding eigenvalues
point

[H.sub.1]               [[lambda].sub.1] = 0
                     [[lambda].sub.2] = -0.001
                 [[lambda].sub.3] = -0.465 + 0.945i
                 [[lambda].sub.4] = -0.465 - 0.945i

[H.sub.2]            [[lambda].sub.1] = -0.001
                     [[lambda].sub.2] = -1.273
                 [[lambda].sub.3] = 0.171 + 1.232i
                 [[lambda].sub.4] = 0.171 - 1.232i
[[DELTA].sub.1] = 2dx - 2cx + (r - 1)m[x.sup.r-2]

[H.sub.3]            [[lambda].sub.1] = -0.001
                     [[lambda].sub.2] = -2.297
                 [[lambda].sub.3] = 0.683 + 1.945i
                 [[lambda].sub.4] = 0.683 - 1.945i
[[DELTA].sub.2] = -2dx - 2cx - (r - 1)m[x.sup.r-2]

Equilibrium     Stability analysis
point

[H.sub.1]          Stable point

[H.sub.2]         Unstable point

[[DELTA].sub.1] = 2dx - 2cx + (r - 1)m[x.sup.r-2]

[H.sub.3]         Unstable point

[[DELTA].sub.2] = -2dx - 2cx - (r - 1)m[x.sup.r-2]

Table 2: The LEs of some typical attractors of system (1).

m      [LE.sub.1]    [LE.sub.2]    [LE.sub.3]    [LE.sub.4]

3.6         0             0           -0.42          0.8
3.8       0.05            0           -0.01         -0.92
6         0.18          0.007           0           -1.10
6.5       0.17          0.01            0           -1.16

m             Attractor

3.6         Periodic orbit
3.8       Chaotic attractor
6       Hyperchaotic attractor
6.5     Hyperchaotic attractor

Table 3: Comparative analysis with related hyperchaotic systems.

Hyperchaotic system        Lyapunov exponents        Kaplan-Yorke
                                                       dimension

Proposed                   [LE.sub.1] = 0.231      [D.sub.KY] = 3.19
hyperchaotic system        [LE.sub.2] = 0.020
                             [LE.sub.3] = 0
                           [LE.sub.4] = -1.169

Piecewise linear           [LE.sub.1] = 0.064      [D.sub.KY] = 3.089
hyperchaotic circuit       [LE.sub.2] = 0.033
[25]                         [LE.sub.3] = 0
                           [LE.sub.4] = -1.098

Hyperchaotic               [LE.sub.1] = 0.142      [D.sub.KY] = 3.134
hyperjerk system           [LE.sub.2] = 0.046
[26]                         [LE.sub.3] = 0
                           [LE.sub.4] = -1.396
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Title Annotation:Research Article
Author:Lassoued, Abir; Boubaker, Olfa
Publication:Complexity
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Date:Jan 1, 2017
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