(15)Using??=��(t,��+E)?��(t,��)+K(��?��)??=��(t,��)+Ke?��(t,��)=��(t,��)?��(t,��)+Ke??as??d����dt��?d����dt�� the first-order Taylor expansion, the function ��(?) is rewritten =?��(t,��)?��E+h.o.t=G(t)E+h.o.t,(16)where h.o.t denotes the?as��(t,��+E)?��(t,��) things higher order terms of the series. We substitute (16) into (15) and yieldd��Edt��=G(t)E+h.o.t+KCTE=[G(t)+KCT]E+h.o.t.(17)We can transfer the (17) intod��Edt��=[G(t)+KCT]E(18)according to Lemma 1, we can know that the error system is asymptotically stable at zero if and only if the following condition is satisfied|arg(eig(G(t)+KCT))|�ݦ���2.(19)4. The Example Analysis and Numerical SimulationsExample 3 (dual synchronization of Van der Pol-Willis systems) ��In the first example, we can use the proposed method to achieve the dual synchronization of the Van der Pol system and the Willis system.

Master 1: Van der Pol systemd��x1dt��=x1?��x13?��x2+f1cost,d��x2dt��=l(x1?mx2+n).(20)Master 2: Willis systemd��y1dt��=y2,d��y2dt��=ay1+by12+cy13+dy2+f2cost.(21)So the corresponding slave systems are as follows:Slave 1:d��X1dt��=X1?��X13?��X2+f1cost+k1e,d��X2dt��=l(X1?mX2+n)+k2e,(22)Slave 2:d��Y1dt��=Y2+k3e,d��Y2dt��=aY1+bY12+cY13+dY2+f2cost+k4e,(23)where e = a1e1 + a2e2 + b1e3 + b2e4,e1 = X1 ? x1,e2 = X2 ? x2,e3 = Y1 ? y1, and e4 = Y2 ? y2.The G(t) matrix of the master systems is achieved asG(t)=?1?3��x12?��00llm00000100a+2by1+3cy12d?,(24)so the corresponding error ��(e1e2e3e4).

(25)We?matrix are as follows:(d��e1dt��d��e2dt��d��e3dt��d��e4dt��)=(1?3��x12+a1k1?��+a2k1b1k1b2k1l+a1k2lm+a2k2b1k2b2k2a1k3a2k3b1k31+b2k3a1k4a2k4a+2by1+3cy12+b1k4d+b2k4) should choose the appropriate parameters so that all the eigenvalues of the Jacobian matrix of (25) satisfy Matignon condition; that is, the eigenvalues evaluated at the equilibrium point are satisfied:|arg(eig(G(t)+KCT))|>����2.(26)The eigenvalue equation of the equilibrium point is locally asymptotically stable. From what we have discussed above, we can know that A and B are two known matrices; the parameter K can be appropriately selected for satisfying the Matignon condition. Dual synchronization of the Van der Pol system and the Willis system is simulated. The system parameters are set to be �� = 1/3, �� = 1, f1 = 0.74, l = 0.1, m = 0.8, n = 0.7, a = ?0.9, b = 3, c = ?2, d = ?0.1, f2 = 0.1, A = [1,1, 1], B = [1,1, 1], and �� = 1, soG(t)+KCT=(1?x12+k1?1+k1k1k10.

1+k2?0.08+k2k2k2k3k3k31+k3k4k4?0.9+6y1?6y12+k4?0.1+k4).(27)If ?295 < k1 < ?130, k2 = ?0.1, k3 = ?1, and k4 = ?400, which satisfy (18), the eigenvalue equation of the equilibrium point is locally asymptotically stable. We choosek1 = ?210, k2 = ?0.1, k3 = ?1, and k4 = ?400. The initial Brefeldin_A conditions of the master system 1 and the master system 2 are taken as x1(0) = 0.1, x2(0) = 0.2 and y1(0) = 0.2,y2(0) = 0.3; the initial conditions of the slave system 1 and the slave system 2 are taken as X1(0) = 0.