Stability Analysis and Control of a New Smooth Chua's System

Abstract

This paper is concerned with the stability analysis and control of a new smooth Chua's system. Firstly, the chaotic characteristic of the system is confirmed with the aid of the Lyapunov exponents. Secondly, it is proved that the system has globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to globally exponentially stabilize the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua's systems are globally synchronized, and the estimation errors of the uncertain parameters converge to zero as 𝑡 tends to infinity. Finally, the numerical simulations are also presented.

1. Introduction

It is well known that Chua’s system is the first analog circuit to realize chaos in experiments. The original Chua’s system is described by the following ordinary differential equations [1]: ̇(𝑥),̇̇𝑥=𝑝𝑦−𝑥−𝑔𝑦=𝑥−𝑦+𝑧,𝑧=−𝑞𝑦, (1) where 𝑥,𝑦,𝑧∈𝑅 are state variables and 𝑔(𝑥)=𝐺_𝑏𝑥+(1/2)(𝐺_𝑎−𝐺_𝑏)(|𝑥−𝐸|−|𝑥−𝐸|), 𝑝>0, 𝑞>0, and 𝐺_𝑎, 𝐺_𝑏 are constants. Due to the form of a simple circuit, there are a large literature on the dynamical behavior of Chua’s system [2–8]. By changing the parameters or the corresponding functions of Chua’s system, the chaotic phenomenon is very rich, and it is more convenient to study the chaotic mechanism and characteristics [6–8].

For the chaotic systems, Lagrange stability, stability of equilibrium points and, synchronization are three important problems which attracted more and more attention (refer to [9–16] and the reference therein). In [11–13], the authors studied the Lagrange stability by applying the attractive set and positive invariant set of the chaotic systems. Moreover, the researchers examined the stabilization of the unstable equilibrium points and the synchronization control for the chaotic systems with linear controllers [7, 14]. Recently, adaptive controllers are used in synchronous control of chaotic systems when the parameters of the systems are uncertain [15–17].

Motivated by the previous results, the main purpose of this paper is to construct a new smooth Chua’s system and investigate the stability and control problems. More precisely, we will consider the following smooth Chua’s system: 𝑥̇=𝑝𝑥+𝑦−𝑥ln√1+𝑥²,̇̇𝑦=𝑥−𝑦+𝑧,𝑧=−𝑞𝑦, (2) where 𝑥,𝑦,𝑧∈𝑅 are state variables and 𝑝>0, 𝑞>0 are constants.

We will show that the chaotic characteristics are depended on the parameters 𝑝 and 𝑞 and the initial state values of the system (2). All equilibrium points of system (2) are examined to be unstable when 𝑝=11 and 𝑞=14.87 (see in Section 3). By computing with MATLAB, the maximum Lyapunov exponent of the system (2) is 0.0021, where the embedding dimension 𝑚 is 3 and the delay time 𝜏 is 5. Since the maximum Lyapunov exponent is greater than 0, the Chua’s system is chaotic. It will be of great significance if the solution of (2) is ultimately bounded (Lagrange asymptotically stable). The chaotic phase diagrams of such system is obtained by simulation with MATLAB. Figure 1 shows the phase diagrams of the system (2) with 𝑥(0)=−3, 𝑦(0)=2, and 𝑧(0)=1, the phase diagrams of Chua’s system exhibits chaotic.

(a)

(b)

(c)

Figure 1 

The phase portraits of Chua’s system (2).

The remains of this paper are organized as follows. The existence of globally exponential attractive set and positive invariant set for the system (2) will be discussed in Section 2. The asymptotic stability of the equilibrium points will be studied in Section 3, and the synchronization control for two similar types of the Chua’s systems will be discussed in Section 4. In Section 5, we will give the numerical simulations to demonstrate the correctness of our results, and finally we will give the conclusions in Section 6.

2. Existence of Globally Exponential Attractive Set and Positive Invariant Set

The Lagrange stability analysis of the system (2) will be studied in this section. To do so, we first give two definitions [7].

Definition 1. If there exists a radially unbounded, positive definite Lyapunov function 𝑉(𝑋(𝑡)) and positive numbers 𝑙>0, 𝛼>0 such that for all 𝑋₀∈𝑅³, when 𝑉(𝑋₀)>𝑙, the solution of the system (2), 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) along 𝑉(𝑋(𝑡)) satisfies |𝑉(𝑋(𝑡))−𝑙|≤|𝑉(𝑥₀)−𝑙|𝑒^{−𝛼(𝑡−𝑡0)}, then system (2) has a globally exponential attractive set Ω≜{𝑋∣𝑉(𝑋(𝑡))<𝑙}.

Definition 2. Let Ω⊆𝑅³, if ∀𝑋₀∈Ω and for all 𝑡≥𝑡₀, 𝑋(𝑡,𝑡₀,𝑋₀)⊆Ω, then Ω is called positive invariant set of the system (2).

It is easy to prove that the globally exponential attractive set is positive invariant. A system with global attractive set is always called Lagrange globally asymptotically stable system or ultimately bounded dissipative system. For the system (2), we will prove the following the Lagrange stability results.

Theorem 3. The system (2) has the following globally exponential attractive and positive invariant set: |𝑥|≤𝑥Ω=^𝑏=min𝑥_𝑏1,𝑥_𝑏2𝑦²+1𝑞𝑧²𝑥−𝜖𝑦𝑧≤5/4𝜂²_𝑏−𝜏₁𝜇_𝑀𝐺⁻₂/𝜆_𝑀𝐺⁺₂𝑥_𝑏1>𝑒^{((2−(𝜖/4))2/(2−𝜖𝑞−(𝜖/4)))+(𝜖/4)+2}𝑥−1_𝑏2=sup_Ω|𝑥|,1Ω≜𝑥∣𝑝𝑥²+𝑦²+1𝑞𝑧²=𝑥−𝜖𝑦𝑧_2𝑏1ln1+𝑥²_𝑏1−𝜇_𝑀𝐺⁻₁𝑥_𝑏1/𝜆_𝑀𝐺⁺₁, (3) where 0<𝜖≪1, 𝐺⁺₁≜1𝑝𝜖0001−2𝜖0−21𝑞,𝐺⁺₂≜𝜖1−2−𝜖21𝑞, (4) are symmetric positive definite matrices, and 𝐺⁻₁≜21−ln1+𝑥²_𝑏1𝜖2−2𝜖2𝜖𝑞−22−𝜖2𝜖2,𝐺−𝜖₋₂≜𝜖𝜖𝑞−22𝜖2,−𝜖 (5) are symmetric negative definite matrices. 𝜆_𝑀(𝐺⁺₁), 𝜇_𝑀(𝐺⁻₁), 𝜆_𝑀(𝐺⁺₂), and 𝜇_𝑀(𝐺⁻₂) are the maximum eigenvalues of 𝐺⁺₁, 𝐺⁻₁, 𝐺⁺₂, and 𝐺⁻₂, respectively. 𝜖,𝑥_𝑏1, and 𝑥_𝑏2 in (3) are chosen to guarantee that 𝐺⁺₁, 𝐺⁺₂ are positive definite and 𝐺⁻₁, 𝐺⁻₂ are negative definite. 𝜏₁, 𝜏₂, and 𝜂 are chosen such that 𝜏₁>0, 𝜏₂>0, 𝜂>0, 𝜏₁+𝜏₂=1, and 𝜏₂𝜇_𝑀(𝐺⁻₂)+𝜂≤0.

Proof. The proof is divided into three steps.
Step I. The existence of 𝜖, 𝑥_𝑏1 such that 𝐺⁺₁ is positive definite and 𝐺⁻₁ is negative definite.
It is well known that 𝐺⁺₁ is positive definite if and only if all the order principal minors of 𝐺⁺₁ are positive [18]. That is, 1𝑝1>0,𝑝0101>0,𝑝𝜖0001−2𝜖0−21𝑞=1𝑝1𝑞−𝜖²4>0. (6) Let |𝜖|<2/√𝑞; it is easy to see that 𝐺⁺₁ is positive definite since 𝑝>0, 𝑞>0.
Next, let 𝑆₁₁=21−ln1+𝑥²_𝑏12,𝑆2𝜖𝑞−2₁₂=−𝜖2𝜖2,𝑆₂₁=−𝜖2,𝜖2,𝑆₂₂=−𝜖. (7) Then, 𝐺⁻₁=𝑆₁₁𝑆₁₂𝑆₂₁𝑆₂₂. (8)
From Schur theorem [18], 𝐺⁻₁ is negative definite if and only if 𝑆₁₁<0, 𝑆₂₂−𝑆₂₁𝑆⁻¹₁₁𝑆₁₂<0. Let |𝜖|, 𝑥_𝑏1 satisfy 𝑥0<𝜖≪1,_𝑏1>e^{((2−(𝜖/4))2/(2−𝜖𝑞−(𝜖/4)))+(𝜖/4)+2}−1. (9) It is easy to verify that 𝐺⁻₁ is negative definite.
Step II. Existence of globally exponential attractive set of 𝑥.
Let 𝜆_𝑚(𝐺⁺₁) and 𝜆_𝑚(𝐺⁻₁) be the minimum eigenvalues of 𝐺⁺₁ and 𝐺⁻₁, respectively. By constructing a radially unbounded Lyapunov function as follows: 1𝑉=𝑝𝑥²+𝑦²+1𝑞𝑧²𝑥𝑦𝑧−𝜖𝑦𝑧=^𝑇𝐺⁺₁𝑥𝑦𝑧, (10)
then, one can obtain that 𝜆_𝑚𝐺⁺₁𝑥²+𝑦²+𝑧²≤𝑉≤𝜆_𝑀𝐺⁺₁𝑥²+𝑦²+𝑧². (11)
The time derivative of 𝑉 along the system (2) is given by 𝑑𝑉𝑑𝑡₍₂₎=2𝑝𝑥̇̇2𝑥+2𝑦𝑦+𝑞𝑧̇̇𝑧̇=2𝑧−𝜖𝑦𝑧−𝜖𝑦𝑝𝑥𝑝𝑥+𝑦−𝑥ln√1+𝑥²+2+2𝑦𝑥−𝑦+𝑧𝑞=𝑥𝑦𝑧−𝑞𝑦−𝜖𝑥−𝑦+𝑧𝑧−𝜖𝑦−𝑞𝑦_𝑇21−ln√1+𝑥²𝜖2−2𝜖2𝜖𝑞−22−𝜖2𝜖2𝑥𝑦𝑧=𝑥𝑦𝑧−𝜖_𝑇𝐺⁻₁𝑥𝑦𝑧−2ln√1+𝑥²−ln1+𝑥²_𝑏1𝑥². (12)
If |𝑥|>|𝑥_𝑏1|, one has 𝑑𝑉𝑑𝑡₍₂₎≤𝑥𝑦𝑧^𝑇𝐺⁻₁𝑥𝑦𝑧≤𝜇_𝑀𝐺⁻₁𝜆_𝑀𝐺⁺₁𝑉. (13) Then, the following inequality holds 𝑉𝑋𝑡,𝑡₀,𝑋₀≤𝑉𝑋₀e^{(𝜇𝑀(𝐺−1)/𝜆𝑀(𝐺+1))(𝑡−𝑡0)}. (14) Thus, the trajectory of the system (2) will exponentially decay into the area 𝑈={𝑥∣|𝑥|≤|𝑥_𝑏1|} if |𝑥|>|𝑥_𝑏1|.
When the trajectory of the system (2) is in the area 𝑈, it holds that 𝑑𝑉𝑑𝑡₍₂₎=𝑥𝑦𝑧^𝑇𝐺⁻₁𝑥𝑦𝑧+2ln1+𝑥²_𝑏1−ln√1+𝑥²𝑥²≤𝜇_𝑀𝐺⁻₁𝜆_𝑀𝐺⁺₁𝑉+2ln1+𝑥²_𝑏1−ln√1+𝑥²𝑥²≤𝜇_𝑀𝐺⁻₁𝜆_𝑀𝐺⁺₁𝑉+𝑥²_𝑏1ln1+𝑥²_𝑏1=𝜇_𝑀𝐺⁻₁𝜆_𝑀𝐺⁺₁𝑉−𝑥²_𝑏1ln1+𝑥²_𝑏1−𝜇_𝑀𝐺⁻₁/𝜆_𝑀𝐺⁺₁. (15)
Let 𝑉_𝑙≜(𝑥²_𝑏1ln(1+𝑥²_𝑏1))/(−𝜇_𝑀(𝐺⁻₁)/𝜆_𝑀(𝐺⁺₁)). If 𝑉(𝑋₀)>𝑉_𝑙 and 𝑉(𝑋(𝑡,𝑡₀,𝑋₀))>𝑉_𝑙, it holds that 𝑉𝑋𝑡,𝑡₀,𝑋₀−𝑉_𝑙≤𝑉𝑋₀−𝑉_𝑙𝑒^{(𝜇𝑀(𝐺−1)/𝜆𝑀(𝐺+1))(𝑡−𝑡0)}. (16)
Let Ω≜{𝑋∣𝑉(𝑋)=𝑉_𝑙}, |𝑥_𝑏2|≜sup_𝑥∈Ω|𝑥|. Then, 𝑉(𝑋(𝑡,𝑡₀,𝑋₀))−𝑉_𝑙 exponentially decays when |𝑥|≥|𝑥_𝑏|≜min{|𝑥_𝑏1|,|𝑥_𝑏2|}. Thus, Ω_𝑥≜{𝑥∣|𝑥|≤|𝑥_𝑏|} is the globally exponential attractive set of 𝑥.
Step III. Existence of globally exponential attractive set of 𝑦 and 𝑧.
Let 𝐺⁺₂≜𝜖1−2−𝜖21𝑞,𝐺⁻₂≜𝜖𝜖𝑞−22𝜖2−𝜖, (17) then, 𝐺⁺₂ is a positive matrix if 𝜖∈(0,2/√𝑞), and 𝐺⁻₂ is a negative matrix if 𝜖∈(0,8/(4𝑞+1)).
At the same time, it is easy to examine that 𝐺⁺₂ is positive and 𝐺⁻₂ is negative if let 𝜖∈(0,8/(4𝑞+1)), since 𝑞>0 and 8/(4𝑞+1)=2/(𝑞+1/4)≤2/√𝑞.
Let 𝜆_𝑚(𝐺⁺₂), 𝜆_𝑀(𝐺⁺₂) be, respectively, minimum and maximum eigenvalues of 𝐺⁺₂, and let 𝜇_𝑚(𝐺⁻₂) and 𝜇_𝑀(𝐺⁻₂) be the minimum and maximum eigenvalues of 𝐺⁻₂, respectively. Since |𝑥|≤|𝑥_𝑏|, a radially unbounded and positive definite Lyapunov function about 𝑦 and 𝑧 is constructed as follows: 𝑊=𝑦²+1𝑞𝑧²𝑦𝑧−𝜖𝑦𝑧=^𝑇𝐺⁺₂𝑦𝑧. (18) Then, the time derivative of 𝑊 along the system (2) yields 𝑑𝑊𝑑𝑡₍₂₎̇2=2𝑦𝑦+𝑞𝑧̇̇𝑧̇=𝑦𝑧𝑧−𝜖𝑦𝑧−𝜖𝑦_𝑇𝐺⁻₂𝑦𝑧+2𝑥𝑦−𝜖𝑥𝑧≤𝜇_𝑀𝐺⁻₂𝑦²+𝑧²+2𝑥_𝑏𝑦+𝜖𝑥_𝑏|𝑧|≤𝜏₁𝜇_𝑀𝐺⁻₂𝑦²+𝑧²+𝜏₂𝜇_𝑀𝐺⁻₂𝑦²+𝑧²+𝜂𝑦²+𝑥²_𝑏𝜂+𝜂𝑧²+𝑥²_𝑏≤𝜏4𝜂₁𝜇_𝑀𝐺⁻₂𝜆_𝑀𝐺⁺₂𝑊+𝜏₂𝜇_𝑀𝐺⁻₂+𝜂𝑦²+𝑧²+5𝑥4𝜂^2𝑏≤𝜏₁𝜇_𝑀𝐺⁻₂𝜆_𝑀𝐺⁺₂𝑥𝑊−5/4𝜂²_𝑏−𝜏₁𝜇_𝑀𝐺⁻₂/𝜆_𝑀𝐺⁺₂, (19) where 𝜏₁,𝜏₂, and 𝜂 are chosen such that 𝜏₁>0, 𝜏₂>0, 𝜂>0, 𝜏₁+𝜏₂=1, and 𝜏₂𝜇_𝑀(𝐺⁻₂)+𝜂≤0.
Let 𝑊_𝑙≜(5/4𝜂)𝑥²_𝑏/(−𝜏₁𝜇_𝑀(𝐺⁻₂)/𝜆_𝑀(𝐺⁺₂)); if 𝑊(𝑋₀)>𝑊_𝑙 and 𝑊(𝑋(𝑡,𝑡₀,𝑋₀))>𝑊_𝑙, it holds that 𝑊𝑋𝑡,𝑡₀,𝑋₀−𝑊_𝑙≤𝑊𝑋₀−𝑊_𝑙𝑒^{(𝜏1𝜇𝑀(𝐺−2)/𝜆𝑀(𝐺+2))(𝑡−𝑡0)}. (20)
Similarly, if 𝑊(𝑋₀)≤𝑊_𝑙, the state trajectory will stay in the area such that 𝑊(𝑋(𝑡,𝑡₀,𝑋₀))≤𝑊_𝑙 holds. Hence, 𝑦 and 𝑧 are exponentially decreased and ultimately enter into the attractive region 𝑊(𝑋(𝑡,𝑡₀,𝑋₀))≤𝑊_𝑙, that is, 𝑦²+1𝑞𝑧²𝑥−𝜖𝑦𝑧≤5/4𝜂²_𝑏−𝜏₁𝜇_𝑀𝐺⁻₂/𝜆_𝑀𝐺⁺₂. (21)
Combining Steps I, II, and III, Theorem 3 is obtained and the proof is completed.

Remark 4. In this section, a constructive method is proposed to prove the main results of existence of globally exponential attractive set and positive invariant set for the Chua’s systems. By constructing the matrices 𝐺⁺₁, 𝐺⁻₁, 𝐺⁺₂, 𝐺⁻₂, and Lyapunov function candidate 𝑉 and 𝑊, the problem is solved ingeniously.

3. Global Linear Stabilization of the Equilibrium Points

In this section, the stability of the equilibrium points for the system (2) will be discussed with the aid of a linear controller.

Firstly, it is easy to examine that the system (2) has three equilibrium points: 𝑆₀=(0,0,0),𝑆₊=√𝑒²−1,0,−√𝑒²,𝑆−1₋=−√𝑒²−1,0,√𝑒²−1. (22) Moreover, the equilibrium points of the system is independent of parameters 𝑝 and 𝑞. It should be mentioned that, however, the stability of the equilibrium points is depended on 𝑝 and 𝑞. In the following, we will design a linear controller to stabilize the unstable equilibrium points.

Let 𝑝=11, 𝑞=14.87, and (𝑥^∗,𝑦^∗,𝑧^∗) be any equilibrium point of the system (2), the corresponding Jacobian matrix is given by 𝐽₀=𝑝−ln1+𝑥^∗2+11+𝑥^∗2𝑝01−110−𝑞0. (23) Then the characteristic equation of the corresponding local linearization system of system (2) is as follows: 𝑎₀𝜆³+𝑎₁𝜆²+𝑎₂𝜆+𝑎₃=0, (24) where 𝑎₀=1, 𝑎₁=1−𝑝𝑤, 𝑎₂=−𝑞−𝑝−𝑝𝑤, 𝑎₃=𝑝𝑞𝑤, and 𝑤=−ln1+𝑥^∗2+(1/(1+𝑥^∗2)).(i)If 𝑥^∗=0, one has 𝑎₀=1>0, 𝑎₁=−10<0. According to Hurwitz stability criterion [19], the necessary condition of stable equilibrium point is the same sign of the coefficients of the characteristic equation. Consequently, the equilibrium 𝑆₀ is unstable.(ii)If 𝑥^∗=±√𝑒²−1, one has 𝑎₀=1, 𝑎₁=10.5113>0, and 𝑎₂=−16.3587<0. Similarly, one can obtain that 𝑆₊ and 𝑆₋ are the unstable equilibrium points.

Now, we will discuss how to design a linear feedback controller such that the unstable equilibrium points are exponentially stable. For this purpose, we add the control terms to the system (2): 𝑥̇=𝑝𝑥+𝑦−𝑥ln√1+𝑥²+𝑢₁,𝑦̇=𝑥−𝑦+𝑧+𝑢₂,𝑧̇=−𝑞𝑦+𝑢₃. (25)

Let 𝑋^∗=(𝑥^∗,𝑦^∗,𝑧^∗) be any of the three unstable equilibrium points, let 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be the solution of the system (25), and 𝑋^∼(𝑡)=(𝑥^∼,𝑦^∼,𝑧^∼)=𝑋(𝑡)−𝑋^∗, then the error system is given by 𝑥̇‌^∼=𝑝𝑥^∼+𝑦^∼−𝑥ln√1+𝑥²+𝑥^∗ln1+𝑥^∗2+𝑢₁,𝑦̇^∼=𝑥^∼−𝑦^∼+𝑧^∼+𝑢₂,𝑧̇^∼=−𝑞𝑦^∼+𝑢₃. (26)

Definition 5. ∀𝑋(0)=(𝑥(0),𝑦(0),𝑧(0))∈𝑅³, if 𝑢_𝑖(𝑖=1,2,3) is appropriately selected such that 𝑋^∼𝑡,𝑡₀,𝑋₀≤𝑋(0)−𝑋^∗𝑒^{−𝛼(𝑡−𝑡0)} (27) holds (𝛼>0). Then, the control input 𝑢_𝑖(𝑖=1,2,3) can globally exponentially stabilize the equilibrium point 𝑋^∗.

Theorem 6. If the following linear controller is added to the error system (26), 𝑢₁=−𝑝𝜎𝑥_𝑥^∼,𝑢₂=𝑢₃=0, (28) where 𝜎_𝑥 is any parameter given beforehand such that 𝜎_𝑥>2; then the equilibrium point 𝑋^∗ is globally exponentially stable.

Proof. The proof is divided into two steps.(1) We will find the existence of 𝜖>0 such that 𝐺⁺₃ is positive definite and 𝐺⁻₃ is negative definite, where 𝐺⁺₃≜1𝑝𝜖0001−2𝜖0−21𝑞,𝐺⁻₃≜21−𝜎_𝑥𝜖2−2𝜖2𝜖𝑞−22−𝜖2𝜖2−𝜖. (29)It is easy to obtain that 𝐺⁺₃ is positive definite if |𝜖|<2/√𝑞. Now we focus on choosing 𝜖>0 such that 𝐺⁻₃ is negative definite. By the Schur theorem [18], 𝐺⁻₃ is negative definite if and only if 𝜖𝜖>0,𝜖𝑞+42𝜎−2<0,_𝑥−𝜖−14𝜖2−𝜖𝑞−4>𝜖2−4², (30) that is 8𝜖>0,𝜖<,𝜎4𝑞+1_𝑥>64−16𝜖+𝜖²𝜖64−32𝜖𝑞−8𝜖+1+8. (31) Obviously, 𝐺⁻₃ is negative definite if 0<𝜖≪1, 𝜎_𝑥>2.(2) We construct a positive definite and radially unbounded Lyapunov function to prove the stability of closed-loop systems (26) with controller (28) which is written as 𝑉𝑋₁^∼𝑡,𝑡₀,𝑋^∼₀=1𝑝^∼𝑥²+𝑦^∼2+1𝑞^∼𝑧²−𝜖𝑦^∼𝑧^∼=^∼𝑥_∼𝑦_∼𝑧^𝑇𝐺⁺₃^∼𝑥_∼𝑦_∼𝑧. (32)
Suppose 𝜆_𝑚(𝐺⁺₃) and 𝜆_𝑀(𝐺⁺₃) are minimum and maximum eigenvalues of the positive definite matrix 𝐺⁺₃, respectively. Then, we have 𝜆_𝑚𝐺𝑥⁺₃^∼2+𝑦^∼2+𝑧^∼2≤𝑉₁≤𝜆_𝑀𝐺𝑥⁺₃^∼2+𝑦^∼2+𝑧^∼2. (33)
Let 𝑓(𝑥^∼)=𝑥ln√1+𝑥²−𝑥^∗ln1+𝑥^∗2. Obviously, 𝑥ln√1+𝑥² is a monotonically increasing odd function. Then, (i)if 𝑥^∼≥0, then 𝑓(𝑥^∼)≥0 and 0≤𝑥^∼𝑓𝑥^∼≤+∞, (34)(ii)if 𝑥^∼≤0, then 𝑓(𝑥^∼)≤0 and 0≤𝑥^∼𝑓𝑥^∼≤+∞. (35)Thus 0≤𝑥^∼𝑓𝑥^∼≤+∞,∀𝑥^∼∈𝑅. (36)
Differentiating 𝑉₁ with respect to time yields 𝑑𝑉₁𝑑𝑡₍₂₆₎=2𝑝^∼𝑥𝑥̇^∼+2𝑦^∼𝑦̇^∼+2𝑞^∼𝑧𝑧̇^∼−𝜖𝑦^∼̇𝑧^∼−𝜖𝑦^∼𝑧̇^∼=^∼𝑥_∼𝑦_∼𝑧^𝑇𝐺⁻₃^∼𝑥_∼𝑦_∼𝑧−2𝑥^∼𝑓𝑥^∼≤𝜇_𝑀𝐺𝑥⁻₃^∼2+𝑦^∼2+𝑧^∼2=𝜇_𝑀𝐺⁻₃𝜆_𝑀𝐺⁺₃𝜆_𝑀𝐺⁺₃^∼𝑥²+𝑦^∼2+𝑧^∼2≤𝜇_𝑀𝐺⁻₃𝜆_𝑀𝐺⁺₃𝑉₁, (37) where 𝜇_𝑀(𝐺⁻₃) is the maximum eigenvalues of the negative definite matrix 𝐺⁻₃. Then, 𝑉𝑋₁^∼𝑡,𝑡₀,𝑋^∼₀≤𝑉𝑋₁^∼₀𝑒^{(𝜇𝑀(𝐺−3)/𝜆𝑀(𝐺+3))(𝑡−𝑡0)},_∼𝑥²(𝑡)+𝑦^∼2(𝑡)+𝑧^∼2(𝑡)≤𝑉𝑋₁^∼𝑡,𝑡₀,𝑋^∼₀𝜆_𝑚𝐺⁺₃≤𝑉𝑋₁^∼₀𝜆_𝑚𝐺⁺₃𝑒^{(𝜇𝑀(𝐺−3)/𝜆𝑀(𝐺+3))(𝑡−𝑡0)}. (38)
Hence, 𝑥^∼2(𝑡), 𝑦^∼2(𝑡), and 𝑧^∼2(𝑡) converge to zero exponentially. According to Definition 5, the equilibrium point 𝑋^∗ is globally exponentially stable. The proof is complete.

Remark 7. A constructive method to stabilize the unstable equilibrium points is proposed in this section, matrices 𝐺⁺₃, 𝐺⁻₃, and Lyapunov function candidate 𝑉₁ are given. Then, a linear controller is obtained to solve the problem. Comparing with nonlinear controller, linear controller is easy to implement in reality.

4. Globally Exponential Synchronization of Two Chua’s Systems

In this section, the globally exponential synchronization of two Chua’s systems will be discussed. The drive system is given by 𝑥̇_𝑑=𝑝𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑,𝑦̇_𝑑=𝑥_𝑑−𝑦_𝑑+𝑧_𝑑,𝑧̇_𝑑=−𝑞𝑦_𝑑, (39) and the response system is described as follows: 𝑥̇_𝑟=𝑝𝑥_𝑟+𝑦_𝑟−𝑥_𝑟ln1+𝑥²_𝑟+𝑢₁,𝑦̇_𝑟=𝑥_𝑟−𝑦_𝑟+𝑧_𝑟+𝑢₂,𝑧̇_𝑟=−𝑞𝑦_𝑟+𝑢₃, (40) where the subscripts 𝑑 and 𝑟 denote the drive and response systems and 𝑢_𝑖(𝑖=1,2,3) is feedback control input which satisfies 𝑢_𝑖(0,0,0)=0.

Let 𝑒_𝑥=𝑥_𝑟−𝑥_𝑑, 𝑒_𝑦=𝑦_𝑟−𝑦_𝑑, and 𝑒_𝑧=𝑧_𝑟−𝑧_𝑑; one obtains that 𝑒_𝑥̇=𝑝𝑒_𝑥+𝑒_𝑦−𝑥_𝑟ln1+𝑥²_𝑟+𝑥_𝑑ln1+𝑥²_𝑑+𝑢₁,𝑒_𝑦̇=𝑒_𝑥−𝑒_𝑦+𝑒_𝑧+𝑢₂,𝑒_𝑧̇=−𝑞𝑒_𝑦+𝑢₃. (41)

Theorem 8. If the following controller is added to the error system (41), 𝑢₁=−𝑝𝛿𝑥_𝑥^∼,𝑢₂=𝑢₃=0, (42) where 𝛿_𝑥 is any parameter given beforehand with 𝛿_𝑥>2; then the zero solution of (41) is globally exponentially stable and the systems (39) and (40) are globally exponentially synchronized.

Proof. Since the proof of this theorem is parallel to that of Theorem 6, we omit it here.

If the parameters 𝑝 and 𝑞 in the drive system are uncertain, one can construct the following controlled response system: 𝑥̇_𝑟=𝑝𝑥_𝑟+𝑦_𝑟−𝑥_𝑟ln1+𝑥²_𝑟+𝑢₁,𝑦̇_𝑟=𝑥_𝑟−𝑦_𝑟+𝑧_𝑟+𝑢₂,𝑧̇_𝑟=−𝑞𝑦_𝑟+𝑢₃, (43) where 𝑝 and 𝑞 are the estimates of the uncertain parameters 𝑝 and 𝑞, respectively. Let 𝑒_𝑥=𝑥_𝑟−𝑥_𝑑, 𝑒_𝑦=𝑦_𝑟−𝑦_𝑑, 𝑒_𝑧=𝑧_𝑟−𝑧_𝑑, 𝑝^∼=𝑝−𝑝, and 𝑞^∼=𝑞−𝑞, then, the error system of (39) and (43) is given by 𝑒_𝑥̇=𝑝𝑒_𝑥+𝑒_𝑦−𝑥_𝑟ln1+𝑥²_𝑟+𝑥_𝑑ln1+𝑥²_𝑑+𝑝^∼𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑+𝑢₁,𝑒_𝑦̇=𝑒_𝑥−𝑒_𝑦+𝑒_𝑧+𝑢₂,𝑒_𝑧̇=−𝑞𝑒_𝑦−𝑞^∼𝑦_𝑑+𝑢₃. (44)

Theorem 9. If the following adaptive controller is added to the error system (44), 𝑢₁=−𝑝𝑒_𝑥+𝑒_𝑦−𝑥_𝑟ln1+𝑥²_𝑟+𝑥_𝑑ln1+𝑥²_𝑑−𝑒_𝑥,𝑢₂=−𝑒_𝑥−𝑒_𝑧,𝑢₃=𝑞𝑒_𝑦−𝑒_𝑧,𝑝̇=𝑝^∼̇=−𝑒_𝑥𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑,𝑞̇=𝑞^∼̇=𝑒_𝑧𝑦_𝑑, (45) then, we have the following.(1)The equilibrium points (𝑝^∼=0, 𝑞^∼=0, 𝑒_𝑥=0, 𝑒_𝑦=0, 𝑒_𝑧=0) of the system (44) with adaptive control law (45) are globally stable; in addition, lim_𝑡→∞𝑒_𝑥=0, lim_𝑡→∞𝑒_𝑦=0, and lim_𝑡→∞𝑒_𝑧=0. And thus, the two systems (39) and (44) are globally synchronized.(2)The parameter estimates 𝑝 and 𝑞 will, respectively, converge to 𝑝 and 𝑞 as 𝑡 tends to infinity.

Proof. The proof contains two steps.
Firstly, a Lyapunov function candidate is constructed as follows: 𝑉₂=12𝑒²_𝑥+𝑒²_𝑦+𝑒²_𝑧+𝑝^∼2+𝑞^∼2. (46)
Then, one has 𝑑𝑉₂𝑑𝑡₍₄₄₎=𝑒_𝑥𝑒_𝑥̇+𝑒_𝑦𝑒_𝑦̇+𝑒_𝑧𝑒_𝑧̇+𝑝^∼𝑝̇^∼+𝑞^∼𝑞̇^∼=𝑒_𝑥𝑝𝑒_𝑥+𝑒_𝑦−𝑥_𝑟ln1+𝑥²_𝑟+𝑥_𝑑ln1+𝑥²_𝑑+𝑢₁+𝑒_𝑦𝑒_𝑥−𝑒_𝑦+𝑒_𝑧+𝑢₂+𝑒_𝑧−𝑞𝑒_𝑦+𝑢₃−𝑞^∼𝑒_𝑧𝑦_𝑑−𝑞̇^∼+𝑝^∼𝑒_𝑥𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑+𝑝̇^∼=−𝑒²_𝑥−𝑒²_𝑦−𝑒²_𝑧≤0. (47)
By LaSalle-Yoshizawa theorem [20], all the equilibrium points of the closed-loop systems are globally stable. Additionally, lim_𝑡→∞𝑒_𝑥=0, lim_𝑡→∞𝑒_𝑦=0, and lim_𝑡→∞𝑒_𝑧=0. And thus, the two systems (39) and (43) are globally synchronized.
By referring to Lemma  4.1 in [21], lim^∼_𝑡→∞𝑝=0, lim^∼_𝑡→∞𝑞=0 if there exist two functions 𝑓₁(𝑡), 𝑓₂(𝑡) which satisfy persistency of excitation condition, such that lim_𝑡→∞𝑝̇‌^∼=0,lim_𝑡→∞𝑞̇‌^∼=0,lim^∼_𝑡→∞𝑝𝑓₁(𝑡)=0,lim^∼_𝑡→∞𝑞𝑓₂(𝑡)=0. (48)
Since lim_𝑡→∞𝑒_𝑥=0, lim_𝑡→∞𝑒_𝑧=0, it is easy to obtain from (45) that lim_𝑡→∞𝑝̇‌^∼=lim_𝑡→∞−𝑒_𝑥𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑=0,lim_𝑡→∞𝑞̇‌^∼=lim_𝑡→∞𝑒_𝑧𝑦_𝑑=0. (49) Add 𝑢₁ and 𝑢₃ from (45) to (44), we have lim_𝑡→∞𝑒_𝑥̇=lim_𝑡→∞−𝑒_𝑥+𝑝^∼𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑=lim^∼_𝑡→∞𝑝𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑=0,lim_𝑡→∞𝑒_𝑧̇=lim_𝑡→∞−𝑒_𝑧−𝑞^∼𝑦_𝑑=lim_𝑡→∞−𝑞^∼𝑦_𝑑=0. (50) Let 𝑓₁(𝑡)≜𝑥_𝑑+𝑦_𝑑−𝑥_𝑑ln1+𝑥²_𝑑 and 𝑓₂(𝑡)≜𝑦_𝑑; it is easy to see that 𝑓₁(𝑡) and 𝑓₂(𝑡) satisfy persistency of excitation condition [21]; thus, we have lim^∼_𝑡→∞𝑝=0,lim^∼_𝑡→∞𝑞=0. (51) The proof is complete.

Remark 10. Synchronization control methods for two Chua’s systems are proposed in this section. A linear controller is given when parameters 𝑝 and 𝑞 are known. In the case of uncertain parameters 𝑝 and 𝑞, an adaptive controller is proposed to solve the problem. Comparing with the results in reference [22, 23], the output tracking error only converges to a small neighborhood of the origin, yet the synchronization errors of Chua’s systems can exponentially approach zero. On the other hand, compared with [23], a novel exponentially convergent method is used to solve the convergence of the estimation error of the uncertain parameters.

5. Numerical Simulations

In this section, several examples of numerical simulations are proposed to illustrate the theoretical results obtained in the previous sections. A fourth-order Runge-Kutta method is used to obtain the simulation results with MATLAB.

Chua’s system (2) and the error systems (26), (41), and (44) are considered in this section for the numerical simulations. Let 𝑝=11, 𝑞=14.87, the initial state 𝑥(0)=−3, 𝑦(0)=2, and 𝑧(0)=1. Figure 2 shows the state trajectories of the closed-loop system (26) with linear control input (28); it is easy to see that the equilibrium point 𝑆₀ is asymptotically stable. Similarly, Figure 3 shows that, with the corresponding control input (28), the equilibrium point 𝑆₋ is asymptotically stable. Figure 4 shows the synchronous errors of the system (39) and (40) with linear controller (42); it is easy to see that the two systems are globally asymptotically synchronized. When parameters 𝑝 and 𝑞 are uncertain, by using adaptive controller (45), the synchronous errors are asymptotically convergent to 𝑒_𝑥=0, 𝑒_𝑦=0, 𝑒_𝑧=0 (see Figure 5), and the estimate values of the uncertain parameters 𝑝 and 𝑞 asymptotically converge to the real values 𝑝=11, 𝑞=14.87 (see Figure 6).

(a)

(b)

(c)

Figure 2 

The asymptotic stability of 𝑆0 with controller (28).

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(b)

(c)

Figure 3 

The asymptotic stability of 𝑆− with controller (28).

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(b)

(c)

Figure 4 

The synchronization errors of the system (39) and (40) with linear controller (42).

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(b)

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Figure 5 

The synchronization errors of the system (39) and (43) with adaptive controller (45).

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(b)

Figure 6 

The convergence of parameters 𝑝 and 𝑞.

6. Conclusions

A new smooth Chua’s system is constructed, and the chaotic characteristics is confirmed by computing the Lyapunov exponents of the system. A Constructive method is used to prove the existence of globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to achieve globally exponential stability of the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua’s systems are globally synchronized, and the estimate errors of the uncertain parameters converge to zero as 𝑡 tends to infinity.

Acknowledgments

The authors want to express their sincere thanks to the editor and the referee for their invaluable comments and suggestions which helped improve the paper greatly. This work was supported by the National Natural Science Foundation of China (50937002 and 51207063) and the Project of the Education Department of Hubei Province (T200910, T201009, and D20132801).