constructing a novel no-equilibrium chaotic system
TRANSCRIPT
International Journal of Bifurcation and Chaosc© World Scientific Publishing Company
CONSTRUCTING A NOVEL NO–EQUILIBRIUM CHAOTIC
SYSTEM
VIET–THANH PHAMSchool of Electronics and Telecommunications
Hanoi University of Science and Technology
01 Dai Co Viet, Hanoi, Vietnam
CHRISTOS VOLOSDepartment of Military Science
Hellenic Army Academy
Athens, GR–16673, Greece
SAJAD JAFARIBiomedical Engineering Department
Amirkabir University of Technology,
Tehran 15875–4413, Iran
ZHOUCHAO WEISchool of Mathematics and Physics
China University of Geosciences
Wuhan, 430074, PR China
XIONG WANG*
Department of Electronic Engineering
City University of Hong Kong
Hong Kong SAR, China
Received (to be inserted by publisher)
This paper introduces a new no–equilibrium chaotic system that is constructed by adding atiny perturbation to a simple chaotic flow having a line equilibrium. Dynamics of the proposedsystem are investigated through Lyapunov exponents, bifurcation diagram, Poincare map andperiod–doubling route to chaos. A circuit realization is also represented. Moreover, two other newchaotic systems without equilibria are also proposed by applying the presented methodology.
Keywords : Chaos; Equilibrium; Hidden attractor; Lyapunov exponent; Bifurcation diagram;Poincare map.
∗Corresponding author.
1
2 V.–T. Pham et al.
1. Introduction
In chaos theory, equilibrium points of a dynamical system, especially a chaotic one, play important roles forstudying its nonlinear behavior. Most reported chaotic systems, i.e. Lorentz system [Lorenz, 1963], Rolssersystem [Rossler, 1976], Chen system [Chen & Ueta, 1999] etc., have a limited number of equilibria. Chaosin 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 havebeen 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 heteroclinicorbits, the Shilnikov method [Shilnikov, 1965; Shilnikov et al., 1998] for verifying chaos cannot be applied tosuch 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 smallneighborhoods of any equilibrium points. Hidden attractors not only make difficulties in simulation ofdrilling systems and phase locked–loop etc. [Leonov et al., 2011a; Leonov & Kuznetsov, 2013] but alsoallow 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 bothacademic significance and practical importance therefore should receive further attentions.
In order to discover new chaotic systems with hidden attractors, an effective approach is based onexisting chaotic systems. Wang and Chen [Wang & Chen, 2012] applied a tiny perturbation to the SprottE system to change the stability of its single equilibrium to a stable one. In the same way, a tiny perturbationmakes 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 noequilibria [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 onesoffers a great challenge.
Motivated by the above research, a new no–equilibrium chaotic system is proposed in this paper. Itworth noting that the new system is constructed by adding a tiny perturbation to a reported system witha line of equilibria. Its dynamics is explored through Lyapunov exponents, bifurcation diagram, Poincaremap and period–doubling route to chaos. In addition to the above analysis, two novel cases are also foundby 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 impossibleto verify the chaotic attractor by choosing an arbitrary initial condition in the vicinity of the unstableequilibria [Jafari & Sprott, 2013]. In this section we first only focus on the Jafari LE1 system:
x = y,
y = −x + yz,
z = −x − axy − bxz,
(1)
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 equilibriumpoints 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,
z = −x − axy − bxz + c,
(2)
where a = 15, b = 1 and c is the real parameter.
Constructing a novel no–equilibrium chaotic system 3
−1
0
1
−1
0
1−2
−1
0
1
xy
z
(a)
−1 −0.5 0 0.5 1−2
−1.5
−1
−0.5
0
0.5
1
y
z
(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 they − z plane.
It is a three–dimensional autonomous flow with quadratic nonlinearities. When c = 0, it becomes theJafari LE1 system; when c 6= 0, however, the new system PNE1 (2) possesses no equilibrium points. Inparticular, 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 hasbeen computed using the algorithm in [Wolf et al., 1985] to verify the chaoticity of system PNE1 whenc = −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
DKY = j +1
∣
∣λLj+1
∣
∣
j∑
i=1
λLi,
where j is the largest integer satisfyingj
∑
i=1
λLi≥ 0 and
j+1∑
i=1
λLi< 0. The calculated dimension of system
PNE1 when c = −0.001 is DKY = 2.1296 > 2. Therefore, it indicates a strange attractor. Further, thePoincare 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 variablez(t). The numerical result of Lyapunov exponents is shown in Fig. 4. Both the bifurcation diagram andthe corresponding Lyapunov spectrum clearly indicate that there are some windows of limit cycles and ofchaotic 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 differentvalues 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 clearlyfrom Fig. 3 and Fig. 5.
3. Circuit implementation
Electronic circuit provides an alternative approach to explore the mathematical model PNE1 (2). The statevariables x, y, and z of the system (2) are scaled up to display in a larger range. Therefore the system (2)
4 V.–T. Pham et al.
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y
z
Fig. 2. Poincare map in the y − z plane when x = 0.
−15 −10 −5 0 5
x 10−3
−0.2
0
0.2
0.4
0.6
0.8
c
z max
Fig. 3. Bifurcation diagram of zmax with c as varying parameter.
will be changed to
X = Y,
Y = −X + 14Y Z,
Z = −X − a4XY − b
4XZ + 4c,
(3)
Constructing a novel no–equilibrium chaotic system 5
−15 −10 −5 0 5
x 10−3
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
c
λ
Fig. 4. 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, operationalamplifiers and multipliers.
The circuit equations, which are derived from Fig.6, have the following form
dvc1
dt= 1
R1C1
R9
R8vC2
,dvc2
dt= − 1
R2C2vC1
+ 1R3C2
vC2vC3
,dvc3
dt= − 1
R4C3vC1
− 1R5C3
vC1vC2
− 1R6C3
vC1vC3
− 1R7C3
Vc,
(4)
where vC1, vC2
, vC3denote 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 ofcomponents 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 illustratethe effectiveness of the approach, which is mentioned in Section 2. In other words, new systems PNE2 andPNE3 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,
z = −y − axy − bxz + c,
(5)
while the PNE3 system is described by
x = y,
y = −x + yz,
z = x2 − axy − bxz + c.
(6)
6 REFERENCES
−1 −0.5 0 0.5 1−1.5
−1
−0.5
0
0.5
1
yz
(a)
−1 −0.5 0 0.5 1−2
−1.5
−1
−0.5
0
0.5
1
y
z
(b)
−1 −0.5 0 0.5 1−2
−1.5
−1
−0.5
0
0.5
1
y
z
(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 = −0.006and (c) period–8 orbit at c = −0.0052.
The corresponding parameters, Lyapunov exponents, Kaplan–Yorke dimensions, and initial conditionsof two new systems are summarized in Table 1. Positive Lyapunov exponents indicate chaos in new intro-duced 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 ofhidden chaotic attractors.
5. Conclusions
A chaotic system without equilibrium has been discovered and analyzed in this paper. In fact the newsystem is obtained by using a simple control parameter, which is applied to a known system with aline equilibrium. In other words, a new hidden chaotic attractor is derived from another hidden chaoticattractor. It is also noted that two additional cases are created by the same approach. Because there are nosinks in the proposed no–equlilibrium chaotic systems, they are appropriate for chaos–based applicationssuch as secure communications.
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REFERENCES 7
Fig. 6. Circuitry of the system (3).
Table 1. Three novel chaotic systems with no equilibria.
Model Parameter LEs DKY (x0, y0, z0)
PNE1 a = 15 0.0708 2.1296 0b = 1 0 0.5
c = −0.001 −0.5461 0.5
PNE2 a = 17 0.0285 2.1089 0b = 1 0 0.4
c = −0.001 −0.2618 0
PNE3 a = 18 0.0274 2.0939 0b = 1 0 −0.4
c = −0.001 −0.2919 0.5
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