Dynamic Analysis, Circuit Implementation and Passive Control of a Novel Four-Dimensional Chaotic System with Multiscroll Attractor and Multiple Coexisting Attractors

Pramana – J. Phys. (2018) 90:33 © Indian Academy of Sciences https://doi.org/10.1007/s12043-018-1525-1

Dynamic analysis, circuit implementation and passive control of a novel four-dimensional chaotic system with multiscroll attractor and multiple coexisting attractors

BANG-CHENG LAI1,∗ and JIAN-JUN HE2

1Institute of Technology, East China Jiaotong University, Nanchang 330100, China 2College of Applied Science, Jiangxi University of Science and Technology, Ganzhou 341000, China ∗Corresponding author. E-mail: [email protected]

MS received 25 August 2017; revised 3 October 2017; accepted 10 October 2017; published online 8 February 2018

Abstract. In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and ﬁve equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.

Keywords. Chaotic system; multiple attractors; multiscroll attractor; circuit implementation; passive control.

PACS No. 05.45

1. Introduction pair of point attractors or limit cycles [7]. By using the polynomial function method, any number of coexist- The multiple coexisting attractors have been the focus ing chaotic attractors were constructed from the Sprott of research topics in nonlinear science in recent years. Bsystem[8]. Braga and Mello investigated the forma- Many academic papers have discussed the multiple tion mechanism of three types of attractors in a simple coexisting attractors in natural and artificial systems [1– chaotic system through rigorous mathematical verifica- 5]. The coexistence of multiple attractors usually means tion [9]. Tamba et al analysed an improved Colpitts that the system has multiple steady states. It provides oscillator that performs a period-1 limit cycle and a the system with the possibility of a variety of normal strange attractor for different initial conditions [10]. operating modes, which will benefit the system perfor- Pham et al put forward a no-equilibrium chaotic sys- mance. tem which coexists with a pair of strange attractors, or a In recent years, the multiple coexisting attractors in pair of limit cycles [11]. Sprott et al presented a special chaotic system has aroused great interest. The existing periodically-forced oscillator which generates an infi- research results indicated that some low-dimensional nite number of coexisting nested attractors, including nonlinear differential equations not only generate chaos, limit cycles, attracting tori and strange attractors [12]. but also generate multiple coexisting attractors. Li and Li et al proposed some variable-boostable chaotic flows Sprott discussed the multistability of Lorenz system with coexisting attractors, and found an interesting off- by computing its basin of attraction and bifurcation set boosting method for diagnosing the multistability diagram [6]. The butterfly attractor in Lorenz system of chaotic systems [13–15]. Moreover, the investiga- can be broken into two coexisting strange attractors tion of the multiple coexisting attractors has also been or two coexisting limit cycles for some special values reported in [16–22]. The study of the chaotic system of parameters. Lai and Chen proposed a new chaotic with multiscroll (or multiwing) strange attractor is also system with four coexisting strange attractors, four an interesting research topic. Previous studies showed coexisting limit cycles, two strange attractors with a that chaotic system with multiscroll attractor can be 33 Page 2 of 12 Pramana – J. Phys. (2018) 90:33

O(0, 0, 0, 0) used to secure communication and can greatly improve √ √ √ √ √ √ the encryption performance. Many effective methods A±(±r b, ±r a, ± ab, ac b/(m b + a)) √ √ √ √ √ √ have been found to construct chaotic system with multi- (± , ± , ± , /( − )), scroll attractor, such as trigonometric function method, B± s b s a ab ac b m b a piecewise-linear function method, coordinate transfor- where mation method, etc. [23–27]. √ √ √ Chaos control has become an important research topic r = cm b/(m b + a) since the famous OGY control method was put for- √ √ √ ward by Ott et al in 1990 [28]. The scientists started to s = cm b/(m b − a). believe that chaos is controllable, and many chaos control methods were presented, including adaptive control, The stability of the equilibria can be determined by impulsive control, passive control, sliding mode con- computing their eigenvalues. It is easy to get that the , − , − , − trol, etc. [29–34]. The passive control has been applied eigenvalues of O are a b c m. Thus, O is an λ in many chaotic systems [35–38]. The studies showed unstable point. The eigenvalues ± of A± should meet that the controller designed on the basis of the passive the following equations: control theory can achieve the desired control effect. 4 3 2 2 λ+ + p1λ+ + p2λ+ + (p3 + 4abr )λ+ + p4 As an effective control method, the passive control has received widespread attention of the scholars. + 4abmr2 = 0, It is extremely interesting and significant to find and 4 3 2 λ + p λ + p λ + p λ− + p = 0, analyse multiple coexisting attractors in chaotic sys- − 1 − 2 − 3 4 tems, and consider their control problems. In this paper, where p1, p2, p3, p4 are given by we present a novel 4D autonomous chaotic system with = + + − , five equilibria and investigate the multiple coexisting p1 b c m a = ( + − ) + ( − )( − 2), attractors of the system. Research results show that p2 b c a m b a √c r 2 2 the system has different types of coexisting attractors, p3 = m(b −√a)(c − r ) − 2r ab, 2 including three coexisting chaotic attractors, three coex- p4 = 2ar ab. isting limit cycles and a chaotic attractor with two limit According to the Routh–Hurwitz criterion, we know that cycles. Numerical simulations are applied to analyse the A+ is stable as long as dynamical behaviours of the system. Based on the pas- ⎧ 2 sive control theory, a controller is proposed for driving ⎪ p1 > 0, p2 > 0, p3 > − 4abr , ⎨ 2 2 the system to the origin. In addition, we also study the p4 > − 4abmr , p1 p2 > p3 + 4abr circuit implementation of the chaotic system. ⎪ ( − − 2)( + 2) ⎩⎪ p1 p2 p3 4abr p3 4abr > 2( + 2) p1 p4 4abmr and A− is stable as long as 2. A novel 4D chaotic system p1 > 0, p2 > 0, p3 > 0, p4 > 0 The chaotic system considered in this paper is described > , > 2 + 2 . p1 p2 p3 p1 p2 p3 p3 p1 p4 by the following autonomous differential equations: ⎧ Similarly, B+ is stable if ⎪ x˙ = ax − yz ⎧ ⎨ ⎪ > , > , > − 2, y˙ =−by + xz ⎨⎪ q1 0 q2 0 q3 4abs (1) > − 2, > + 2 ⎩⎪ z˙ =−cz + xy + w q4 4abms q1q2 q3 4abs 2 ⎪ ( − + 2)( + 2) w˙ =−mw + y ⎩⎪ q1q2 q3 4abs q3 4abs > 2( + 2) q1 q4 4abms where x, y, z,ware the state variables, and a, b, c, m ∈ + and B− is stable if R are the parameters. System (1) is generated from the 3D chaotic system which was proposed by Liu and Chen q1 > 0, q2 > 0, q3 > 0, q4 > 0 [39] by introducing a variable w with w˙ =−mw + y2. > , > 2 + 2 q1q2 q3 q1q2q3 q3 q1 q4 If a < b + c + m, then system (1) is dissipative with its 2 divergence ∇V = ∂x˙/∂x +∂ y˙/∂y+∂z˙/∂z+∂w/∂w˙ = where q1 = p1, q2 = (b + c −√a)m + (b − a)(√c − s ), 2 2 2 a − b − c − m < 0. The equilibria of system (1) can be q3 = m(b − a)(c − s ) − 2s ab, p4 = 2as ab. obtained√ by letting x˙ =˙y =˙z =˙w = 0. Suppose that It is well known that the number and stability of the m > a/b, then system (1) will have five equilibria equilibria play a crucial role on the dynamic behaviour as follows: of the system. The existence of multiple equilibria is an Pramana – J. Phys. (2018) 90:33 Page 3 of 12 33

100 (a) 80 (a) 80 60 60

40 40 20 20 0 y

−20 z 0 −40 −20 −60 −80 −40 −100 −60 −150 −100 −50 0 50 100 150 10 12 14 16 18 20 x (b) 100 (b) 5 80 0 60

40 −5 20

z 0 −10 −20 −15 −40 Lyapunov exponents −60 −20 −80 −150 −100 −50 0 50 100 150 −25 5 10 15 x b 100 (c) 5 80 (c) 60 0 40 −5 20 z 0 −10 −20 −15 −40 Lyapunov exponents −60 −20 −80 −100 −50 0 50 100 −25 y 5 10 15 b (d) 250 Figure 2. The bifurcation diagrams (a) and Lyapunov expo- 200 nents from initial values X0 (b), Y0 (c) with the parameter b ∈ (10, 20). 150 w 100 O(0, 0, 0, 0),

50 A±(±26.1128, ±11.6780, ±8.9443, ±17.0470), B±(±27.6159, ±12.3502, ±8.9443, ±19.0658). 0 −100 −50 0 50 100 y It can be veriﬁed that all the equilibria O, A±, B± are unstable. The Matlab simulation shows that sys- Figure 1. The four-scroll attractors of system (1) with tem (1) exists as a four-scroll chaotic attractor. The a = 4, b = 20, c = 36, m = 8: (a) x–y;(b) x–z;(c) y–z; phase portraits of system (1) plotted in ﬁgure 1 directly w (d) y– . illustrate the attractor with a = 4, b = 20, c = 36, m = 8. By using the Wolf method [40] with step size t = 0.01 and time interval t ∈[0, 500], the Lay- important factor in the generation of multiscroll attractor punov exponents of system (1) with given parameters and multiple coexisting attractors. are calculated as l1 = 1.7359, l2 = 0, l3 = 8.5942, When a = 4, b = 20, c = 36, m = 8and l4 =−53.1417. Therefore, the attractor is fractal as initial value X0 = (1, 1, 1, 1),system(1)hasthe its Lyapunov dimension D = 3 − l1/(l3 + l4) = following equilibria: 3.0281. 33 Page 4 of 12 Pramana – J. Phys. (2018) 90:33

(a) 60 (a) 60

40 50

40 20

30 x

z 0 20 −20 10 −40 0

−60 −50 0 50 −10 5 10 15 20 25 30 35 40 x m (b) 70 (b) 2

60 0

50 −2

40 −4 w 30 −6

20 −8

10 Lyapunov exponents −10

0 −60 −40 −20 0 20 40 60 −12 5 10 15 20 25 30 35 40 z m (c) 40 (c) 16 30 15 20

14 10 z

0 x 13

−10 12

−20 11 −30 −40 −30 −20 −10 0 10 20 30 10 x 5 10 15 20 25 30 35 40 m (d) 60 (d) 2 50

40 0

30 −2 w 20 −4

10 −6 0 Lyapunov exponents −8 −10 −30 −20 −10 0 10 20 30 40 −10 z 5 10 15 20 25 30 35 40 m Figure 3. The phase portraits (a), (b) and Poincaré section on y = 0(c), (d) of the coexisting two double-scroll chaotic Figure 4. The bifurcation diagrams and Lyapunov expo- attractors. nents with the parameter m ∈ (5, 40) and initial values: (a), (b) X0;(c), (d) Y0, Z0.

3. Analysis of multiple attractors different initial values will eventually lead to different The coexistence of multiple attractors in a system is attractors. A system with multiple attractors generally an extremely interesting phenomenon. It implies that has a variety of performance status for given parameters. Pramana – J. Phys. (2018) 90:33 Page 5 of 12 33

Here we shall give detailed investigation of the multiple (a) 50 attractors in system (1). It shows that system (1)has 40 abundant multiple attractors for different parameters and initial values. 30 Consider the dynamical evolution of system (1) with 20 z regard to the parameter b ∈ (10, 20).Leta = 4, c = 10 4, m = 8. We can plot the bifurcation diagrams of sys- 0 tem (1) with respect to initial values X0(1, 1, 1, 1) and −10 Y0(−1, −1, −1, −1). As shown in figure 2a, the different coloured branches of X (green colour) and Y (pink −20 0 0 −50 −40 −30 −20 −10 0 10 20 30 40 colour) imply that system (1) performs two coexisting x attractors. (b) 40 The Lyapunov exponents in figures 2band2cwhich 30 are respectively generated from X0, Y0 indicate that sys- b tem (1) is chaotic for most of the values of in the 20 interval (10, 20). We can show the coexisting attractors more clearly by using the phase portraits and Poincaré z 10 = sections. For b 12, two double-scroll strange attrac- 0 tors are observed in system (1), as shown in figure 3. It is easy to calculate that their Lyapunov dimensions −10 = . = . Dg 3 0683 and Dp 3 0612 are fractal. −20 = , = , = −40 −30 −20 −10 0 10 20 30 40 Suppose that a 4 b 9 c 4, then the bifurca- x tion diagrams and Lyapunov exponents vs. m ∈ (5, 40) 25 can be numerically obtained. Figures 4aand4b show the (c) 20 bifurcation diagram and Lyapunov exponents of sys- 15 tem (1) with initial value X . It is clear that system 0 10 (1) is chaotic for most of the values of m ∈ (5, 40) 5 with initial value X . Figures 4cand4d show the bifur- z 0 0 cation diagrams and Lypunov exponents of system (1) −5 with initial values Y (pink colour) and Z (1, 1, −1, −1) 0 0 −10 (blue-green colour). We only show the first three Lya- −15 punov exponents in figures 4band4d. The minimum −20 Lyapunov exponents are less than −10 with m ∈ (5, 40). −25 −20 −15 −10 −5 0 5 10 15 20 25 As m increases, system (1) will generate periodic and x chaotic attractors from initial values Y0, Z0.Foragiven (d) 25 parameter m, different attractors are observed in the sys- 20 tem with initial values Y0, Z0.Whenm = 6, 11, 20, 35, 15 system (1) appears respectively as a strange attractor 10 with a symmetric limit cycle, a strange attractor with 5 z two period-1 limit cycles, a strange attractor with two 0 period-2 limit cycles and three coexisting strange attrac- −5 tors, as shown in figure 5. −10 System (1) has various types of multiple coexist- −15 −20 ing attractors. We also can observe some examples of −20 −15 −10 −5 0 5 10 15 20 multiple coexisting attractors in the system for differ- x ent parameters and initial values. Here we give some types of multiple attractors of system (1) as follows: Figure 5. The multiple coexisting attractors in system (1) a = , b = , c = a m = b m = a = , b = , c = , m = with 4 9 4 and ( ) 6; ( ) 11; (i) Three limit cycles with 4 8 4 12 (c) m = 20; (d) m = 35. and initial values (±1, ±1, ±1, ±1), (1, 1, 1, −1) in figure 6a; (ii) two limit cycles with a = 4, b = 7, c = 4, m = 20 and initial values (±1, ±1, ±1, ±1) in a strange attractor and two limit cycles with a = 4, figure 6b; (iii) two strange attractors and a limit cycle b = 26, c = 4, m = 10 and initial values (±1, ±1, with a = 4, b = 23, c = 4, m = 10 and initial val- ±1, ±1), (1, 1, 1, −1) in figure 6d; (v) three strange ues (±1, ±1, ±1, ±1), (1, 1, 1, −1) in figure 6c; (iv) attractors and a limit cycle with a = 4, b = 35, 33 Page 6 of 12 Pramana – J. Phys. (2018) 90:33

(a) 15 (e) 100 80 10 60 5 40 z 0 z 20

0 −5 −20 −10 −40

−15 −60 −15 −10 −5 0 5 10 15 −100 −80 −60 −40 −20 0 20 40 60 80 x x

(b) 15 (f) 60

10 40

5 20 0 z z 0 −5 −20 −10

−15 −40

−20 −60 −20 −15 −10 −5 0 5 10 15 20 −50 0 50 x x

10 0 z z −5 0 −10 −10 −15 −20 −20 −25 −30 −30 −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 x x

(d) 40 (h) 30 30 20 20 10 10 0 0 z −10 z −20 −10

−30 −20 −40 −30 −50 −60 −40 −60 −40 −20 0 20 40 60 −50 0 50 x x

Figure 6. The multiple coexisting attractors in system (1) with different parameters and initial values. c = 4, m = 6 and initial values (±1, ±1, ±1, ±1), c = 12, m = 20 and initial values (±1, ±1, ±1, ±1) in (1, 1, ±1, −1) in figure 6e; (vi) two strange attractors figure 6g; (viii) two strange attractors with a = 4, b = 9, and two limit cycles with a = 4, b = 34, c = 4, m = 9 c = 13, m = 20 and initial values (±1, ±1, ±1, ±1) and initial values (±1, ±1, ±1, ±1), (1, 1, ±1, −1) in in figure 6h. Table 1 summarises the corresponding figure 6f; (vii) two limit cycles with a = 4, b = 9, parameter values, initial values, maximum Lyapunov Pramana – J. Phys. (2018) 90:33 Page 7 of 12 33

Table 1. Multiple attractors of system (1) with different parameters and initial values. Parameters Initial values Types of attractors MLEs Figures

a = 4, b = 8 (±1, ±1, ±1, ±1) Three limit cycles Lp = Lg = Lbg = 0 Figure 6a c = 4, m = 12 (1, 1, 1, −1) a = 4, b = 7 (±1, ±1, ±1, ±1) Two limit cycles Lp = Lg = 0 Figure 6b c = 4, m = 20 a = 4, b = 23 (±1, ±1, ±1, ±1) Two strange attractors Lp = 0 Figure 6c c = 4, m = 10 (1, 1, 1, −1) and a limit cycle Lg = Lbg = 0.3032 a = 4, b = 26 (±1, ±1, ±1, ±1) One strange attractor and Lg = Lbg = 0 Figure 6d c = 4, m = 10 (1, 1, 1, −1) two limit cycles Lp = 0.7061 a = 4, b = 35 (±1, ±1, ±1, ±1) Three strange attractors Lg = 0, Lbg = 0.473 Figure 6e c = 4, m = 6 (1, 1, ±1, −1) and a limit cycle Lp = Lb = 0.2752 a = 4, b = 34 (±1, ±1, ±1, ±1) Two strange attractors Lg = Lbg = 0 Figure 6f c = 4, m = 9 (1, 1, ±1, −1) and two limit cycles Lp = Lb = 0.2962 a = 4, b = 9 (±1, ±1, ±1, ±1) Two limit cycles Lp = Lg = 0 Figure 6g c = 12, m = 20 a = 4, b = 9 (±1, ±1, ±1, ±1) Two strange attractors Lp = Lb = 0.4695 Figure 6h c = 13, m = 20

Figure 7. The circuit diagram of system (1). 33 Page 8 of 12 Pramana – J. Phys. (2018) 90:33 exponents (MLEs) of the coexisting attractors of system (1) that are shown in ﬁgure 6,whereLp, Lg, Lbg, Lb respectively represent the MLEs of the pink colour, green colour, blue-green colour and blue colour attractors.

4. Electronic circuit implementation

This section will consider the electronic circuit implementation of system (1). A circuit diagram containing four channels with respect to the variables x, y, z,wof system (1) is given in figure 7. The circuit is constructed by applying the chaotic circuit theory and modular design method. All the resistors, capacitors, operational amplifiers and analog multipliers are combined together to realise the addition, subtraction and multiplication operations of system (1). The values of the circuit elements are related to the parameters a, b, c, m of system (1). After scaling up system (1) and doing some simple calculations, the values of the circuit elements can be determined. C1 = C2 = C3 = C4 = 10 nF, R1 = R2 = R3 = R4 = 10 k Figure 8. The four-scroll attractors of system (1) with R5 = R6 = R7 = R8 = R13 = 10 k, a = 4, b = 20, c = 36, m = 8 in the oscilloscope: R9 = R10 = R11 = R14 = 100 k, (a) x–z;(b) y–z. R16 = R17 = R18 = R19 = 10 k, R22 = R23 = R24 = R25 = 0.5k, and R12 = 10/36 ≈ 0.2778 k,R15 = 10/8 = 1.25 k R20 = 10/4 = 2.5k,R21 = 10/20 = 0.5k. Running the circuit, we can see a four-scroll attractor from the oscilloscope as shown in figure 8. Obviously, the oscilloscope figures are consistent with the numerical simulation figures in figure 1. Thus, the four-scroll chaotic attractor of system (1) with the parameters a = 4, b = 20, c = 36, m = 8 is realised by the designed circuit. Reset R12=10/4=2.5k,R15 = 10/8 = 1.25 k, R20 = 10/4 = 2.5k,R21= 10/12 ≈ 0.8333 k, we can observe two coexisting attractors from the oscilloscope as shown in figure 9. The oscilloscope figures are consistent with the numerical simulation figures in figure 3a. It means that the coexisting attractors of = = = = system (1) with a 4, b 12, c 4, m 8 are exper- Figure 9. The coexisting attractors in the oscilloscope of sys- imentally realised. By resetting R12 = 10/4 = 2.5k, tem (1) with a = 4, b = 12, c = 4, m = 8 and initial values: R15 = 10/6 ≈ 1.6667 k,R21= 10/9 ≈ 1.1111 k (a) (1, 1, 1, 1);(b) (−1, −1, −1, −1). and R20 = 10/4 = 2.5k, we can realise a strange attractor and a limit cycle of system (1) with a = 4, b = 9, c = 4, m = 6 as shown in figure 10. Obviously, the oscil- R15 = 10/35 ≈ 0.2835 k,R21= 10/9 ≈ 1.1111 k, loscope figures in figure 10 isthesameasfigure5a. Simi- R20 = 10/4 = 2.5k, three coexisting strange attrac- larly, if we select the resistors as R12 = 10/4 = 2.5k, tors of system (1) with a = 4, b = 9, c = 4, m = 35 can Pramana – J. Phys. (2018) 90:33 Page 9 of 12 33

well known that the passive characteristics of the system can maintain the internal stability of the system. Thus, a controller can be used to force the closed-loop system to be passive for obtaining the stability of the system. In essence, the passive control is a smooth nonlinear feedback method which has some notable advantages including simple control mode, ease in implementation, clear physical explanation, etc. The passive control can also achieve stabilisation more quickly than some other control methods. For the following nonlinear affine system ξ˙ = f (ξ) + g(ξ)u (2) ς = h(ξ) with the state variable ξ ∈ Rn, the input u ∈ Rm and output ς ∈ Rm. f and g are smooth vector fields, h is a smooth mapping. Based on the passive theory [35,36], system (2) is said to be passive if there exists a real number β such that the following inequality holds: t Figure 10. The coexisting attractors in the oscilloscope of uT (τ)y(τ)dτ ≥ β (3) system (1) with a = 4, b = 9, c = 4, m = 6 and initial 0 values: (a) (1, 1, 1, 1);(b) (−1, −1, −1, −1). or there exists a real number β and ρ>0 such that t t T (τ) (τ) τ + β ≥ ρ T (τ) (τ) τ be realised as the oscilloscope figures are shown in fig- u y d y y d (4) 0 0 ure 11 which is consistent with figure 5d. for any t ≥ 0. It is well known that the passive system (1) can be stabilised at the origin ξ = 0byusingthe controller u =−ϕ(ξ). 5. Passive control of the system If system (2) has relative degree [1, 1,...,1] at ξ = 0 (L h(0) is non-singular), then system (2) can be rewrit- In order to control the chaos of system (1), we shall g ten as follows [35,36]: design a controller according to the passive control theory. The designed controller will eliminate the chaos γ˙ = f0(γ ) + d(γ, ς)ς (5) of system (1) and stabilise the system to the origin. It is ς˙ = φ(γ,ς) + ψ(γ,ς)u.

Figure 11. The coexisting attractors in the oscilloscope of system (1) with a = 4, b = 9, c = 4, m = 35 and initial values: (a) (1, 1, 1, 1);(b) (−1, −1, −1, −1);(c) (1, 1, −1, −1). 33 Page 10 of 12 Pramana – J. Phys. (2018) 90:33

By using the feedback input u = π(ξ)+ κ(ξ)v,system (a) 100 (5) can be rendered passive. 80 Applying the control input, then the controlled system 60 (1) can be given by 40 ⎧ 20

y 0 ⎪ x˙ = ax − yz + u1 ⎨ −20 y˙ =−by + xz + u2 ⎪ ˙ =− + + w (6) −40 ⎩ z cz xy −60 2 w˙ =−mw + y . −80 −100 It is easy to know that the controlled system (1) can be −150 −100 −50 0 50 100 150 T x rewritten as the normal form (5) with γ =[γ1,γ2] , γ = γ = w ς =[ς ,ς ]T ς = ς = 1 z, 2 , 1 2 , 1 x, 2 y, (b) 250 =[ , ]T x u u1 u2 and 200 y z T w f0(γ ) =[−cγ1 + γ2, −mγ2] , 150

φ(γ,ς) =[ ς − ς γ , − ς + ς γ ]T , 100 a 1 2 1 b 2 11 ς 0 10 50 (γ, ς) = 2 ,ψ(γ,ς)= . state variables d ς 0 2 01 0 Our primary objective is to design a suitable control −50 T =[ , ] −100 input u u1 u2 such that system (6) is passive. Let 0 2 4 6 8 10 the Lyapunov function be as follows: time 1 V (γ, ς) = W(γ ) + ς T ς, (7) Figure 12. The chaotic attractors of system (6) with 2 u1 = u2 = 0: (a) x–y;(b) time series. where W(γ ) is the Lyapunov function of f0(γ ) which can be expressed as d V (γ, ς) ≤ νT ς − αςT ς. (9) 1 dt W(γ ) = (γ 2 + γ 2). (8) 2 1 2 By integrating inequality (9), one has (γ, ς) − (γ ( ), ς( )) It is easy to verify that V V 0 0 t t ≤ νT (τ)ς(τ) τ − α ς T (τ)ς(τ) τ. d 1 2 1 d d (10) (γ ) =− γ − γ − − γ 2 ≤ 0 0 W c 1 2 m 2 0 dt 2c 4c Let β = V (γ (0), ς(0)), then the inequality becomes with 4mc ≥ 1. Thus, the zero dynamics γ˙ = f0(γ ) of t t system (5) is stable. It follows that νT (τ)ς(τ)dτ +β ≥ α ς T (τ)ς(τ)dτ +V (γ, ς) 0 0 d ∂ t V (γ, ς) = W(γ )γ˙ +˙ςς ≥ α ς T (τ)ς(τ)dτ. (11) dt ∂γ 0 ∂W(γ ) Consequently, system (6) with the following control ≤ d(γ, ς)ς ∂γ input +[φ(γ,ς)+ψ(γ,ς) ]ς. u T u =[−(a + α)x + v1,(b − α)y − xz − yw + v2] If we let the control input (12) T is rendered to be passive. It means that system (1) can 10 ∂W(γ ) u = −φ(γ,ς)− d(γ, ς) −ας+v be stabilised at the origin O by using the controller (12). 01 ∂γ Some simulations are given to verify the effectiveness T = , = = −(a+α)ς1 +v1,(b−α)ς2 −ς1γ1 − ς2γ2 + v2 , of the designed controller. In system (6), let a 4 b 20, c = 36, m = 8 and initial value X0 = (1, 1, 1, 1).If T where α>0 is a real number and v =[v1,v2] is an there is no control input, i.e. u1 = u2 = 0, then system external signal with respect to the reference input, then (6) has a four-scroll chaotic attractor (ﬁgure 12)which we have is in accordance with ﬁgure 1. If the control input is Pramana – J. Phys. (2018) 90:33 Page 11 of 12 33

(a) 1.5 References

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