A chaotic jerk system with non-hyperbolic equilibrium: Dynamics, effect of time delay and circuit realisation
KARTHIKEYAN RAJAGOPAL1, VIET-THANH PHAM2,∗, FADHIL RAHMA TAHIR3, AKIF AKGUL4, HAMID REZA ABDOLMOHAMMADI5and SAJAD JAFARI6
1Center for Nonlinear Dynamics, College of Engineering, Defence University, Bishoftu, Ethiopia
2Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3Electrical Engineering Department, University of Basrah, Basra, Iraq
4Department of Electrical and Electronic Engineering, Faculty of Technology, Sakarya University, Adapazarı, Turkey
5Department of Electrical Engineering, Golpayegan University of Technology, Golpayegan, Iran
6Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran
∗Corresponding author. E-mail: phamvietthanh@tdt.edu.vn
MS received 7 July 2017; revised 3 October 2017; accepted 12 November 2017; published online 9 March 2018 Abstract. The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic 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 time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.
Keywords. Chaos; jerk system; equilibrium; time delay; circuit.
PACS Nos 05.45.Ac; 02.30.Ks
1. Introduction
In the past decades, chaotic phenomenon in nature, eco- nomics, 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 [5], simple chaotic flows [6], memristor-based systems [7], Lorenz-type systems with multiwing but- terfly chaotic attractors [8], systems with multiscroll attractors [9–11], with multiple attractors [12–14], with extreme multistability [15–18], with megastability [19], with hidden attractors [20–26] and so on. Some chaotic systems are special because of their equilibria [27–30].
Systems with no equilibria [31,32], stable equilib- ria [33], line of equilibria [34], plane of equilibria [35], curve of equilibria [36–38] and surface of equilibria [39]
are such systems.
Equilibrium points of chaotic systems are often inves- tigated because they can be used to study the types of systems, the shapes of attractors, or amplitude control
[28,40–42]. Conventional three-dimensional systems often have hyperbolic equilibrium, of which the real parts of eigenvalues are non-zero. Chaos in such hyper- bolic system is proved by Silnikov criterion [43,44].
It is interesting that there are a few chaotic systems having non-hyperbolic equilibria [45,46], 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 non-hyperbolic equilibria have been the source of inspiration for designing some very rare chaotic flows [47,48], and this indicates the importance of knowing them.
Motivated by the special features of non-hyperbolic system, in this work we present a new three-dimensional autonomous system with only one non-hyperbolic equi- librium. In the next section, the new system is intro- duced and its dynamics are investigated. To explore the complex dynamics of the system, we study the stabil- ity of equilibrium, bifurcation diagram and Lyapunov exponents. The delay effects on the novel system are
discussed in §3. Section4presents an electronic circuit implementation of the system to verify its feasibility.
Finally, §5concludes our work.
2. Novel chaotic jerk system
In physics, time derivative of acceleration is called the jerk. Any dynamical system which can be defined by an equation like...
x = j(x,x˙,x¨)is called a jerk system.
By changing the variables, the equation can be written as a set of first-order differential equations [49]. Such systems are very important in nonlinear dynamics. For example, the simplest possible chaotic system is in this form [50]. Here we propose a novel jerk system with seven parameters and three nonlinear terms as follows:
˙
x =a1y,
˙
y =a2z,
˙
z=a3x +a4z2+a5x y+a6x z+a7, (1) where ai for i ∈ [1,7] are the parameters of the sys- tem. Figure1shows the 2D phase portraits of the novel jerk system for a1 = 1,a2 = 1,a3 = −1,a4 =
−2.69,a5 =1,a6 =1,a7 = −1 and initial conditions (−3,0,1). 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 = −a7/a3,y=0,z =0 and its charac- teristic equation is
λ3+a6a7
a3 λ2+a2a5a7
a3 λ−a1a2a3=0.
According to the Routh–Hurwitz criterion, all the prin- cipal minors need to be positive in order to have stable equilibria. The principal minors are
1 =δ0 >0, 2= δ1 δ0
δ3 δ2
>0, 3 =>
δ1 δ0 0 δ3 δ2 δ1
0 0 δ3
>0, (2)
where
δ0 =1, δ1 = a6a7
a3 , δ2 = a2a5a7
a3 , δ3= −a1a2a3. The condition for having unstable equilibria is that any one of the principal minors 1, 2 and 3 be nega- tive. For parametersa1 = 1,a2 = 1,a3 = −1,a4 =
−2.69,a5 = 1,a6 = 1,a7 = −1 the principal minors are1 =1>0, 2 =0 showing that the equilibrium may be unstable. The eigenvalues for the given param- eter values areλ1 = −1, λ2,3 = ±i. Table1shows the range of parameters for stable and unstable regions of the equilibrium in system (1) and figure2shows the sta- ble and unstable regions for the real part of eigenvalue λ2 (λ1 always has a negative real value while λ2 and λ3are complex conjugate pairs). The red marker in the figures shows the transition point from stable to unsta- ble region. As can be seen from table1, the parameters a2 anda4 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 diver- gence of the states. The LEs of system (1) are calculated as L1 = 0.05852,L2 = 0,L3 = −1.7471 and the Kaplan–Yorke dimension is 2.033. 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 dif- ferent values [51–54]. In this paper, we have used the method proposed in [55].
To study the effect of the parameters on system (1), we plot the bifurcation diagram and LE diagrams when parametera6changes (figure3).
Figure 1. 2D phase portraits of the novel jerk system fora1=1,a2=1,a3= −1,a4= −2.69,a5=1,a6=1,a7= −1 and initial conditions(−3,0,1).
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 a1 0.98≤a1<1 1>a1>1.006 a1=1 a3 −1>a3>−0.994 −1.0017≤a3<−1 a3= −1 a7 −1.011>a7>−1 −1≤a7<−0.997 a7= −1 a5 1>a5>1.012 0.997>a5>1 a5=1 a6 1>a6>1.03 0.99>a6>1 a6=1
a2 No change in real part
a4 No change in real part
Figure 2. (a–f) Stable and unstable regions for parameters with real part of eigenvalue (λ2).
Figure 3. (a) Bifurcation of system (1) with changinga6with initial conditions of(−3,0,1)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 ofa6(the third LE is out of scale).
It can be seen from figure3that 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. Time-delay jerk system (TDJS)
Time-delayed differential equation is important in real- time engineering applications [56,57]. For example, both integer- and fractional-order memristor
time-delayed chaotic systems and their dynamic proper- ties have been discussed in [58,59], or a parameter iden- tification problem for a general time-delayed chaotic system is considered and analysed in [60].
Motivated by the above discussions, we are interested in investigating the time-delay effects on system (1) and hence we introduce multiple time delays in the third equation of (1) and the dimensionless model of the time- delay jerk system is given by
˙
x =a1y,
˙
y =a2z,
˙
z =a3x(t−τ1)+a4z(t−τ3)2 +a5x(t−τ1)y(t−τ2)
+a6x(t−τ1)z(t−τ3)+a7, (3) whereai fori ∈ [1,7]are the parameters of the TDJS andτi fori ∈ [1,3]are the multiple time delays of the system.
The equilibrium points of the delayed and non- delayed 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∗+xi,y = y∗+yi,z = z∗+zi and derive the Jacobian matrix as
JE =
0 1 0
0 0 e−λτ1
a3e−λτ1+a5y∗e−λτ1+a6z∗e−λτ1 a5x∗e−λτ2 a6x∗e−λτ3+2a4z∗e−λτ3
, (4)
where x∗,y∗ andz∗ are the equilibrium points of the TDJS systems.
The generalised characteristic equation of the TDJS with time delaysτ1, τ2, τ3is
λ3+a6a7
a3 λ2e−λτ3+a2a5a7
a3 λe−λτ2−a1a2a3e−λτ1. (5)
For the parameter values a1 = 1,a2 = 1,a3 =
−1,a4 = −2.69,a5 = 1,a6 = 1,a7 = −1, the char- acteristic equation has an absolute minimum of 1 at λ=0, and so the characteristic equation has only imag- inary solutions. Hence, assuming that the eigenvalues are purely imaginary, we useλ=iθ withθ >0 in (4), (iθ)3+a6a7
a3 (iθ)2e−iθτ3 +a2a5a7 a3
iθe−iθτ2
−a1a2a3e−iθτ1. (6) Using commensurate time delayτ, using the parameter values and equating real and imaginary terms,
θ2cos(τθ)−θsin(τθ)+cos(τθ)=0
θ3−θ2sin(τθ)−θcos(τθ)−cos(τθ)=0. (7) Solving eq. (7) for τ = 0.06, we get −235.54 as the real part and±1.357 as the imaginary part. Hence the eigenvalues are complex conjugate pair with negative real parts (stable focus). Figure4 shows the 2D phase portraits of the TDJS forτ1 =0.05, τ2=0.18, τ3 =0.1 and initial conditions(−3,0,1).
Various algorithms based on chaos synchronisation are proposed for the estimation of Lyapunov exponent of time-delayed dynamical systems [61–63]. In this paper,
we adopted the technique by employing the synchroni- sation of identical systems coupled by linear negative feedback mechanism [63] for finding the exact Lya- punov exponents of the TDJS. The calculated Lyapunov exponents areL1 =0.05314,L2 =0,L3 = −1.7867.
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 parametera6 as
Figure 4. 2D phase portraits of the TDJS fora1=1,a2=1,a3= −1,a4= −2.69,a5=1,a6=1,a7= −1,τ1=0.05, τ2=0.18, τ3=0.1 and initial conditions(−3,0,1).
the bifurcation parameter and the other parameters are taken asa1 =1,a2 = 1,a3 = −1,a4 = −2.69,a5 = 1,a7 = −1 and time delays as τ1 = 0.05, τ2 = 0.18, τ3 =0.1. Figure5shows the bifurcation plot for the parametera6.
Figure 5. Bifurcation of TDJS witha6for the initial con- ditions(−3,0,1)which is reinitialised after every iteration with the end values of the state trajectories.
To plot the time-delay impact, we plot the bifurca- tion of TDJS with the time delays (figure 6) and the parameter values fixed as a1 = 1,a2 = 1,a3 =
−1,a4 = −2.69,a5 = 1,a6 = 1,a7 = −1 with the initial conditions(−3,0,1)and varying the respective time delays to get the bifurcation plots while the others are fixed atτ1=0.05, τ2 =0.18, τ3 =0.1.
4. The electronic circuit implementation
In this section, we present a realisation of theoretical sys- tem (1) by using electronic components [64]. It should be noted that theyandzoutputs in the jerk system have noise-like behaviours because their signal values are very low. They are in the interval of−1.5 and 1. There- fore, we must scale them to increase their amplitude values. For scale process, letX =x,Y =5y,Z =5z.
The new variables X;Y;Z become the following after scale process:
X˙ = a1
5 y, Y˙ =a2z, Z˙ =5a3x+a4
5 z2+a5x y+a6x z+5a7. (8)
Figure 6. Bifurcation of TDJS withτ1, τ2, τ3. The parameter values area1 =1,a2 =1,a3 = −1,a4 = −2.69,a5 =1, a6=1,a7= −1,τ1=0.05,τ2=0.18,τ3=0.1 and initial conditions are(−3,0,1).
OPA404/BB 12 + 13 -
4 V+
11 V- OUT 14
0
-Y
R5
100k Vn
Vp R6 100k
OPA404/BB 10 +
9 -
4 V+
11 V- OUT 8
0 Vp
Vn C2 1n
-Z
Y
R4 400k
U60
AD633/AD 1 X1
2 X2 3 Y 1 4 Y 2 6 Z
W 7 V+8 V-5
0 Vn Vp X
Y
V V
V
OPA404/BB 12 + 13 -
4 V+
11 V- OUT 14
0
-X
R2
100k Vn
Vp R3 100k
OPA404/BB 12 + 13 -
4 V+
11 V- OUT 14
-Z
0 R12 100k
Vp Vn R13 100k
R9 1200k Vp
OPA404/BB 10 +
9 -
4 V+
11 V- OUT 8
0 Vp
Vn C3 1n
X
Z
R8 80k
U56
AD633/AD 1 X1
2 X2 3 Y 1 4 Y 2 6 Z
W 7 V+8 V-5
0 Vn Vp X
Z
R7 40k
R10 40k
U57
AD633/AD X1 1 X2 2 Y 1 3 Y 2 4 Z 6 7 W85 V+V- 0
Vn Vp
Z Z R11
74k Vp
V3 15Vdc
Vn V4 15Vdc
0
OPA404/BB 10 +
9 -
4 V+
11 V- OUT 8
Vn
0 Vp
C1 1n X
R1 2000k -Y
Figure 7. The circuit schematic of the scaled jerk system.
Figure 8. 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 8. We selected C1 = C2 = C3 =
1 nF, R1 = 2000 k,R2 = R3 = R5 = R6 = R12 = R13 = 100 k, R4 = 400 k,R7 = R10 = 40 k, R8 = 80 k, R9 = 1200 k, R11 = 74 k.
Figure 9 displays the obtained phase portraits on the oscilloscope, which agree with numerical results in figure1.
Figure 9. Experimental phase portraits of scaled jerk system on the oscilloscope.
5. Conclusion
A three-dimensional autonomous chaotic system with non-hyperbolic equilibrium has been studied in this paper and its dynamics has been discovered. To under- stand the effect of time delays on this new system, we introduced multiple time delays into the third state equa- tion of the system. It is interesting that time-delay jerk system displays chaotic behaviour. Moreover, an analog circuit was built to realise the theoretical system. Future research should concentrate on the investigation of sys- tem’s applications because of its complex behaviour and feasibility.
References
[1] S H Strogatz, Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engi- neering (Westview Press, 2014)
[2] A T Azar, S Vaidyanathan and A DeMarco,Handbook of research on advanced intelligent control engineering and automation(Engineering Science Reference, 2015) [3] E N Lorenz,J. Atmos. Sci.20(2), 130 (1963)
[4] O E Rössler,Phys. Lett. A57(5), 397 (1976)
[5] G Chen and T Ueta,Int. J. Bifurc. Chaos9(07), 1465 (1999)
[6] J C Sprott,Elegant chaos: Algebraically simple chaotic flows(World Scientific, 2010)
[7] R Wu and C Wang,Int. J. Bifurc. Chaos26(09), 1650145 (2016)
[8] S Yu, W K Tang, J Lü and G Chen, Int. J. Bifurc.
Chaos20(01), 29 (2010)
[9] J Ma, X Wu, R Chu and L Zhang, Nonlinear Dynam.76(4), 1951 (2014)
[10] E Tlelo-Cuautle, J Rangel-Magdaleno, A Pano- Azucena, P Obeso-Rodelo and J Nunez-Perez,Commun.
Nonlinear. Sci.27(1), 66 (2015)
[11] S Jafari, V-T Pham and T Kapitaniak, Int. J. Bifurc.
Chaos26(02), 1650031 (2016)
[12] J Kengne, Z Njitacke and H Fotsin, Nonlinear Dynam.83(1–2), 751 (2016)
[13] Q Lai and S Chen,Int. J. Bifurc. Chaos26(11), 1650177 (2016)
[14] J Kengne, A N Negou and D Tchiotsop, Nonlinear Dynam.88(4), 2589 (2017)
[15] B Bao, T Jiang, Q Xu, M Chen, H Wu and Y Hu,Non- linear Dynam.86(3), 1711 (2016)
[16] B-C Bao, Q Xu, H Bao and M Chen, Electron.
Lett.52(12), 1008 (2016)
[17] B Bao, H Bao, N Wang, M Chen and Q Xu, Chaos Solitons Fractals94, 102 (2017)
[18] B Bao, T Jiang, G Wang, P Jin, H Bao and M Chen, Nonlinear Dynam.89(2), 1157 (2017)
[19] J C Sprott, S Jafari, A J M Khalaf and T Kapitaniak, Euro. Phys. J. Special Topics226(9), 1979 (2017) [20] P R Sharma, M D Shrimali, A Prasad, N Kuznetsov and
G Leonov,Int. J. Bifurc. Chaos25(04), 1550061 (2015) [21] M-F Danca and N Kuznetsov, Chaos Solitons Frac-
tals103, 144 (2017)
[22] M-F Danca, N Kuznetsov and G Chen, Nonlinear Dynam.88(1), 791 (2017)
[23] N Kuznetsov, G Leonov, M Yuldashev and R Yuldashev, Commun. Nonlinear Sci.51, 39 (2017)
[24] D Dudkowski, S Jafari, T Kapitaniak, N V Kuznetsov, G A Leonov and A Prasad,Phys. Rep.637, 1 (2016) [25] Y Feng and W Pan,Pramana – J. Phys.88(4), 62 (2017) [26] W Pan and L Li,Pramana – J. Phys.88(6), 87 (2017) [27] K Rajagopal, A Akgul, S Jafari, A Karthikeyan and I
Koyuncu,Chaos Solitons Fractals103, 476 (2017) [28] E Tlelo-Cuautle, L G de la Fraga, V-T Pham, C Volos,
S Jafari and A de Jesus Quintas-Valles, Nonlinear Dynam.89(2), 1129 (2017)
[29] V-T Pham, X Wang, S Jafari, C Volos and T Kapitaniak, Int. J. Bifurc. Chaos27(06), 1750097 (2017)
[30] V-T Pham, S Jafari, C Volos and T Kapitaniak,Int. J.
Bifurc. Chaos27(09), 1750138 (2017)
[31] V-T Pham, C Volos, S Jafari and T Kapitaniak,Nonlinear Dynam.87(3), 2001 (2017)
[32] V-T Pham, S T Kingni, C Volos, S Jafari and T Kapita- niak,AEU – Int. J. Electron. C78, 220 (2017)
[33] V-T Pham, S Jafari, T Kapitaniak, C Volos and S T Kingni,Int. J. Bifurc. Chaos27(04), 1750053 (2017) [34] S Jafari and J C Sprott,Chaos Solitons Fractals57, 79
(2013)
[35] S Jafari, J C Sprott and M Molaie, Int. J. Bifurc.
Chaos26(06), 1650098 (2016)
[36] V-T Pham, S Jafari, C Volos, T Gotthans, X Wang and D V Hoang,Optik130, 365 (2017)
[37] V-T Pham, S Jafari and C Volos,Optik131, 343 (2017) [38] V-T Pham, C Volos, T Kapitaniak, S Jafari and X Wang,
Int. J. Electron.105(3), 305 (2018)
[39] S Jafari, J C Sprott, V-T Pham, C Volos and C Li,Non- linear Dynam.86(2), 1349 (2016)
[40] C Shen, S Yu, J Lü and G Chen, IEEE T. Circuits-I:
Regular Papers61(8), 2380 (2014)
[41] M F Tolba, A M AbdelAty, N S Soliman, L A Said, A H Madian, A T Azar and A G Radwan,AEU – Int. J.
Electron. C78, 162 (2017)
[42] C Li, K Su and J Zhang, Appl. Math. Model.39(18), 5392 (2015)
[43] L Šil’Nikov,Math. USSR-Sb.,10(1), 91 (1970) [44] B Chen, T Zhou and G Chen, Int. J. Bifurc.
Chaos19(05), 1679 (2009)
[45] Z Wei, J Sprott and H Chen,Phys. Lett. A379(37), 2184 (2015)
[46] Z Wei, W Zhang and M Yao,Nonlinear Dynam.82(3), 1251 (2015)
[47] Z Wei,Phys. Lett. A376(2), 102 (2011)
[48] X Wang and G Chen,Commun. Nonlinear. Sci.17(3), 1264 (2012)
[49] J Sprott,Am. J. Phys.65(6), 537 (1997) [50] J Sprott,Phys. Lett. A228(4–5), 271(1997)
[51] G A Leonov and N V Kuznetsov, Int. J. Bifurc.
Chaos17(04), 1079 (2007)
[52] N Kuznetsov, T Alexeeva and G Leonov, arXiv:1410.2016(2014)
[53] N Kuznetsov, T Mokaev and P Vasilyev,Commun. Non- linear. Sci.19(4), 1027 (2014)
[54] N Kuznetsov,Phys. Lett. A380(25), 2142 (2016) [55] A Wolf, J B Swift, H L Swinney and J A Vastano,Physica
D16(3), 285 (1985)
[56] J K Hale and S M V Lunel,Introduction to functional differential equations (Springer Science & Business Media, 2013) Vol. 99
[57] W Deng, Y Wu and C Li,Int. J. Bifurc. Chaos16(02), 465 (2006)
[58] V-T Pham, S Vaidyanathan, C Volos, S Jafari, N Kuznetsov and T Hoang, Euro. Phys. J. Special Topic225(1), 127 (2016)
[59] W Hu, D Ding, Y Zhang, N Wang and D Liang, Optik130, 189 (2017)
[60] Y Tang, M Cui, L Li, H Peng and X Guan,Chaos Solitons Fractals41(4), 2097 (2009)
[61] A Stefanski, A Dabrowski and T Kapitaniak,Chaos Soli- tons Fractals23(5), 1651 (2005)
[62] A Dabrowski,Nonlinear Dynam.3(78), 1601 (2014) [63] A Stefanski, T Kapitaniak and A Dabrowski, IUTAM
Symp. Chaotic Dynam.(2005)
[64] Y Lin, C Wang, H He and L L Zhou, Pramana – J.
Phys.86(4), 801 (2016)