A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation

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1 Pramana J. Phys. (21) 9:2 Indian Academy of Sciences A chaotic jerk system with nonhyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation KARTHIKEYAN RAJAGOPAL 1, VIETTHANH PHAM 2,, FADHIL RAHMA TAHIR 3, AKIF AKGUL 4, HAMID REA ABDOLMOHAMMADI and SAJAD JAFARI 6 1 Center for Nonlinear Dynamics, College of Engineering, Defence University, Bishoftu, Ethiopia 2 Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Electrical Engineering Department, University of Basrah, Basra, Iraq 4 Department of Electrical and Electronic Engineering, Faculty of Technology, Sakarya University, Adapazarı, Turkey Department of Electrical Engineering, Golpayegan University of Technology, Golpayegan, Iran 6 Biomedical Engineering Department, Amirkabir University of Technology, Tehran , Iran Corresponding author. phamvietthanh@tdt.edu.vn MS received 7 July 217; revised 3 October 217; accepted 12 November 217; published online 9 March 21 Abstract. The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with nonhyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the timedelay effects on the proposed system. Realisation of such a system is presented to verify its feasibility. Keywords. PACS Nos Chaos; jerk system; equilibrium; time delay; circuit..4.ac; 2.3.Ks 1. Introduction In the past decades, chaotic phenomenon in nature, economics, physics and especially engineering has attracted the interest of researchers [1,2]. After the studies about Lorenz s system [3] and Rössler s system [4], there are many researches on chaotic systems such as Chen s system [], simple chaotic flows [6], memristorbased systems [7], Lorenztype systems with multiwing butterfly chaotic attractors [], systems with multiscroll attractors [9 ], with multiple attractors [12 14], with extreme multistability [1 1], with megastability [19], with hidden attractors [2 26] and so on. Some chaotic systems are special because of their equilibria [27 3]. Systems with no equilibria [31,32], stable equilibria [33], line of equilibria [34], plane of equilibria [3], curve of equilibria [36 3] and surface of equilibria [39] are such systems. Equilibrium points of chaotic systems are often investigated because they can be used to study the types of systems, the shapes of attractors, or amplitude control [2,4 42]. Conventional threedimensional systems often have hyperbolic equilibrium, of which the real parts of eigenvalues are nonzero. Chaos in such hyperbolic system is proved by Silnikov criterion [43,44]. It is interesting that there are a few chaotic systems having nonhyperbolic equilibria [4,46], of which the real part of the eigenvalues is zero. Such chaotic systems are abnormal and it is difficult to apply the Silnikov criterion to verify the emergence of chaos in them. In fact, systems with nonhyperbolic equilibria have been the source of inspiration for designing some very rare chaotic flows [47,4], and this indicates the importance of knowing them. Motivated by the special features of nonhyperbolic system, in this work we present a new threedimensional autonomous system with only one nonhyperbolic equilibrium. In the next section, the new system is introduced and its dynamics are investigated. To explore the complex dynamics of the system, we study the stability of equilibrium, bifurcation diagram and Lyapunov exponents. The delay effects on the novel system are

2 2 Page 2 of Pramana J. Phys. (21) 9:2 discussed in 3. Section 4 presents an electronic circuit implementation of the system to verify its feasibility. 1 = δ >, 2 = δ 1 δ >, δ 3 δ 2 Finally, concludes our work. δ 1 δ 3 => δ 3 δ 2 δ 1 >, (2) δ 3 2. Novel chaotic jerk system where In physics, time derivative of acceleration is called the δ = 1, δ 1 = a 6a 7, δ 2 = a 2a a 7, δ 3 = a 1 a 2. jerk. Any dynamical system which can be defined by an equation like... x = j (x, ẋ, ẍ) is called a jerk system. By changing the variables, the equation can be written as a set of firstorder differential equations [49]. Such systems are very important in nonlinear dynamics. For example, the simplest possible chaotic system is in this form []. Here we propose a novel jerk system with seven parameters and three nonlinear terms as follows: ẋ = a 1 y, ẏ = a 2 z, ż = x + a 4 z 2 + a xy + a 6 xz + a 7, (1) where a i for i [1, 7] are the parameters of the system. Figure 1 shows the 2D phase portraits of the novel jerk system for a 1 = 1, a 2 = 1, = 1, a 4 = 2.69, a = 1, a 6 = 1, a 7 = 1 and initial conditions ( 3,, 1). Note that there is nothing special about this initial condition. It has been mentioned only to allow readers the possibility of reproducing the results. It is simple to verify that the equilibrium point of system (1)isx = a 7 /, y =, z = and its characteristic equation is λ 3 + a 6a 7 λ 2 + a 2a a 7 λ a 1 a 2 =. According to the Routh Hurwitz criterion, all the principal minors need to be positive in order to have stable equilibria. The principal minors are The condition for having unstable equilibria is that any one of the principal minors 1, 2 and 3 be negative. For parameters a 1 = 1, a 2 = 1, = 1, a 4 = 2.69, a = 1, a 6 = 1, a 7 = 1 the principal minors are 1 = 1 >, 2 = showing that the equilibrium may be unstable. The eigenvalues for the given parameter values are λ 1 = 1,λ 2,3 =±i. Table1 shows the range of parameters for stable and unstable regions of the equilibrium in system (1) and figure 2 shows the stable and unstable regions for the real part of eigenvalue λ 2 (λ 1 always has a negative real value while λ 2 and λ 3 are complex conjugate pairs). The red marker in the figures shows the transition point from stable to unstable region. As can be seen from table 1, the parameters a 2 and a 4 do not have any effect on the real part of the complex eigenvalues. The calculation of Lyapunov exponents (LEs) of a nonlinear system defines the convergence and divergence of the states. The LEs of system (1) are calculated as L 1 =.2, L 2 =, L 3 = and the Kaplan Yorke dimension is The sum of LEs is negative showing that the system is dissipative. Note that there are different methods for calculating Lyapunov exponents. These methods can sometimes result in different values [1 4]. In this paper, we have used the method proposed in []. To study the effect of the parameters on system (1), we plot the bifurcation diagram and LE diagrams when parameter a 6 changes (figure 3). Figure 1. 2D phase portraits of the novel jerk system for a 1 = 1, a 2 = 1, = 1, a 4 = 2.69, a = 1, a 6 = 1, a 7 = 1 and initial conditions ( 3,, 1).

3 Pramana J. Phys. (21) 9:2 Page 3 of 2 Table 1. Range of parameters for stable and unstable eigenvalues. Parameter Stable eigenvalues (negative real part) Unstable eigenvalues (positive real part) Eigenvalues with no real part a 1.9 a 1 < 1 1 > a 1 > 1.6 a 1 = 1 1 > > < 1 = 1 a 7 1. > a 7 > 1 1 a 7 <.997 a 7 = 1 a 1 > a > > a > 1 a = 1 a 6 1 > a 6 > > a 6 > 1 a 6 = 1 a 2 No change in real part No change in real part a 4 Figure 2. (a f) Stable and unstable regions for parameters with real part of eigenvalue (λ 2 ). Figure 3. (a) Bifurcation of system (1) with changing a 6 with initial conditions of ( 3,, 1) and reinitialising the initial conditions for every iteration with the ending value of the states from previous iteration. (b) First and second Lyapunov exponents (LEs) in the same range of a 6 (the third LE is out of scale). It can be seen from figure 3 that the system displays the routine period doubling route to chaos. In this range, the stability of the equilibrium changes, while the chaotic attractor exists in both conditions. Such a feature is very rare in dynamical systems. 3. Timedelay jerk system (TDJS) Timedelayed differential equation is important in realtime engineering applications [6,7]. For example, both integer and fractionalorder memristor

4 2 Page 4 of Pramana J. Phys. (21) 9:2 timedelayed chaotic systems and their dynamic properties have been discussed in [,9], or a parameter identification problem for a general timedelayed chaotic system is considered and analysed in [6]. Motivated by the above discussions, we are interested in investigating the timedelay effects on system (1)and hence we introduce multiple time delays in the third equation of (1) and the dimensionless model of the timedelay jerk system is given by ẋ = a 1 y, ẏ = a 2 z, ż = x(t τ 1 ) + a 4 z(t τ 3 ) 2 + a x(t τ 1 )y(t τ 2 ) + a 6 x(t τ 1 )z(t τ 3 ) + a 7, (3) where a i for i [1, 7] are the parameters of the TDJS and τ i for i [1, 3] are the multiple time delays of the system. The equilibrium points of the delayed and nondelayed systems will be the same as at equilibrium points the effect of time delays is zero. For linearising the TDJS system, let us replace x = x + x i, y = y + y i, z = z + z i and derive the Jacobian matrix as For the parameter values a 1 = 1, a 2 = 1, = 1, a 4 = 2.69, a = 1, a 6 = 1, a 7 = 1, the characteristic equation has an absolute minimum of 1 at λ =, and so the characteristic equation has only imaginary solutions. Hence, assuming that the eigenvalues are purely imaginary, we use λ = iθ with θ>in(4), (iθ) 3 + a 6a 7 (iθ) 2 e iθτ 3 + a 2a a 7 iθe iθτ 2 a 1 a 2 e iθτ 1. (6) Using commensurate time delay τ, using the parameter values and equating real and imaginary terms, θ 2 cos(τθ) θ sin(τθ) + cos(τθ) = θ 3 θ 2 sin(τθ) θ cos(τθ) cos(τθ) =. (7) Solving eq. (7) forτ =.6, we get 23.4 as the real part and ± 1.37 as the imaginary part. Hence the eigenvalues are complex conjugate pair with negative real parts (stable focus). Figure 4 shows the 2D phase portraits of the TDJS for τ 1 =.,τ 2 =.1,τ 3 =.1 and initial conditions ( 3,, 1). Various algorithms based on chaos synchronisation are proposed for the estimation of Lyapunov exponent of timedelayed dynamical systems [61 63]. In this paper, 1 J E = e λτ 1 e λτ 1 + a y e λτ 1 + a 6 z e λτ 1 a x e λτ 2 a 6 x e λτ 3 + 2a 4 z e λτ 3, (4) where x, y and z are the equilibrium points of the TDJS systems. The generalised characteristic equation of the TDJS with time delays τ 1,τ 2,τ 3 is λ 3 + a 6a 7 λ 2 e λτ 3 + a 2a a 7 λe λτ 2 a 1 a 2 e λτ 1. () we adopted the technique by employing the synchronisation of identical systems coupled by linear negative feedback mechanism [63] for finding the exact Lyapunov exponents of the TDJS. The calculated Lyapunov exponents are L 1 =.314, L 2 =, L 3 = To investigate the impact of the parameters and time delays on the TDJS system, we derive the bifurcation plots. In our discussion, we choose the parameter a 6 as Figure 4. 2D phase portraits of the TDJS for a 1 = 1, a 2 = 1, = 1, a 4 = 2.69, a = 1, a 6 = 1, a 7 = 1, τ 1 =., τ 2 =.1,τ 3 =.1 and initial conditions ( 3,, 1).

5 Pramana J. Phys. (21) 9:2 the bifurcation parameter and the other parameters are taken as a1 = 1, a2 = 1, a3 = 1, a4 = 2.69, a = 1, a7 = 1 and time delays as τ1 =., τ2 =.1, τ3 =.1. Figure shows the bifurcation plot for the parameter a6. Page of 2 To plot the timedelay impact, we plot the bifurcation of TDJS with the time delays (figure 6) and the parameter values fixed as a1 = 1, a2 = 1, a3 = 1, a4 = 2.69, a = 1, a6 = 1, a7 = 1 with the initial conditions ( 3,, 1) and varying the respective time delays to get the bifurcation plots while the others are fixed at τ1 =., τ2 =.1, τ3 = The electronic circuit implementation Figure. Bifurcation of TDJS with a6 for the initial conditions ( 3,, 1) which is reinitialised after every iteration with the end values of the state trajectories. In this section, we present a realisation of theoretical system (1) by using electronic components [64]. It should be noted that the y and z outputs in the jerk system have noiselike behaviours because their signal values are very low. They are in the interval of 1. and 1. Therefore, we must scale them to increase their amplitude values. For scale process, let X = x, Y = y, = z. The new variables X ; Y ; become the following after scale process: a1 X = y, Y = a2 z, a4 = a3 x + z 2 + a x y + a6 x z + a7. () Figure 6. Bifurcation of TDJS with τ1, τ2, τ3. The parameter values are a1 = 1, a2 = 1, a3 = 1, a4 = 2.69, a = 1, a6 = 1, a7 = 1, τ1 =., τ2 =.1, τ3 =.1 and initial conditions are ( 3,, 1).

6 V V V 2 Page 6 of Pramana J. Phys. (21) 9:2 C1 1n X 1k X R3 V3 1Vdc V4 1Vdc Y R1 2k 9 V Vn 1 OPA44/BB C2 1n Y 1k R2 13 V Vn OPA44/BB 12 1k R6 14 Y X Y X Vn U6 1 2 X1 3 X2 7 4 Y1 W 6 Y2 V Vn AD633/AD X U6 1 2 X1 3 X2 7 4 Y1 W 6 Y2 V Vn AD633/AD R4 4k R7 4k R k R9 12k R1 4k 9 V Vn 1 OPA44/BB C3 1n 9 V Vn 1 OPA44/BB R 74k AD633/AD 1k R 1k R12 U7 1 X1 2 7 X2 3 W Y1 4 Y2 6 V Vn 13 V Vn OPA44/BB k R13 13 V Vn OPA44/BB Figure 7. The circuit schematic of the scaled jerk system. Figure. The experimental circuit of the scaled jerk system. A circuit for realising the scaled jerk system is designed by using electronic components (see figure 7). The circuit includes thirteen resistors, three capacitors, six operational amplifiers and three analog multipliers. The circuit was implemented on the electronic card as shown in figure. We selected C1 = C2 = C3 = 1nF, R1 = 2 k, R2 = R3 = R = R6 = R12 = R13 = 1 k, R4 = 4 k, R7 = R1 = 4 k, R = k, R9 = 12 k, R = 74 k. Figure 9 displays the obtained phase portraits on the oscilloscope, which agree with numerical results in figure 1.

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