# Synchronization of Two Rank-One Chaotic Systems without and with Delay via Linear Delayed Feedback Control

Authors: Hui Fang
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

This paper illustrates the presence of chaos in rank-one chaotic systems with delay via a binary
test (called 0-1 test) for chaos. Chaotic synchronization between two rank-one chaotic systems
without and with delay is achieved by means of Lyapunov functional and linear delayed feedback
control method. Numerical simulations are implemented to verify the effectiveness of the proposed
chaos synchronization scheme.


---

## Body

## 1. Introduction

The introduction of the notion of chaotic synchronization by Pecora and Carroll [1] has stimulated a wide range of research activity in various fields. In 1999, Mainieri and Rehacek proposed a new method of synchronization, that is, projective synchronization [2]. Since then, many papers have been published on the projective synchronization of chaotic systems (see [3–14] and references cited therein). For some recent works on the synchronization of delayed chaotic systems, we refer the reader to [14–16].Wang and Young first developed a general theory of rank-one attractors, that is, attractors with a single direction of instability and strong contraction in all other directions, then applied it to prove the existence of rank-one attractors in certain systems of differential equations [17–19]. In 2005, Wang and Oksasoglu presented a new mechanism of producing rank-one chaos in Chua’s system [20]. In 2009, Chen and Han studied the existence of rank-one chaos in the neighborhood of a periodically kicked stable limit cycle close to a heteroclinic cycle of a planar equation [21]. However, to our knowledge, few papers have been published on the synchronization of rank-one chaotic systems.In this paper, we study the synchronization between rank-one chaotic systems without and with delay using linear control method. The detailed arrangement is as follows: in Section2, the rank-one chaotic systems proposed in [20, 21] are briefly described. The presence of chaos in rank-one chaotic systems with delay are tested using the 0-1 test for chaos recently proposed in [22]. In Section 3, the linear delayed feedback control method is adopted to realize the synchronization between two rank-one chaotic systems without and with delay. In Section 4, numerical simulations are shown to verify the validity and feasibility of the proposed method, and finally some conclusions are given in Section 5.


## 2. Systems Description

Let(2.1)PT,p(t)=1p∑n=-∞+∞Fn,T,p(t),
where(2.2)Fn,T,p(t)={1,nT≤t<nT+p,0,elsewhere.PT,p(t) is a periodic pulse train with a pulse width of p and a period of T.The rank-one chaotic system proposed in [20] can be described as follows:(2.3)ẋ=α[y-β1x-β3x3]+ηxPT,p(t),ẏ=x-y+z,ż=-βy,
where η is used for controlling the magnitude of the forcing. When the system parameters are taken as the following values:(2.4)α=2,β=2.31135,β1=0.2495,β3=-1,η=0.15,p=0.5,
there exist rank-one attractors in system (2.3) for large enough T, such as T=95 [20].The rank-one chaotic system proposed in [21] can be described as follows:(2.5)ẋ=2Bxy+ε(x+ax2)-ε1x(t)PT,p(t),ẏ=1-2Ax-By2,
where ε1 is used to control the magnitude of the forcing. When A=1, B=1, ε=-1, a=-1.23, p=0.5, ε1=0.4, there exist rank-one attractors in system (2.5) for T=10 and T=8 [21].Without the particular statement, the above parameter values are adopted in this whole paper.Incorporating delays into (2.3), (2.5) due to the effect of propagation delay in practical circuits, we consider the following systems:(2.6)ẋ=α[y(t-τ)-β1x(t)-β3x3(t)]+ηx(t)PT,p(t),ẏ=x(t-τ)-y(t)+z(t-τ),ż=-βy(t-τ);(2.7)ẋ=2Bx(t)y(t)+ε[x(t)+ax2(t)]-ε1x(t)PT,p(t),ẏ=1-2Ax(t-τ)-By2(t).The presence of chaos in systems (2.6), (2.7) will be confirmed via the 0-1 test for chaos. For convenience, we briefly describe the test as follows. For more details, we refer the reader to [22–24].Consider a set of discrete dataϕ(n)(n=1,2,3,…), representing a one-dimensional observable dataset obtained from the underlying dynamics, such as a set of experimental data or a set of discrete data obtained from numerical solutions. For c∈(0,π), define the translation variables as follows:(2.8)pc(n)=∑j=1nϕ(j)cos⁡jc,qc(n)=∑j=1nϕ(j)sin⁡jc,n=1,2,3,…,N.
To determine the growth of pc and qc, it is convenient to look at the mean square displacement (MSD), defined as(2.9)Mc(n)=lim⁡N→∞1N∑j=1N[(pc(j+n)-pc(j))2+(qc(j+n)-qc(j))2].
The test for chaos is based on the asymptotic growth rate kc of Mc(n) as a function of n, which can be estimated as follows:(2.10)kc=lim⁡n→∞log⁡Mc(n)log⁡n.
Numerically, kc is determined by fitting a straight line to the graph of logMc(n) versus logn through minimizing the absolute deviation.For differentc values, different kc values can be calculated. The underlying dynamics should be examined by whether the median k of kc values approach 0 or 1. Specifically, k is close to 0, signifying nonchaotic dynamics (i.e., periodic or quasi-periodic), or k is close to 1, signifying chaotic dynamics. Moreover, the dynamics of the translation components (p,q) are corresponding to the underlying dynamics. Bounded trajectories are corresponding to periodic or quasi-periodic dynamics and Brownian-like (unbounded) trajectories are corresponding to chaotic dynamics.For a sampled time seriesx(t) for system (2.6), we have the following results:(1)
k=0.9980 for τ=0.001, k=0.9974 for τ=0.003, k=0.9978 for τ=0.01, indicating that the dynamics of system (2.6) are chaotic. (a), (b), and (c) in Figure 1 depict Brownian-like trajectories, confirming that the dynamics of system (2.6) are chaotic for τ=0.001, τ=0.003, τ=0.01;(2)
k=0.0561 for τ=0.06, indicating that the dynamics of system (2.6) are nonchaotic. (d) in Figure 1 depicts a bounded trajectory, confirming that the dynamics of system (2.6) is nonchaotic for τ=0.06.Dynamics of the translation components(p,q) of system (2.6): (a), (b), and (c) show Brownian-like trajectories for τ=0.001, τ=0.003, τ=0.01, respectively; (d) shows a bounded trajectory for τ=0.06.
(a)(b)(c)(d)For a sampled time seriesx(t) for system (2.7), we have the following results: k=0.9977 for τ=0.03,T=8, and k=0.9978 for τ=0.03, T=10, indicating that the dynamics of system (2.7) are chaotic. (a) and (b) in Figure 2 depict Brownian-like trajectories, confirming that the dynamics of system (2.7) are chaotic for T=8, T=10, and τ=0.03.Dynamics of the translation components(p,q) of system (2.7) with τ=0.03: (a) and (b) show Brownian-like trajectories for T=8, T=10, respectively.
(a)(b)


## 3. Synchronization between Two Rank-one Chaotic Systems without and with Delay

The aim of this section is to achieve chaos synchronization for two rank-one chaotic systems without and with delay using linear delayed feedback control method.We define the drive and response systems as follows:(3.1)Ẋd=F(t,Xd(t)),(3.2)Ẋr=G(t,Xr(t),Xr(t-τ))+U(t),
where Xd(t)=[Xd1(t),…,Xdm(t)]T∈Rm, Xr(t)=[Xr1(t),…,Xrn(t)]T∈Rn, τ≥0, and U(t)=[u1(t),…,un(t)]T is a controller and that can be called the control function.For brevity, let us define the synchronous error as follows:(3.3)[e1(t),…,en(t)]T=Xr(t)-Kn×mXd(t),
where Kn×m is a constant scaling matrix.Before presenting the main results, we introduce the following Barbalat’s lemma and definition.Lemma 3.1 (Barbalat’s Lemma, [25, Lemma  1.2.2, page 4]).
Letf be a nonnegative function defined on [0,+∞) such that f is integrable on [0,+∞) and is uniformly continuous on [0,+∞). Then, limt→+∞f(t)=0.Definition 3.2 (See [2]).
The drive system (3.1) and the response system (3.2) are said to realize projective synchronization, if there exists a tracking control U(t)=[u1(t),…,un(t)]T such that the synchronous errors satisfy
(3.4)lim⁡t→+∞ei(t)=0(i=1,2,…,n).
Consider the drive system (3.5) and the response system (3.6):
(3.5)ẋ1=α[y1(t)-β1x1(t)-β3x13(t)]+ηx1(t)PT,p(t),ẏ1=x1(t)-y1(t)+z1(t),ż1=-βy1(t),(3.6)ẋ2=α[y2(t-τ)-β1x2(t)-β3x23(t)]+ηx2(t)PT,p(t)+u1(t),ẏ2=x2(t-τ)-y2(t)+z2(t-τ)+u2(t),ż2=-βy2(t-τ)+u3(t),
where u(t)=[u1(t),u2(t),u3(t)]T is the control function.In this case, we define the synchronous errors as follows:(3.7)e1(t)=x2(t)-k1x1(t),e2(t)=y2(t)-k2y1(t),e3(t)=z2(t)-k3z1(t).Now, we present the main result for the synchronization between system (3.5) and system (3.6).Theorem 3.3.
If the linear controlleru(t)=[u1(t),u2(t),u3(t)]T is given as follows:
(3.8)u1(t)=α[k1y1(t)-k2y1(t-τ)]-m1[x2(t)-k1x1(t)],u2(t)=k2x1(t)-k1x1(t-τ)+k2z1(t)-k3z1(t-τ)-m2[y2(t)-k2y1(t)],u3(t)=β[k2y1(t-τ)-k3y1(t)]-m3[z2(t)-k3z1(t)],
where k1=±1, m1>1+η/p-αβ1-0.1875αβ3, m2>α+β-1, m3>1. Then, the drive system (3.5) and the response system (3.6) realize projective synchronization.Proof.
Because|x1(t)|<0.25 for sufficiently large t≥t0* [20], we make the following prior assumption:
(3.9)|x2(t)|<0.25,t≥t0(t0≥t0*).
Consider the Lyapunov functional defined by
(3.10)V(t)=|e1(t)|+|e2(t)|+|e3(t)|+∫t-τt[|e1(s)|+(α+β)|e2(s)|+|e3(s)|]ds,t≥t0.
Calculating the upper right Dini derivative D+V(t) of V(t) along the solutions of system (3.5) and system (3.6), we get
(3.11)D+V(t)≤D̃1(t)+D̃2(t)+D̃3(t)+|e1(t)|-|e1(t-τ)|+(α+β)[|e2(t)|-|e2(t-τ)|]+|e3(t)|-|e3(t-τ)|,
where
(3.12)D̃i(t)=D+|ei(t)|={ėi(t),ei(t)>0,-ėi(t),ei(t)<0,|ėi(t)|,ei(t)=0,(i=1,2,3).
There are the following three cases to consider forD̃1(t).
(i)
Ife1(t)>0 (i.e., x2(t)>k1x1(t)), then
(3.13)D̃1(t)=-(αβ1+m1)e1(t)+αe2(t-τ)-αβ3[x23(t)-k1x13(t)]+ηe1(t)PT,p(t)≤-(αβ1-ηp+m1)|e1(t)|+α|e2(t-τ)|-αβ3|e1(t)||x22(t)+k1x1(t)x2(t)+k12x12(t)|≤-[αβ1-ηp+0.1875αβ3+m1]|e1(t)|+α|e2(t-τ)|.(ii)
Ife1(t)<0 (i.e., x2(t)<k1x1(t)), then
(3.14)D̃1(t)=(αβ1+m1)e1(t)-αe2(t-τ)+αβ3[x23(t)-k13x13(t)]-ηe1(t)PT,p(t)≤-[αβ1-ηp+0.1875αβ3+m1]|e1(t)|+α|e2(t-τ)|.(iii)
Ife1(t)=0 (i.e., x2(t)=k1x1(t)), then
(3.15)D̃1(t)=α|e2(t-τ)|=-[αβ1-ηp+0.1875αβ3+m1]|e1(t)|+α|e2(t-τ)|.From (i)–(iii), we have
(3.16)D̃1(t)≤-[αβ1-ηp+0.1875αβ3+m1]|e1(t)|+α|e2(t-τ)|,t≥t0.
It is easy to see that
(3.17)D̃2(t)={e1(t-τ)-(1+m2)e2(t)+e3(t-τ),e2(t)>0,-e1(t-τ)+(1+m2)e2(t)-e3(t-τ),e2(t)<0,|e1(t-τ)+e3(t-τ)|,e2(t)=0,D̃3(t)={-βe2(t-τ)-m3e3(t),e3(t)>0,βe2(t-τ)+m3e3(t),e3(t)<0,β|e2(t-τ)|,e3(t)=0.
Considering D̃2(t), D̃3(t) in the same way, we can obtain
(3.18)D̃2(t)≤-(1+m2)|e2(t)|+|e1(t-τ)|+|e3(t-τ)|,t≥t0,D̃3(t)≤-m3|e3(t)|+β|e2(t-τ)|,t≥t0.
Hence, we have
(3.19)D+V(t)≤D̃1(t)+D̃2(t)+D̃3(t)+|e1(t)|-|e1(t-τ)|+(α+β)[|e2(t)|-|e2(t-τ)|]+|e3(t)|-|e3(t-τ)|≤-[αβ1-ηp+0.1875αβ3+m1-1]|e1(t)|-(1+m2-α-β)|e2(t)|-(m3-1)|e3(t)|,t≥t0.
So there exists c>0 such that
(3.20)D+V(t)≤-c(|e1(t)|+|e2(t)|+|e3(t)|),t≥t0.
Integrating on both sides of the above inequality leads to
(3.21)V(t)+c∫t0t[|e1(s)|+|e2(s)|+|e3(s)|]ds≤V(t0)<+∞,t≥t0,
which implies
(3.22)|e1(t)|+|e2(t)|+|e3(t)|∈L1[0,+∞).
From the boundedness of xi(t), yi(t), and zi(t)(i=1,2), it follows that e1(t),e2(t),e3(t), and their derivatives remain bounded on [0,+∞). Hence, |e1(t)|+|e2(t)|+|e3(t)| is uniformly continuous. By Lemma 3.1, it follows that
(3.23)lim⁡t→+∞(|e1(t)|+|e2(t)|+|e3(t)|)=0.
Hence,
(3.24)lim⁡t→+∞|e1(t)|=0,lim⁡t→+∞|e2(t)|=0,lim⁡t→+∞|e3(t)|=0.
The proof is complete.Remark 3.4.
Whenk1=k2=k3=1, τ=0, the linear controller u(t)=[u1(t),u2(t),u3(t)]T is given as follows:
(3.25)u1(t)=-m1[x2(t)-x1(t)],u2(t)=-m2[y2(t)-y1(t)],u3(t)=-m3[z2(t)-z1(t)].
In this case, the drive system (3.5) and the response system (3.6) realize complete synchronization.Remark 3.5.
Whenk1=k2=k3=-1, τ=0, the linear controller u(t)=[u1(t),u2(t),u3(t)]T is given as follows:
(3.26)u1(t)=-m1[x2(t)+x1(t)],u2(t)=-m2[y2(t)+y1(t)],u3(t)=-m3[z2(t)+z1(t)].
In this case, the drive system (3.5) and the response system (3.6) realize antisynchronization.Consider the drive system (3.27) and the response system (3.28):(3.27)ẋ1=2Bx1(t)y1(t)+ε[x1(t)+ax12(t)]-ε1x1(t)PT,p(t),ẏ1=1-2Ax1(t)-By12(t),(3.28)ẋ2=2Bx2(t)y2(t)+ε[x2(t)+ax22(t)]-ε1x2(t)PT,p(t)+u1(t),ẏ2=1-2Ax2(t-τ)-By22(t)+u2(t),
where u=[u1(t),u2(t)]T is the control function.Our goal is to determine the control functionu=[u1(t),u2(t)]T. For brevity, let us define(3.29)e1(t)=x2(t)-x1(t),e2(t)=y2(t)-y1(t).
Now, we present the main result for the synchronization between system (3.27) and system (3.28).Theorem 3.6.
If the linear controlleru=[u1(t),u2(t)]T is given as follows:
(3.30)u1(t)=-m1[x2(t)-x1(t)],u2(t)=2A[x1(t-τ)-x1(t)]-m2[y2(t)-y1(t)],
where m1>2A+εa, m2>4B. Then, e1, e2 converge to zero as time t tends to infinity, that is, the drive system (3.27) and the response system (3.28) asymptotically synchronize.Proof.
Because0<x1(t)<1, |y1(t)|<1 for sufficiently large t≥t0* [21], we make the following prior assumption:
(3.31)0<x2(t)<1,|y2(t)|<1,t≥t0(t0≥t0*).
Consider the Lyapunov functional defined by(3.32)V(t)=|ln⁡x2(t)-ln⁡x1(t)|+|y2(t)-y1(t)|+2A∫t-τt|x2(s)-x1(s)|ds,t≥t0.
Calculating the upper right Dini derivative D+V(t) of V(t) along the solutions of system (3.27) and system (3.28), we get
(3.33)D+V(t)≤D̃1(t)+D̃2(t)+2A[|x2(t)-x1(t)|-|x2(t-τ)-x1(t-τ)|],
where
(3.34)D̃1(t)={ẋ2x2-ẋ1x1=2B[y2(t)-y1(t)]+[εa-m1x2(t)][x2(t)-x1(t)],x2(t)>x1(t),ẋ1x1-ẋ2x2=-2B[y2(t)-y1(t)]-[εa-m1x2(t)][x2(t)-x1(t)],x2(t)<x1(t),|ẋ2x2-ẋ1x1|=2B|y2(t)-y1(t)|,x2(t)=x1(t),D̃2(t)={ẏ2-ẏ1=-2A[x2(t-τ)-x1(t-τ)]-[B(y2(t)+y1(t))+m2]e2(t),y2(t)>y1(t),ẏ1-ẏ2=2A[x2(t-τ)-x1(t-τ)]+[B(y2(t)+y1(t))+m2]e2(t),y2(t)<y1(t),|ẏ2-ẏ1|=2A|x2(t-τ)-x1(t-τ)|,y2(t)=y1(t).
There are the following three cases to consider for D̃1(t).(i)
Ifx2(t)>x1(t), then
(3.35)D̃1(t)=2B[y2(t)-y1(t)]+[εa-m1x2(t)][x2(t)-x1(t)]≤2B|y2(t)-y1(t)|+(εa-m1)|x2(t)-x1(t)|.(ii)
Ifx2(t)<x1(t), then
(3.36)D̃1(t)=-2B[y2(t)-y1(t)]-[εa-m1x2(t)][x2(t)-x1(t)]≤2B|y2(t)-y1(t)|+(εa-m1)|x2(t)-x1(t)|.(iii)
Ifx2(t)=x1(t), then
(3.37)D̃1(t)=2B|y2(t)-y1(t)|=2B|y2(t)-y1(t)|+(εa-m1)|x2(t)-x1(t)|.From (i)–(iii), we have
(3.38)D̃1(t)≤(εa-m1)|x2(t)-x1(t)|+2B|y2(t)-y1(t)|,t≥t0.
Considering D̃2(t) in the same way, we can obtain
(3.39)D̃2(t)≤2A|x2(t-τ)-x1(t-τ)|+(2B-m2)|y2(t)-y1(t)|,t≥t0.
Hence, we have
(3.40)D+V(t)≤D̃1(t)+D̃2(t)+2A[|x2(t)-x1(t)|-|x2(t-τ)-x1(t-τ)|]≤(2A+εa-m1)|x2(t)-x1(t)|+(4B-m2)|y2(t)-y1(t)|,t≥t0.
So there exists α>0 such that
(3.41)D+V(t)≤-α(|x2(t)-x1(t)|+|y2(t)-y1(t)|),t≥t0.
Integrating on both sides of the above inequality leads to
(3.42)V(t)+α∫t0t(|x2(s)-x1(s)|+|y2(s)-y1(s)|)ds≤V(t0)<+∞,t≥t0,
which implies
(3.43)|x2(t)-x1(t)|+|y2(t)-y1(t)|∈L1[0,+∞).
From the boundedness of xi(t) and yi(t)(i=1,2), it follows that x2(t)-x1(t),y2(t)-y1(t) and their derivatives remain bounded on [0,+∞). Hence, |x2(t)-x1(t)|+|y2(t)-y1(t)| is uniformly continuous. By Lemma 3.1, it follows that
(3.44)lim⁡t→+∞(|x2(t)-x1(t)|+|y2(t)-y1(t)|)=0.
Hence,
(3.45)lim⁡t→+∞|x2(t)-x1(t)|=0,lim⁡t→+∞|y2(t)-y1(t)|=0.
This result implies that e1(t), e2(t) converge to zero as time t tends to infinity, that is, the drive system (3.27) and the response system (3.28) asymptotically synchronize. The proof is complete.


## 4. Numerical Simulations

In this section, simulation results are given below to illustrate the validity of the proposed method. Numerical simulations are carried out using MATLAB. DDE23 solver is adopted to solve delay differential equations in Section3.The initial values of the drive and response systems in simulations are chosen as follows:(1)
the initial values of systems (3.5) and (3.6): x1(s)=0.01, y1(s)=-0.01, z1(s)=-0.018 and x2(s)=0.02, y2(s)=-0.03, z2(s)=-0.01, for all s∈[-τ,0];(2)
the initial values of systems (3.27) and (3.28): (i) when T=8, x1(s)=0.85, y1(s)=0.5 and x2(s)=-0.65, y2(s)=-0.5, for all s∈[-τ,0]. (ii) when T=10, x1(s)=0.85, y1(s)=-0.5 and x2(s)=-0.65, y2(s)=0.5, for all s∈[-τ,0].The dynamics of synchronization errors for the drive and response systems are shown in Figures3 and 4. It can be seen in these figures that the synchronization errors will rapidly converge to zero, that is to say, with the passage of time, by means of the linear control method, rank-one chaotic systems without and with delay are ultimately synchronous.Synchronization errorse1(t), e2(t), and e3(t) for the drive system (3.5) and response system (3.6). (a) τ=0.001, m1=1.1761, m2=3.31136, m3=1.001, k1=1, k2=-3, k3=-3. (b) τ=0.003, m1=1.1761, m2=3.31136, m3=1.001, k1=-1, k2=3, k3=3. (c) τ=0.01, m1=1.1761, m2=3.31136, m3=1.001, k1=1, k2=-1, k3=2.
(a)(b)(c)Synchronization errorse1(t), e2(t) for the drive system (3.27) and response system (3.28). (a) τ=0.03, T=8, m1=3.301, m2=4.1. (b) τ=0.03, T=10, m1=3.231, m2=4.01.
(a)(b)


## 5. Conclusions

In this paper, the presence of chaos in rank-one chaotic systems with delay are confirmed via a binary test (called 0-1 test) for chaos. The synchronization problems for rank-one chaotic systems are discussed. The linear delayed feedback control is used to achieve the synchronization between two rank-one chaotic systems without and with delay. The numerical simulations are in excellent agreement with the analytical results.



---

*Source: 325131-2012-03-15.xml*