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

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.

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 Section 2, 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

Let𝑃_𝑇,𝑝(𝑡)=1𝑝^+∞_𝑛=−∞𝐹_{𝑛,𝑇,𝑝}(𝑡), (2.1) where𝐹_{𝑛,𝑇,𝑝}(𝑡)=1,𝑛𝑇≤𝑡<𝑛𝑇+𝑝,0,elsewhere. (2.2)𝑃_𝑇,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:𝑥̇=𝛼𝑦−𝛽₁𝑥−𝛽₃𝑥³+𝜂𝑥𝑃_𝑇,𝑝(𝑡),̇𝑦̇=𝑥−𝑦+𝑧,𝑧=−𝛽𝑦, (2.3) where 𝜂 is used for controlling the magnitude of the forcing. When the system parameters are taken as the following values:𝛼=2,𝛽=2.31135,𝛽₁=0.2495,𝛽₃=−1,𝜂=0.15,𝑝=0.5, (2.4) 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:𝑥̇=2𝐵𝑥𝑦+𝜀𝑥+𝑎𝑥²−𝜀₁𝑥(𝑡)𝑃_𝑇,𝑝(𝑡),𝑦̇=1−2𝐴𝑥−𝐵𝑦², (2.5) 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:𝑥̇=𝛼𝑦(𝑡−𝜏)−𝛽₁𝑥(𝑡)−𝛽₃𝑥³(𝑡)+𝜂𝑥(𝑡)𝑃_𝑇,𝑝(𝑡),()(𝑡)(),̇();(𝑡)𝑦(𝑡)𝑥(𝑡)𝑦̇=𝑥𝑡−𝜏−𝑦+𝑧𝑡−𝜏𝑧=−𝛽𝑦𝑡−𝜏𝑥̇=2𝐵𝑥+𝜀+𝑎𝑥²(𝑡)−𝜀₁𝑥(𝑡)𝑃_𝑇,𝑝(𝑡),𝑦̇=1−2𝐴𝑥(𝑡−𝜏)−𝐵𝑦²(𝑡). (2.6)(2.7)

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 data 𝜙(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:𝑝_𝑐(𝑛)=^𝑛_𝑗=1𝜙𝑗cos𝑗𝑐,𝑞_𝑐(𝑛)=^𝑛_𝑗=1𝜙𝑗sin𝑗𝑐,𝑛=1,2,3,…,𝑁. (2.8) To determine the growth of 𝑝_𝑐 and 𝑞_𝑐, it is convenient to look at the mean square displacement (MSD), defined as𝑀_𝑐(𝑛)=lim_𝑁→∞1𝑁^𝑁_𝑗=1𝑝_𝑐𝑗+𝑛−𝑝_𝑐𝑗²+𝑞_𝑐𝑗+𝑛−𝑞_𝑐𝑗². (2.9) The test for chaos is based on the asymptotic growth rate 𝑘_𝑐 of 𝑀_𝑐(𝑛) as a function of 𝑛, which can be estimated as follows:𝑘_𝑐=lim_𝑛→∞log𝑀_𝑐(𝑛)log𝑛. (2.10) Numerically, 𝑘_𝑐 is determined by fitting a straight line to the graph of log𝑀_𝑐(𝑛) versus log𝑛 through minimizing the absolute deviation.

For different 𝑐 values, different 𝑘_𝑐 values can be calculated. The underlying dynamics should be examined by whether the median 𝑘 of 𝑘_𝑐 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 series 𝑥(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.

(a)

(b)

(c)

(d)

Figure 1 

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.

For a sampled time series 𝑥(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.

(a)

(b)

Figure 2 

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.

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:𝑋̇_𝑑=𝐹𝑡,𝑋_𝑑(𝑡),𝑋̇_𝑟=𝐺𝑡,𝑋_𝑟(𝑡),𝑋_𝑟(𝑡−𝜏)+𝑈(𝑡), (3.1)(3.2) where 𝑋_𝑑(𝑡)=[𝑋¹_𝑑(𝑡),…,𝑋^𝑚_𝑑(𝑡)]^𝑇∈𝑅^𝑚, 𝑋_𝑟(𝑡)=[𝑋¹_𝑟(𝑡),…,𝑋^𝑛_𝑟(𝑡)]^𝑇∈𝑅^𝑛, 𝜏≥0, and 𝑈(𝑡)=[𝑢₁(𝑡),…,𝑢_𝑛(𝑡)]^𝑇 is a controller and that can be called the control function.

For brevity, let us define the synchronous error as follows:𝑒₁(𝑡),…,𝑒_𝑛(𝑡)^𝑇=𝑋_𝑟(𝑡)−𝐾_𝑛×𝑚𝑋_𝑑(𝑡), (3.3) where 𝐾_𝑛×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]). Let 𝑓 be a nonnegative function defined on [0,+∞) such that 𝑓 is integrable on [0,+∞) and is uniformly continuous on [0,+∞). Then, lim_𝑡→+∞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),…,𝑢_𝑛(𝑡)]^𝑇 such that the synchronous errors satisfy lim_𝑡→+∞𝑒_𝑖(𝑡)=0(𝑖=1,2,…,𝑛). (3.4) Consider the drive system (3.5) and the response system (3.6): 𝑥̇₁=𝛼𝑦₁(𝑡)−𝛽₁𝑥₁(𝑡)−𝛽₃𝑥³₁(𝑡)+𝜂𝑥₁(𝑡)𝑃_𝑇,𝑝(𝑡),𝑦₁̇=𝑥₁(𝑡)−𝑦₁(𝑡)+𝑧₁(𝑡),𝑧₁̇=−𝛽𝑦₁(𝑡),𝑥̇₂=𝛼𝑦₂(𝑡−𝜏)−𝛽₁𝑥₂(𝑡)−𝛽₃𝑥³₂(𝑡)+𝜂𝑥₂(𝑡)𝑃_𝑇,𝑝(𝑡)+𝑢₁(𝑡),𝑦₂̇=𝑥₂(𝑡−𝜏)−𝑦₂(𝑡)+𝑧₂(𝑡−𝜏)+𝑢₂(𝑡),𝑧₂̇=−𝛽𝑦₂(𝑡−𝜏)+𝑢₃(𝑡), (3.5)(3.6) where 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡)]^𝑇 is the control function.

In this case, we define the synchronous errors as follows:𝑒₁(𝑡)=𝑥₂(𝑡)−𝑘₁𝑥₁(𝑡),𝑒₂(𝑡)=𝑦₂(𝑡)−𝑘₂𝑦₁(𝑡),𝑒₃(𝑡)=𝑧₂(𝑡)−𝑘₃𝑧₁(𝑡). (3.7)

Now, we present the main result for the synchronization between system (3.5) and system (3.6).

Theorem 3.3. If the linear controller 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡)]^𝑇 is given as follows: 𝑢₁(𝑡)=𝛼𝑘₁𝑦₁(𝑡)−𝑘₂𝑦₁(𝑡−𝜏)−𝑚₁𝑥₂(𝑡)−𝑘₁𝑥₁(𝑡,𝑢₂)(𝑡)=𝑘₂𝑥₁(𝑡)−𝑘₁𝑥₁(𝑡−𝜏)+𝑘₂𝑧₁(𝑡)−𝑘₃𝑧₁(𝑡−𝜏)−𝑚₂𝑦₂(𝑡)−𝑘₂𝑦₁(𝑡,𝑢₃)(𝑡)=𝛽𝑘₂𝑦₁(𝑡−𝜏)−𝑘₃𝑦₁(𝑡)−𝑚₃𝑧₂(𝑡)−𝑘₃𝑧₁(𝑡), (3.8) 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≥𝑡^∗₀ [20], we make the following prior assumption: 𝑥₂(𝑡)<0.25,𝑡≥𝑡₀𝑡₀≥𝑡^∗₀. (3.9) Consider the Lyapunov functional defined by 𝑉(𝑡)=𝑒₁(𝑡)+𝑒₂(𝑡)+𝑒₃(𝑡)+^𝑡_𝑡−𝜏𝑒₁(𝑠)+𝛼+𝛽𝑒₂(𝑠)+𝑒₃(𝑠)𝑑𝑠,𝑡≥𝑡₀. (3.10) 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 𝐷⁺𝑉(𝑡)≤𝐷₁(𝑡)+𝐷₂(𝑡)+𝐷₃(𝑡)+𝑒₁(𝑡)−𝑒₁(+𝑒𝑡−𝜏)𝛼+𝛽₂(𝑡)−𝑒₂(𝑡−𝜏)+𝑒₃(𝑡)−𝑒₃(𝑡−𝜏), (3.11) where 𝐷_𝑖(𝑡)=𝐷⁺𝑒_𝑖(𝑡)=𝑒_𝑖̇(𝑡),𝑒_𝑖(𝑡)>0,−𝑒_𝑖̇(𝑡),𝑒_𝑖(𝑡)<0,𝑒_𝑖̇(𝑡),𝑒_𝑖(𝑡)()=0,𝑖=1,2,3. (3.12)
There are the following three cases to consider for 𝐷₁(𝑡).
(i)If 𝑒₁(𝑡)>0 (i.e., 𝑥₂(𝑡)>𝑘₁𝑥₁(𝑡)), then 𝐷₁(𝑡)=−𝛼𝛽₁+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏)−𝛼𝛽₃𝑥³₂(𝑡)−𝑘₁𝑥³₁(𝑡)+𝜂𝑒₁(𝑡)𝑃_𝑇,𝑝(𝑡)≤−𝛼𝛽₁−𝜂𝑝+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏)−𝛼𝛽₃𝑒₁(𝑡)𝑥²₂(𝑡)+𝑘₁𝑥₁(𝑡)𝑥₂(𝑡)+𝑘²₁𝑥²₁(𝑡)≤−𝛼𝛽₁−𝜂𝑝+0.1875𝛼𝛽₃+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏). (3.13)(ii)If 𝑒₁(𝑡)<0 (i.e., 𝑥₂(𝑡)<𝑘₁𝑥₁(𝑡)), then 𝐷₁(𝑡)=𝛼𝛽₁+𝑚₁𝑒₁(𝑡)−𝛼𝑒₂(𝑡−𝜏)+𝛼𝛽₃𝑥³₂(𝑡)−𝑘³₁𝑥³₁(𝑡)−𝜂𝑒₁(𝑡)𝑃_𝑇,𝑝(𝑡)≤−𝛼𝛽₁−𝜂𝑝+0.1875𝛼𝛽₃+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏). (3.14)(iii)If 𝑒₁(𝑡)=0 (i.e., 𝑥₂(𝑡)=𝑘₁𝑥₁(𝑡)), then 𝐷₁(𝑡)=𝛼𝑒₂(𝑡−𝜏)=−𝛼𝛽₁−𝜂𝑝+0.1875𝛼𝛽₃+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏). (3.15)From (i)–(iii), we have 𝐷₁(𝑡)≤−𝛼𝛽₁−𝜂𝑝+0.1875𝛼𝛽₃+𝑚₁𝑒₁(𝑡)+𝛼𝑒₂(𝑡−𝜏),𝑡≥𝑡₀. (3.16) It is easy to see that 𝐷₂(𝑡)=𝑒₁(𝑡−𝜏)−1+𝑚₂𝑒₂(𝑡)+𝑒₃(𝑡−𝜏),𝑒₂(𝑡)>0,−𝑒₁(𝑡−𝜏)+1+𝑚₂𝑒₂(𝑡)−𝑒₃(𝑡−𝜏),𝑒₂(𝑡)𝑒<0,₁(𝑡−𝜏)+𝑒₃(𝑡−𝜏),𝑒₂(𝑡)𝐷=0,₃(𝑡)=−𝛽𝑒₂(𝑡−𝜏)−𝑚₃𝑒₃(𝑡),𝑒₃(𝑡)>0,𝛽𝑒₂(𝑡−𝜏)+𝑚₃𝑒₃(𝑡),𝑒₃(𝑡)𝛽𝑒<0,₂(𝑡−𝜏),𝑒₃(𝑡)=0. (3.17) Considering 𝐷₂(𝑡), 𝐷₃(𝑡) in the same way, we can obtain 𝐷₂(𝑡)≤−1+𝑚₂𝑒₂(𝑡)+𝑒₁(𝑡−𝜏)+𝑒₃(𝑡−𝜏),𝑡≥𝑡₀,𝐷₃(𝑡)≤−𝑚₃𝑒₃(𝑡)+𝛽𝑒₂(𝑡−𝜏),𝑡≥𝑡₀. (3.18) Hence, we have 𝐷⁺𝑉(𝑡)≤𝐷₁(𝑡)+𝐷₂(𝑡)+𝐷₃(𝑡)+𝑒₁(𝑡)−𝑒₁(+𝑒𝑡−𝜏)𝛼+𝛽₂(𝑡)−𝑒₂(𝑡−𝜏)+𝑒₃(𝑡)−𝑒₃(𝑡−𝜏)≤−𝛼𝛽₁−𝜂𝑝+0.1875𝛼𝛽₃+𝑚₁−1𝑒₁(𝑡)−1+𝑚₂−𝛼−𝛽𝑒₂(𝑡−𝑚₃)−1𝑒₃(𝑡),𝑡≥𝑡₀. (3.19) So there exists 𝑐>0 such that 𝐷⁺𝑉(𝑡)≤−𝑐𝑒₁(𝑡)+𝑒₂(𝑡)+𝑒₃(𝑡),𝑡≥𝑡₀. (3.20) Integrating on both sides of the above inequality leads to 𝑉(𝑡)+𝑐^𝑡_𝑡0𝑒₁(𝑠)+𝑒₂(𝑠)+𝑒₃(𝑠)𝑑𝑠≤𝑉𝑡₀<+∞,𝑡≥𝑡₀, (3.21) which implies 𝑒₁(𝑡)+𝑒₂(𝑡)+𝑒₃(𝑡)∈𝐿¹[0,+∞). (3.22) From the boundedness of 𝑥_𝑖(𝑡), 𝑦_𝑖(𝑡), and 𝑧_𝑖(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 lim_𝑡→+∞𝑒₁(𝑡)+𝑒₂(𝑡)+𝑒₃(𝑡)=0. (3.23) Hence, lim_𝑡→+∞𝑒₁(𝑡)=0,lim_𝑡→+∞𝑒₂(𝑡)=0,lim_𝑡→+∞𝑒₃(𝑡)=0. (3.24) The proof is complete.

Remark 3.4. When 𝑘₁=𝑘₂=𝑘₃=1, 𝜏=0, the linear controller 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡)]^𝑇 is given as follows: 𝑢₁(𝑡)=−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡),𝑢₂(𝑡)=−𝑚₂𝑦₂(𝑡)−𝑦₁(𝑡),𝑢₃(𝑡)=−𝑚₃𝑧₂(𝑡)−𝑧₁(𝑡). (3.25) In this case, the drive system (3.5) and the response system (3.6) realize complete synchronization.

Remark 3.5. When 𝑘₁=𝑘₂=𝑘₃=−1, 𝜏=0, the linear controller 𝑢(𝑡)=[𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡)]^𝑇 is given as follows: 𝑢₁(𝑡)=−𝑚₁𝑥₂(𝑡)+𝑥₁(𝑡),𝑢₂(𝑡)=−𝑚₂𝑦₂(𝑡)+𝑦₁(𝑡),𝑢₃(𝑡)=−𝑚₃𝑧₂(𝑡)+𝑧₁(𝑡). (3.26) 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):𝑥̇₁=2𝐵𝑥₁(𝑡)𝑦₁(𝑡)+𝜀𝑥₁(𝑡)+𝑎𝑥²₁(𝑡)−𝜀₁𝑥₁(𝑡)𝑃_𝑇,𝑝(𝑡),𝑦₁̇=1−2𝐴𝑥₁(𝑡)−𝐵𝑦²₁(𝑡),𝑥̇₂=2𝐵𝑥₂(𝑡)𝑦₂(𝑡)+𝜀𝑥₂(𝑡)+𝑎𝑥²₂(𝑡)−𝜀₁𝑥₂(𝑡)𝑃_𝑇,𝑝(𝑡)+𝑢₁(𝑡),𝑦₂̇=1−2𝐴𝑥₂(𝑡−𝜏)−𝐵𝑦²₂(𝑡)+𝑢₂(𝑡), (3.27)(3.28) where 𝑢=[𝑢₁(𝑡),𝑢₂(𝑡)]^𝑇 is the control function.

Our goal is to determine the control function 𝑢=[𝑢₁(𝑡),𝑢₂(𝑡)]^𝑇. For brevity, let us define𝑒₁(𝑡)=𝑥₂(𝑡)−𝑥₁(𝑡),𝑒₂(𝑡)=𝑦₂(𝑡)−𝑦₁(𝑡). (3.29) Now, we present the main result for the synchronization between system (3.27) and system (3.28).

Theorem 3.6. If the linear controller 𝑢=[𝑢₁(𝑡),𝑢₂(𝑡)]^𝑇 is given as follows: 𝑢₁(𝑡)=−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡,𝑢₂)(𝑡)=2𝐴𝑥₁(𝑡−𝜏)−𝑥₁(𝑡)−𝑚₂𝑦₂(𝑡)−𝑦₁(𝑡), (3.30) 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. Because 0<𝑥₁(𝑡)<1, |𝑦₁(𝑡)|<1 for sufficiently large 𝑡≥𝑡^∗₀ [21], we make the following prior assumption: 0<𝑥₂(𝑡)<1,𝑦₂(𝑡)<1,𝑡≥𝑡₀𝑡₀≥𝑡^∗₀. (3.31)
Consider the Lyapunov functional defined by𝑉(𝑡)=ln𝑥₂(𝑡)−ln𝑥₁(𝑡)+𝑦₂(𝑡)−𝑦₁(𝑡)+2𝐴^𝑡_𝑡−𝜏𝑥₂(𝑠)−𝑥₁(𝑠)𝑑𝑠,𝑡≥𝑡₀. (3.32) 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 𝐷⁺𝑉(𝑡)≤𝐷₁(𝑡)+𝐷₂(𝑡)+2𝐴𝑥₂(𝑡)−𝑥₁(𝑡)−𝑥₂(𝑡−𝜏)−𝑥₁(𝑡−𝜏), (3.33) where 𝐷₁(𝑡)=𝑥̇₂𝑥₂−𝑥̇₁𝑥₁=2𝐵𝑦₂(𝑡)−𝑦₁(𝑡+𝑚)𝜀𝑎−¹𝑥₂(𝑡)𝑥₂(𝑡)−𝑥₁(𝑡),𝑥₂(𝑡)>𝑥₁(𝑡),𝑥̇₁𝑥₁−𝑥̇₂𝑥₂=−2𝐵𝑦₂(𝑡)−𝑦₁(𝑡−𝑚)𝜀𝑎−¹𝑥₂(𝑡)𝑥₂(𝑡)−𝑥₁(𝑡),𝑥₂(𝑡)<𝑥₁(𝑡),𝑥̇₂𝑥₂−𝑥̇₁𝑥₁=2𝐵𝑦₂(𝑡)−𝑦₁(𝑡),𝑥₂(𝑡)=𝑥₁(𝑡),𝐷₂(𝑡)=𝑦₂̇−𝑦₁̇=−2𝐴𝑥₂(𝑡−𝜏)−𝑥₁(𝑡−𝜏)−𝐵𝑦₂(𝑡)+𝑦₁(𝑡)+𝑚₂𝑒₂(𝑡),𝑦₂(𝑡)>𝑦₁(𝑡),𝑦₁̇−𝑦₂̇=2𝐴𝑥₂(𝑡−𝜏)−𝑥₁(𝑡−𝜏)+𝐵𝑦₂(𝑡)+𝑦₁(𝑡)+𝑚₂𝑒₂(𝑡),𝑦₂(𝑡)<𝑦₁(𝑡),𝑦₂̇−𝑦₁̇=2𝐴𝑥₂(𝑡−𝜏)−𝑥₁(𝑡−𝜏),𝑦₂(𝑡)=𝑦₁(𝑡). (3.34) There are the following three cases to consider for 𝐷₁(𝑡).(i)If 𝑥₂(𝑡)>𝑥₁(𝑡), then 𝐷₁(𝑡)=2𝐵𝑦₂(𝑡)−𝑦₁(𝑡+𝑚)𝜀𝑎−¹𝑥₂(𝑡)𝑥₂(𝑡)−𝑥₁(𝑡𝑦)≤2𝐵₂(𝑡)−𝑦₁(𝑡)+𝜀𝑎−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡). (3.35)(ii)If 𝑥₂(𝑡)<𝑥₁(𝑡), then 𝐷₁(𝑡)=−2𝐵𝑦₂(𝑡)−𝑦₁(𝑡−𝑚)𝜀𝑎−¹𝑥₂(𝑡)𝑥₂(𝑡)−𝑥₁(𝑡𝑦)≤2𝐵₂(𝑡)−𝑦₁(𝑡)+𝜀𝑎−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡). (3.36)(iii)If 𝑥₂(𝑡)=𝑥₁(𝑡), then 𝐷₁(𝑡)=2𝐵𝑦₂(𝑡)−𝑦₁(𝑡𝑦)=2𝐵₂(𝑡)−𝑦₁(𝑡)+𝜀𝑎−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡). (3.37)From (i)–(iii), we have 𝐷₁(𝑡)≤𝜀𝑎−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡)+2𝐵𝑦₂(𝑡)−𝑦₁(𝑡),𝑡≥𝑡₀. (3.38) Considering 𝐷₂(𝑡) in the same way, we can obtain 𝐷₂(𝑡)≤2𝐴𝑥₂(𝑡−𝜏)−𝑥₁(𝑡−𝜏)+2𝐵−𝑚₂𝑦₂(𝑡)−𝑦₁(𝑡),𝑡≥𝑡₀. (3.39) Hence, we have 𝐷⁺𝑉(𝑡)≤𝐷₁(𝑡)+𝐷₂(𝑡)+2𝐴𝑥₂(𝑡)−𝑥₁(𝑡)−𝑥₂(𝑡−𝜏)−𝑥₁(≤𝑡−𝜏)2𝐴+𝜀𝑎−𝑚₁𝑥₂(𝑡)−𝑥₁(𝑡)+4𝐵−𝑚₂𝑦₂(𝑡)−𝑦₁(𝑡),𝑡≥𝑡₀. (3.40) So there exists 𝛼>0 such that 𝐷⁺𝑉(𝑡)≤−𝛼𝑥₂(𝑡)−𝑥₁(𝑡)+𝑦₂(𝑡)−𝑦₁(𝑡),𝑡≥𝑡₀. (3.41) Integrating on both sides of the above inequality leads to 𝑉(𝑡)+𝛼^𝑡_𝑡0𝑥₂(𝑠)−𝑥₁(𝑠)+𝑦₂(𝑠)−𝑦₁(𝑠)𝑑𝑠≤𝑉𝑡₀<+∞,𝑡≥𝑡₀, (3.42) which implies 𝑥₂(𝑡)−𝑥₁(𝑡)+𝑦₂(𝑡)−𝑦₁(𝑡)∈𝐿¹[0,+∞). (3.43) From the boundedness of 𝑥_𝑖(𝑡) and 𝑦_𝑖(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 lim_𝑡→+∞𝑥₂(𝑡)−𝑥₁(𝑡)+𝑦₂(𝑡)−𝑦₁(𝑡)=0. (3.44) Hence, lim_𝑡→+∞𝑥₂(𝑡)−𝑥₁(𝑡)=0,lim_𝑡→+∞𝑦₂(𝑡)−𝑦₁(𝑡)=0. (3.45) 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 Section 3.

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 Figures 3 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.

(a)

(b)

(c)

Figure 3 

Synchronization errors 𝑒1(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)

Figure 4 

Synchronization errors 𝑒1(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.

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.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant nos. 10971085, 11061016). The author would like to thank Dr. Paul Matthews, University of Nottingham, for providing MATLAB codes used for computing the median 𝑘 of 𝑘_𝑐 values. The author also would like to thank the referees for their valuable comments.