International Journal of Bifurcation and Chaos c World Scientific Publishing Company
CONSTRUCTING A NOVEL NO–EQUILIBRIUM CHAOTIC SYSTEM VIET–THANH PHAM School of Electronics and Telecommunications Hanoi University of Science and Technology 01 Dai Co Viet, Hanoi, Vietnam
[email protected] CHRISTOS VOLOS Department of Military Science Hellenic Army Academy Athens, GR–16673, Greece
[email protected] SAJAD JAFARI Biomedical Engineering Department Amirkabir University of Technology, Tehran 15875–4413, Iran
[email protected] ZHOUCHAO WEI School of Mathematics and Physics China University of Geosciences Wuhan, 430074, PR China
[email protected] XIONG WANG* Department of Electronic Engineering City University of Hong Kong Hong Kong SAR, China
[email protected] Received (to be inserted by publisher)
This paper introduces a new no–equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. Dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincar´e map and period–doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology. Keywords: Chaos; Equilibrium; Hidden attractor; Lyapunov exponent; Bifurcation diagram; Poincar´e map.
∗
Corresponding author. 1
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V.–T. Pham et al.
1. Introduction In chaos theory, equilibrium points of a dynamical system, especially a chaotic one, play important roles for studying its nonlinear behavior. Most reported chaotic systems, i.e. Lorentz system [Lorenz, 1963], R¨ olsser system [R¨ ossler, 1976], Chen system [Chen & Ueta, 1999] etc., have a limited number of equilibria. Chaos in these systems can be proved by using conventional Shilnikov criteria [Shilnikov, 1965; Shilnikov et al., 1998], in which at least one unstable equilibrium for emergence of chaos is required. Recently, a few chaotic systems without equilibrium points or with only one stable equilibrium have been introduced [Wei, 2011; Jafari et al., 2013; Wang et al., 2012; Wang & Chen, 2012, 2013; Molaei et al., 2013; Wei & Pehlivan, 2012; Wei et al., 2013]. Because they can have neither homoclinic nor heteroclinic orbits, the Shilnikov method [Shilnikov, 1965; Shilnikov et al., 1998] for verifying chaos cannot be applied to such systems. Chaotic system without equilibrium is categorized as chaotic system with hidden attraction [Leonov et al., 2011a,b, 2012] due to the fact that its basin of attraction does not intersect with small neighborhoods of any equilibrium points. Hidden attractors not only make difficulties in simulation of drilling systems and phase locked–loop etc. [Leonov et al., 2011a; Leonov & Kuznetsov, 2013] but also allow unexpected responses to perturbations in a structure like a bridge or an airplane wing [Jafari & Sprott, 2013]. As a result, investigation of systems with hidden attractors is an interesting topic of both academic significance and practical importance therefore should receive further attentions. In order to discover new chaotic systems with hidden attractors, an effective approach is based on existing chaotic systems. Wang and Chen [Wang & Chen, 2012] applied a tiny perturbation to the Sprott E system to change the stability of its single equilibrium to a stable one. In the same way, a tiny perturbation makes the Sprott D system with a degenerate equilibrium to have no equilibria [Wei, 2011]. In addition, Jafari et al. has implemented a systematic search algorithm to find a catalog of chaotic flows with no equilibria [Jafari et al., 2013], or a list of simple chaotic flows with a line of equilibria [Jafari & Sprott, 2013]. Investigations on hidden attractors are still going on and converting them into new hidden ones offers a great challenge. Motivated by the above research, a new no–equilibrium chaotic system is proposed in this paper. It worth noting that the new system is constructed by adding a tiny perturbation to a reported system with a line of equilibria. Its dynamics is explored through Lyapunov exponents, bifurcation diagram, Poincar´e map and period–doubling route to chaos. In addition to the above analysis, two novel cases are also found by using the same methodology.
2. Dynamics of a new chaotic system without equilibrium Recently nine simple chaotic flows with a line equilibrium have introduced by Jafari and Sprott [Jafari & Sprott, 2013] through an exhaustive computer search. These attractors are hidden because it is impossible to verify the chaotic attractor by choosing an arbitrary initial condition in the vicinity of the unstable equilibria [Jafari & Sprott, 2013]. In this section we first only focus on the Jafari LE1 system: x˙ = y, y˙ = −x + yz, (1) z˙ = −x − axy − bxz,
where a = 15, b = 1. It is easy to see that the chaotic system (1) has a line of equilibria E (0, 0, z). It is also easy to imagine that a tiny perturbation to the system (1) may be able to change the uncountable number of equilibrium points while preserving its chaotic dynamics. Therefore a simple parameter c is added to the Jafari LE1 system in order to obtain the following new system (denoted as the model PNE1 ): x˙ = y, y˙ = −x + yz, (2) z˙ = −x − axy − bxz + c, where a = 15, b = 1 and c is the real parameter.
Constructing a novel no–equilibrium chaotic system
3
1 0.5
1
0
z
z
0
−0.5
−1 −1 −2 1 1 0
y
0 −1 −1
−1.5 −2 −1
−0.5
0
0.5
1
y
x
(a)
(b)
Fig. 1. Chaotic attractor with no equilibrium in the novel system PNE1 (2) for c = −0.001 (a) in the 3–D space, (b) in the y − z plane.
It is a three–dimensional autonomous flow with quadratic nonlinearities. When c = 0, it becomes the Jafari LE1 system; when c 6= 0, however, the new system PNE1 (2) possesses no equilibrium points. In particular, when c = −0.001 and the initial conditions (x0 , y0 , z0 ) = (0, 0.5, 0.5), the new system PNE1 exhibits a chaotic attractor with no equilibria, as shown in Fig. 1 . The Lyapunov exponents measure the exponential rates of the divergence and convergence of nearby trajectories in the phase space of the chaotic system. Thus Lyapunov exponents of the system PNE1 has been computed using the algorithm in [Wolf et al., 1985] to verify the chaoticity of system PNE1 when c = −0.001. Here Lyapunov exponents are denoted by λLi , i = 1, 2, 3, with λL1 > λL2 > λL3 . Obviously, the system PNE1 is chaotic because λL1 = 0.0708 > 0, λL2 = 0 and λL3 = −0.5461 with |λL1 | < |λL3 |. The fractional dimension, which presents the complexity of attractor, is defined by j
DKY
where j is the largest integer satisfying
j P
i=1
X 1 λL , = j + λLj+1 i=1 i
λLi ≥ 0 and
j+1 P
λLi < 0. The calculated dimension of system
i=1
PNE1 when c = −0.001 is DKY = 2.1296 > 2. Therefore, it indicates a strange attractor. Further, the Poincar´e map of system PNE1 also reflects properties of chaos (see Fig. 2). To better understand of the new system PNE1 (2), its behavior with respect to the control parameter c is discovered. The bifurcation diagram (Fig. 3) is obtained by plotting the local maxima of the state variable z(t). The numerical result of Lyapunov exponents is shown in Fig. 4. Both the bifurcation diagram and the corresponding Lyapunov spectrum clearly indicate that there are some windows of limit cycles and of chaotic behavior. Obviously, the bifurcation diagram agrees well with the Lyapunov exponent spectrum. It can be seen from Fig. 4 that chaos occurs for −0.0043 < c < 0.002. Fig. 5 illustrates the phase plane representation in the y − z plane of the system PNE1 for different values of the control parameter c. Typical period–2 orbit, period–4 orbit, and period–8 orbit are obtained, as shown in Fig. 5a, 5b, 5c, respectively. Therefore, a period-doubling route to chaos is observed clearly from Fig. 3 and Fig. 5.
3. Circuit implementation Electronic circuit provides an alternative approach to explore the mathematical model PNE1 (2). The state variables x, y, and z of the system (2) are scaled up to display in a larger range. Therefore the system (2)
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V.–T. Pham et al.
0.7 0.6 0.5
z
0.4 0.3 0.2 0.1 0 −1
−0.5
0
0.5
1
y Fig. 2.
Poincar´e map in the y − z plane when x = 0.
0.8
z
max
0.6 0.4 0.2 0 −0.2 −15
−10
−5
c Fig. 3.
0
5 −3
x 10
Bifurcation diagram of zmax with c as varying parameter.
will be changed to X˙ = Y, Y˙ = −X + 41 Y Z, ˙ Z = −X − a4 XY − 4b XZ + 4c,
(3)
Constructing a novel no–equilibrium chaotic system
5
0.1 0 −0.1
λ
−0.2 −0.3 −0.4 −0.5 −0.6 −15
−10
−5
c Fig. 4.
0
5 −3
x 10
The Lyapunov exponents λL1 , λL2 , λL3 (solid line, dot line, and dash-dot line, respectively) versus c ∈ [−0.015, 0.005].
where X = 4x, Y = 4y and Z = 4z. A possible electronic circuit to realize (3) is proposed in Fig. 6. The circuit consists of common off–the–shelf discrete components such as resistors, capacitors, operational amplifiers and multipliers. The circuit equations, which are derived from Fig.6, have the following form dvc 1 R9 1 dt = R1 C1 R8 vC2 , dvc2 1 1 (4) dt = − R2 C2 vC1 + R3 C2 vC2 vC3 , dvc3 1 1 1 1 dt = − R4 C3 vC1 − R5 C3 vC1 vC2 − R6 C3 vC1 vC3 − R7 C3 Vc , where vC1 , vC2 , vC3 denote voltages of capacitors C1 , C2 and C3 , respectively. It is noted that each state variable in (3) , i.e. X, Y , Y , is implemented as the votage across a corresponding capacitor, C1 , C2 , C3 , respectively. Components of the circuit have been selected to match Eqs. (4). Hence, the values of components are as follows: R1 = R2 = R4 = R8 = R9 = 10kΩ, R3 = R6 = 40kΩ, R5 = 2.666kΩ, R7 = 2.5kΩ, C1 = C2 = C3 = 10nF and Vc = 1mVDC .
4. Discussion Two new chaotic no–equilibrium systems (called PNE2 , PNE3 ) are presented in this section to illustrate the effectiveness of the approach, which is mentioned in Section 2. In other words, new systems PNE2 and PNE3 are also obtained by adding a tiny control parameter c to the Jafari LE2 and Jafari LE3 systems, respectively. As a result, the PNE2 system has the following form x˙ = y, y˙ = −x + yz, (5) z˙ = −y − axy − bxz + c, while the PNE3 system is described by
x˙ = y, y˙ = −x + yz, z˙ = x2 − axy − bxz + c.
(6)
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REFERENCES
1 0.5
z
0 −0.5 −1 −1.5 −1
−0.5
0
0.5
1
y
1
1
0.5
0.5
0
0
−0.5
−0.5
z
z
(a)
−1
−1
−1.5
−1.5
−2 −1
−0.5
0
y (b)
0.5
1
−2 −1
−0.5
0
0.5
1
y (c)
Fig. 5. Phase portrait in y−z plane for different values of c. (a) Period–2 orbit at c = −0.012, (b) period–4 orbit at c = −0.006 and (c) period–8 orbit at c = −0.0052.
The corresponding parameters, Lyapunov exponents, Kaplan–Yorke dimensions, and initial conditions of two new systems are summarized in Table 1. Positive Lyapunov exponents indicate chaos in new introduced systems PNE2 , PNE3 . To the best of our knowledge, there is a little information about the conversion of a hidden attractor, which is rarely found, into a new hidden attractor. For this reason, our work has expanded the list of hidden chaotic attractors.
5. Conclusions A chaotic system without equilibrium has been discovered and analyzed in this paper. In fact the new system is obtained by using a simple control parameter, which is applied to a known system with a line equilibrium. In other words, a new hidden chaotic attractor is derived from another hidden chaotic attractor. It is also noted that two additional cases are created by the same approach. Because there are no sinks in the proposed no–equlilibrium chaotic systems, they are appropriate for chaos–based applications such as secure communications.
References Chen, G. R. & Ueta, T. [1999] “Yet another chaotic attractor,” Int. J. Bifurcation and Chaos 9, 1465–1466. Jafari, S. & Sprott, J. [2013] “Simple chaotic flows with a line equilibrium,” Chaos Solitons Fractals 57, 79–84.
REFERENCES
Fig. 6. Table 1.
Circuitry of the system (3).
Three novel chaotic systems with no equilibria.
Model
Parameter
LEs
DKY
(x0 , y0 , z0 )
PNE1
a = 15 b=1 c = −0.001 a = 17 b=1 c = −0.001 a = 18 b=1 c = −0.001
0.0708 0 −0.5461 0.0285 0 −0.2618 0.0274 0 −0.2919
2.1296
0 0.5 0.5 0 0.4 0 0 −0.4 0.5
PNE2
PNE3
7
2.1089
2.0939
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