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

and Applied Analysis 3 where 0 < ε ≪ 1, G + 1 ≜ [ [ [ [ [ [ [

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][10][11][12][13][14][15][16] and the reference therein).In [11][12][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][16][17].
Motivated by the previous results, the main purpose of this paper is to construct a new smooth Chua's system and Figure 1: The phase portraits of Chua's system (2).
investigate the stability and control problems.More precisely, we will consider the following smooth Chua's system: where , ,  ∈  are state variables and  > 0,  > 0 are constants.We will show that the chaotic characteristics are depended on the parameters  and  and the initial state values of the system (2).All equilibrium points of system (2) are examined to be unstable when  = 11 and  = 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  is 3 and the delay time  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 (0) = −3, (0) = 2, and (0) = 1, the phase diagrams of Chua's system exhibits chaotic.
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.

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 2. Let Ω ⊆  3 , if ∀ 0 ∈ Ω and for all  ≥  0 , (,  0 ,  0 ) ⊆ 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: Abstract and Applied Analysis 3 where 0 <  ≪ 1, are symmetric positive definite matrices, and Proof.The proof is divided into three steps.
Step I.The existence of ,  1 such that  + 1 is positive definite and  − 1 is negative definite.It is well known that  + 1 is positive definite if and only if all the order principal minors of  + 1 are positive [18].That is, Let || < 2/ √ ; it is easy to see that  + 1 is positive definite since  > 0,  > 0.
Next, let Then, From Schur theorem [18],  − 1 is negative definite if and only if It is easy to verify that  − 1 is negative definite.
Step II.Existence of globally exponential attractive set of .
Let   ( + 1 ) and   ( − 1 ) be the minimum eigenvalues of  + 1 and  − 1 , respectively.By constructing a radially unbounded Lyapunov function as follows: then, one can obtain that The time derivative of  along the system ( 2) is given by Then, the following inequality holds Thus, the trajectory of the system (2) will exponentially decay into the area When the trajectory of the system (2) is in the area , it holds that ) . ( Let Step III.Existence of globally exponential attractive set of  and . Let then,  + 2 is a positive matrix if  ∈ (0, 2/ √ ), and  − 2 is a negative matrix if  ∈ (0, 8/(4 + 1)).
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  + 1 ,  − 1 ,  + 2 ,  − 2 , and Lyapunov function candidate  and , the problem is solved ingeniously.

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: Abstract and Applied Analysis 5 Moreover, the equilibrium points of the system is independent of parameters  and .It should be mentioned that, however, the stability of the equilibrium points is depended on  and .In the following, we will design a linear controller to stabilize the unstable equilibrium points.Let  = 11,  = 14.87, and ( * ,  * ,  * ) be any equilibrium point of the system (2), the corresponding Jacobian matrix is given by Then the characteristic equation of the corresponding local linearization system of system ( 2) is as follows: where  0 = 1,  1 = 1 − ,  2 = − −  − ,  3 = , and  = − ln √ 1 +  * 2 + (1/(1 +  * 2 )).
(i) If  * = 0, one has  0 = 1 > 0,  1 = −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  0 is unstable.
(ii) If  * = ± √  2 − 1, one has  0 = 1,  1 = 10.5113> 0, and  2 = −16.3587< 0. Similarly, one can obtain that  + and  − 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): Let  * = ( * ,  * ,  * ) be any of the three unstable equilibrium points, let () = ((), (), ()) be the solution of the system (25), and X() = ( x, ỹ, z) = () −  * , then the error system is given by holds ( > 0).Then, the control input   ( = 1, 2, 3) can globally exponentially stabilize the equilibrium point  * .Theorem 6.If the following linear controller is added to the error system (26), where   is any parameter given beforehand such that   > 2; then the equilibrium point  * is globally exponentially stable.
Proof.The proof is divided into two steps.
(1) We will find the existence of  > 0 such that  + 3 is positive definite and  − 3 is negative definite, where It is easy to obtain that  + 3 is positive definite if || < 2/ √ .Now we focus on choosing  > 0 such that  − 3 is negative definite.By the Schur theorem [18],  − 3 is negative definite if and only if that is Obviously,  − 3 is negative definite if 0 <  ≪ 1,   > 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 Suppose   ( + 3 ) and   ( + 3 ) are minimum and maximum eigenvalues of the positive definite matrix  + 3 , respectively.Then, we have Let ( x) =  ln √ 1 +  2 −  * ln √ 1 +  * 2 .Obviously,  ln √ 1 +  2 is a monotonically increasing odd function.Then, (i) if x ≥ 0, then ( x) ≥ 0 and (ii) if x ≤ 0, then ( x) ≤ 0 and Thus Differentiating  1 with respect to time yields where   ( − 3 ) is the maximum eigenvalues of the negative definite matrix  − 3 .Then, Hence, x2 (), ỹ2 (), and z2 () converge to zero exponentially.According to Definition 5, the equilibrium point  * 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  + 3 ,  − 3 , and Lyapunov function candidate  1 are given.Then, a linear controller is obtained to solve the problem.Comparing with nonlinear controller, linear controller is easy to implement in reality.

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 and the response system is described as follows: where the subscripts  and  denote the drive and response systems and   ( = 1, 2, 3) is feedback control input which satisfies   (0, 0, 0) = 0.
Let   =   −  ,   =   −  , and   =   −  ; one obtains that Theorem 8.If the following controller is added to the error system (41), where   is any parameter given beforehand with   > 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.
(2) The parameter estimates p and q will, respectively, converge to  and  as  tends to infinity.
Proof.The proof contains two steps.Firstly, a Lyapunov function candidate is constructed as follows: Then, one has By LaSalle-Yoshizawa theorem [20], all the equilibrium points of the closed-loop systems are globally stable.Additionally, lim  → ∞   = 0, lim  → ∞   = 0, and lim  → ∞   = 0.And thus, the two systems (39) and (43) are globally synchronized.
Remark 10.Synchronization control methods for two Chua's systems are proposed in this section.A linear controller is given when parameters  and  are known.In the case of uncertain parameters  and , 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.and the estimate values of the uncertain parameters  and  asymptotically converge to the real values  = 11,  = 14.87 (see Figure 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  tends to infinity.