Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable

Accepted Manuscript Regular paper Dynamics and circuit realization of a no–equilibrium chaotic system with a boostable variable Viet-Thanh Pham, Akif Akgul, Christos Volos, Sajad Jafari, Tomasz Kapitaniak PII: DOI: Reference:

S1434-8411(17)30926-3 http://dx.doi.org/10.1016/j.aeue.2017.05.034 AEUE 51905

To appear in:

International Journal of Electronics and Communications

Received Date: Accepted Date:

19 April 2017 20 May 2017

Please cite this article as: V-T. Pham, A. Akgul, C. Volos, S. Jafari, T. Kapitaniak, Dynamics and circuit realization of a no–equilibrium chaotic system with a boostable variable, International Journal of Electronics and Communications (2017), doi: http://dx.doi.org/10.1016/j.aeue.2017.05.034

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Dynamics and circuit realization of a no–equilibrium chaotic system with a boostable variable Viet–Thanh Phama,e,∗, Akif Akgulb , Christos Volosc , Sajad Jafarid , and Tomasz Kapitaniake a

School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam b Department of Electric and Electronic Engineering, University of Sakarya, Sakarya, Turkey c Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, GR-54124, Greece d Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran e Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland

Abstract Recent evidence suggests that there exists chaos in a few no–equilibrium systems. A chaotic system without equilibrium is proposed and studied in this work. It is worth noting that due to the absence of equilibrium, such a system belongs to a class of systems with hidden attractor. Dynamics properties and the feasibility of the system are investigated by using numerical simulations and circuit implementation. Interestingly, this no–equilibrium system has one variable with the freedom of offset boosting. Keywords: Chaos, Equilibrium, Hidden attractor, Boostable variable, Circuit

Corresponding author Email address: [email protected], [email protected] (Viet–Thanh Pham) ∗

Preprint submitted to Elsevier

May 23, 2017

1. Introduction After the discovery of Lorenz’s system [1], a considerable amount of chaotic systems have been reported in the literature [2–7]. Several attempts have been made to explore special chaotic systems such as memristive chaotic system with heart–shaped attractors [8], simplest chaotic circuits [9–15], electro–mechanical oscillator [16], chaotic flow with a continuously adjustable attractor dimension [17], systems with multi–wing butterfly chaotic attractors [18], or systems with multi–scroll chaotic attractors [19–21]. In recent years, there has been an increasing interest in special chaotic systems without equilibrium [22–24]. From the viewpoint of computation, numerical localization of the attractors in such no–equilibrium systems is challenging due to the absence of transient processes leading to them from the neighbourhoods of unstable equilibrium points [25–27]. In other words, attractors of systems without equilibrium are “hidden” [28, 29]. Wei proposed a no–equilibrium chaotic system with six terms by applying a constant to the Sprott D system [30]. In oder to find simple quadratic flows with no equilibria, Jafari et al. developed a systematic search routine [22]. When constructing chaotic systems with any number of equilibria, Wang and Chen found a new no–equilibrium system [23]. The presence of multiple attractors in a three–dimensional (3D) system without equilibrium point was observed in [31]. In addition, a random number generator based on a 3D chaotic system with no equilibrium was implemented by Akgul et al. [32]. Moreover, four–dimensional hyperchaotic systems without equilibrium were aslo reported in [33–35]. However, there are still various attractive features related to no–equilibrium systems which should be studied further [36–39]. 2

Motivated by considerable interest in discovering chaotic systems with hidden attractors, this work makes a contribution to the known list of systems with hidden attractors by investigating a chaotic system without equilibrium. In the next section, the description of the no–equilibrium system is presented briefly. Dynamical properties of such a system without equilibrium are studied in Sect. 3. Realization of the theoretical system is discussed and experimental results are reported in Sect. 4. Finally, conclusion remarks are drawn in Sect. 5. 2. Description of the system without equilibrium In this work we consider a 3D autonomous system given by    x˙ = y + a,   y˙ = −x + z,     z˙ = −bx2 + z 2 + c,

(1)

in which x, y, z are three state variables while a, b, c are three positive parameters. In order to find the equilibrium of system (1), we solve x˙ = 0, y˙ = 0, z˙ = 0, or y + a = 0,

(2)

−x + z = 0,

(3)

−bx2 + z 2 + c = 0.

(4)

x = z.

(5)

From Eq. (3), we have

3

By substituting Eq. (5) into Eq. (4), we get x2 =

c , b−1

(6)

for b 6= 1. Moreover, it is trivial to verify that Eq. (6) is insistent for b < 1.

(7)

In other words, system (1) has no any equilibrium for b < 1. The well–known example of systems without equilibrium is the conservative Sprott A system [40]. The conservative Sprott A system is important because it is a special case of the Nose–Hoover oscillator, which describes several natural phenomena [41–43]. Therefore, it suggests that such systems may have practical and theoretical importance. Interestingly, system (1) generates chaotic behavior despite the absence of equilibrium. For example, chaotic attractors of system (1) are shown in Fig. 1 for a = 1, b = 0.8, c = 2 and initial conditions (x(0), y(0), z(0)) = (0, 3, 0). Calculated Lyapunov exponents and Kaplan–York dimension of the system without equilibrium (1) are L1 = 0.026, L2 = 0, L3 = −6.8624, and DKY = 2.0038, respectively. It is worth noting that from the computation point of view, system (1) is a system exhibiting hidden attractors, which play important roles in a wide range of scientific and engineering processes [44–47]. Remarkably, optimizing the maximum Lyapunov exponent of the chaotic system is not a trivial task and should be studied further [48, 49]. Metaheuristics can help to find the higher maximum Lyapunov exponent [49].

4

2

2

0

0

−2

−2

−4

z

y

4

−4

−6

−6

−8

−8 −15

−10

−5

0

−10 −15

5

−10

x

−5

0

5

x

(a)

(b)

2 0

5 0

z

z

−2 −4

−5 −6 −10 10

−8

10 0

−10 −10

−5

0

0 −10

5

y

y

(c)

−10 −20

x

(d)

Figure 1: Chaotic attractors without equilibrium of system (1) in (a) x − y plane, (b) x − z plane, (c) y − z plane, and (d) x − y − z space, for a = 1, b = 0.8, c = 2, and initial conditions (x(0), y(0), z(0)) = (0, 3, 0).

5

3. Dynamical properties of the system without equilibrium System without equilibrium (1) has been investigated by varying the bifurcation parameter c in the range from 1.9 to 2.9. Figures 2, 3 present the bifurcation diagram and the diagram of maximal Lyapunov exponents (MLEs) of system (1), respectively. As can be seen from Figs. 2, 3, there is the presence of a classical period–doubling route to chaos when decreasing the value of the parameter c. For c ≥ 2.07, system (1) generates periodical oscillations. For instance, period–1, period–2 and period–4 oscillations of system (1) are illustrated in Figs. 4a, 4b, 4c, respectively. For c < 2.07, system (1) displays more complex behaviors, for example chaos (see Fig. 4d). It is noted that the system is unbounded for c < 1.887. Similarly, we have changed the value of the parameter b to discover the dynamics of system (1). The bifurcation diagram of system (1) for b ∈ [0.6, 0.82] is reported in Fig. 5. As shown in Fig. 5, the presence of a classical period–doubling route to chaos is observed when increasing the value of the parameter b. As others have highlighted [39], researchers have shown an increasing interest in chaotic flows that provide offset boosting by a single constant. It is interesting that system without equilibrium (1) is also a variable–boostable chaotic one. From Eq. (1), it is simple to verify that we can boost the variable y by varying the parameter a. It means that the parameter a boosts the amplitude of the variable y [39]. We fix the initial conditions (x(0), y(0), z(0)) = (0, 3, 0) while increasing the boosting controller a from 0 to 4. Phase portraits of system (1) are adjusted according to the boosting controller as illustrated in Fig. 6. When increasing the boosting controller a, the chaotic signal y is boosted from a bipolar signal to a unipolar one as 6

y

2.5

2

1.5

2

2.2

2.4

2.6

2.8

c Figure 2: Bifurcation diagram of the system without equilibrium (1), for a = 1, b = 0.8 and c ∈ [1.9, 2.9].

0.04

MLE

0.03 0.02 0.01 0 2

2.2

2.4

2.6

2.8

c Figure 3: Maximum Lyapunov exponents of the system without equilibrium (1) for a = 1, b = 0.8 when varying the value of the parameter c (c ∈ [1.9, 2.9]).

7

2

2

0

0

−2

−2

−4

−4

−6

−6

−8 −15

y

y

4

−10

−5

0

−8 −15

5

(a)

(b) 4

2

2

0

0

−2

−2

−4

−4

−6

−6 −10

−5

x

4

−8 −15

−10

x

y

y

4

−5

0

−8 −15

5

−10

−5

x

x

(c)

(d)

0

5

0

5

Figure 4: Four views of limit cycles and chaos of the system without equilibrium (1): (a) period–1 oscillation (c = 2.9), (b) period–2 oscillation (c = 2.4), (c) period–4 oscillation (c = 2.1), (d) chaotic oscillation (c = 2.05) for initial conditions (x(0), y(0), z(0)) = (0, 3, 0), a = 1, and b = 0.8.

8

3.5 3

y

2.5 2 1.5 1 0.6

0.65

0.7

0.75

0.8

b Figure 5: Bifurcation diagram of the system without equilibrium (1), for a = 1, c = 2 and b ∈ [0.6, 0.82].

illustrated in Fig. 7. It is easy to see that the shapes of the signals y are similar. The average value of the variable y is reduced when increasing the boosting controller a as shown in Fig. 8a. In contrast, the boosting controller does not have an influence on the average values of the variables x, z. As can be seen in Fig. 8b, the relatively same maximal Lyapunov exponents are displayed for different values of the boosting controller.

9

4

2

2

0

0

−2

z

y

−2 −4

−4 −6

−6

−8

−8 −10 −15

−10

−5

0

−10 −10

5

−5

0

x

y

(a)

(b)

5

Figure 6: Phase portraits in (a) x–y plane and (b) y–z plane for initial conditions (x(0), y(0), z(0)) = (0, 3, 0), b = 0.8, c = 2, and different values of the boosting controller a: black for a = 0, blue for a = 2, red for a = 4.

4 2 0

y

−2 −4 −6 −8 −10 500

600

700

800

900

1000

t Figure 7: Signals y(t) for initial conditions (x(0), y(0), z(0)) = (0, 3, 0), b = 0.8, c = 2, and different values of the boosting controller a: black for a = 0, blue for a = 2, red for a = 4.

10

1

0.1

mean(x) mean(y) mean(z)

0

0.05

MLE

M

−1 −2 −3

0

−4 −5 0

1

2

3

−0.05 0

4

a

1

2

3

4

a

(a)

(b)

Figure 8: Presentations of (a) the average values and (b) maximal Lyapunov exponents for initial conditions (x(0), y(0), z(0)) = (0, 3, 0), b = 0.8, c = 2, when increasing the value of the boosting controller a from 0 to 4.

4. Circuit realization of the proposed system without equilibrium Extensive researches have shown that complex behavior of chaos is an appropriate feature for using in various engineering applications from true and pseudo random bit generation [50–53], chaotic video communication scheme via WAN remote transmission [54], audio encryption scheme [55], autonomous mobile robots [56], chaotic communication systems [57] to image encryption [58–61] etc. It is worth noting that the ability of realizing theoretical chaotic models plays very important roles in practical applications [62–67]. The implementation of chaotic systems is a major area of interest within the field of chaos [68–70]. Analog and digital approaches have been applied to realize chaotic oscillators by using different kinds of electronic devices such as common off–the–shelf electronic components [71, 72], integrated circuit technology [73, 74], micro–controller [75] or field–programmable gate array 11

Figure 9: Schematic of the circuit including eleven resistors, three capacitors (C1 = C2 = C3 = C), five operational amplifiers and two analog multipliers. Here the power supplies of all operational amplifiers and analog multipliers are ±15 VDC .

(FPGA) [76–78]. When designing chaotic oscillators with operational amplifiers, frequency limitations should be considered carefully [79, 80]. It is interesting that FPGA provides a fast prototype for verifying chaotic system [81]. The physical implementation of the mathematical system (1) is discussed in this section. The aim of our work is to realize system (1) with electronic components. Figure 9 shows the schematic of the designed circuit. As shown in Fig. 9, the circuit includes eleven resistors, three capacitors, five operational amplifiers and two analog multipliers. By applying Kirchhoff’s circuit laws into the designed circuit, we get the

12

following circuital equation   dvC1 R9 R 1  = RC R vC2 +    dt  8 R1 dvC2 dt

   

dvC3 dt

= =

1 RC 1 RC

− RR3 vC1

R V R2 2



,

R11 R v R10 R4 C3

+  − R5R10V vC2 1 +

R R6 10V



(8)

,

vC2 3 +

R V R7 2



,

in which vC1 , vC2 , vC3 are the voltages on the capacitors C1 , C2 , and C3 , respectively. By normalizing circuital equation (8) with τ =

t , RC

we obtain the dimen-

sionless system  V2  9 R  X˙ = R Y + RR2 1V ,  R8 R1  11 R Y˙ = − RR3 X + R Z, R10 R4     Z˙ = − R X 2 + R Z 2 + 10R5 10R6

(9) R V2 R7 1V

.

In system (9), the three variables X, Y and Z correspond to the voltages vC1 , vC2 , vC3 , respectively. It is simple to verify that system (9) is equivalent to the proposed system without equilibrium (1) with a = and c =

R V2 R7 1V

R V2 R2 1V

, b=

R , 10R5

.

Here we have selected the values of electronic components to realize a = 1, b = 0.8 and c = 2. Therefore, the electronic components in Fig. 9 are: R1 = R3 = R4 = R = 400 kΩ, R2 = 6 MΩ, R5 = 50 kΩ, R6 = 40 kΩ, R7 = 3 MΩ, R8 = R9 = R10 = R11 = 100 kΩ, C1 = C2 = C3 = C = 1 nF, and V1 = V2 = 15 VDC . The designed circuit has been implemented in OrCAD–PSpice and PSpice results are reported in Fig. 10. Finally, we have realized the circuit on a breadboard with off–the–shelf electronic components such as resistors, capacitors, operational amplifiers and analog multipliers. The TL081 operational amplifiers and the AD633 multipliers have been used in the electronic circuit. 13

(a)

(b)

(c)

Figure 10: PSpice chaotic attractors of the designed circuit in (a) X − Y plane, (b) X − Z plane, and (c) Y − Z plane.

However, it should be noted that the proposed circuit can be implemented with integrated circuit technology [73]. In addition, the proposed system can be verified by using a fast prototype based on FPGA and VHDL descriptions [81]. Signals of the real circuit are measured and displayed on an oscilloscope as shown in Fig. 11. It is easy to see the agreement between the experimental results and numerical results (Fig. 1) as well as PSpice results (Fig. 10).

14

(a)

(b)

(c)

Figure 11: Captured chaotic attractors of the implemented circuit in (a) X − Y plane, (b) X − Z plane, and (c) Y − Z plane.

15

5. Conclusions The present study has introduced an additional rare system without equilibrium, which has just discovered recently. Dynamics of the proposed system are studied via phase portrait, bifurcation diagram, maximal Lyapunov exponent. In spite of the absence of equilibrium, the system still generates chaotic signals. It is interesting that the system has a boostable variable. Therefore, we can obtain a unipolar chaotic signal by changing the boosting controller a. Remarkably, the no–equilibrium system has been realized conveniently by using only common electronic components. The system can be used in different chaos–based applications such as random signal generator, image encryption and especially secure communications. We believe that the system without equilibrium is suitable for chaos–based communications because there is no limitation of equilibrium. We will assess the practical applications of the new system in our future works. Acknowledgements The authors acknowledge Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong for suggesting many helpful references. This work has been supported by the Polish National Science Centre, MAESTRO Programme - Project No 2013/08/A/ST8/00/780. References [1] E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130– 141.

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