3(0)) = (0.2,0.3,0.4), respectively.Figure blog post 5Modified projective synchronization between systems (10) and (11) can be realized for the scaling matrix �� = diag (1,1.5,2) when k = ?0.5. The initial values of the drive and response systems are chosen as (x1(0), x2(0), x3(0)) …4. DiscussionIn this section, another example is provided to compare the results obtained by the projective system approach with those obtained by Lyapunov method. Consider the following coupled Lorenz systemsx�B1=?��x1+��x2,x�B2=(��?x3)x1?x2,x�B3=x1x2?��x3,y�B1=?��y1+��y2,y�B2=(��?x3)y1?y2,(15)where ��, ��, and �� are system parameters. Two-variable partially projective synchronization has been found in system (15) [21]. That is, lim t����||y1,2 ? ��x1,2|| = 0 holds under certain conditions, in which �� R is the scaling factor.

Next, the synchronization conditions for system (15) are separately derived by Lyapunov method and the projective system approach.According to [21], lim t����||y1,2 ? ��x1,2|| = 0 is equivalent to lim t����(x1y2 ? y1x2) = 0. Denote the error vector by e = x1y2 ? y1x2; then error system can be written ase�B=x�B1y2+x1y�B2?y�B1x2?y1x�B2.(16)Lyapunov function is chosen as V(t) = (1/2)e2; thenV�B=ee�B=e(x�B1y2+x1y�B2?y�B1x2?y1x�B2)=?(��+1)e2.(17)Obviously, V�B<0 as long as �� > ?1, which is the condition for two-variable partially projective synchronization in system (15). It is worth pointing out that condition �� > ?1 also is necessary for the occurrence of the projective synchronization since V�B>0 if �� < ?1. However, it is generally difficult to find such proper Lyapunov function for any two coupled chaotic systems.

In the following, the projective system approach is applied to get the condition for synchronization. Comparing system (1) with system (15), the controller u can be expressed byu1=0,u2=?y1x3.(18)From (7), one hash01=[0,0,0]T,h02=[����(��?1),����(��?1),��?1]T.(19)From (9), the discriminant P(h02,h03)=[?�̦�1?1].(20)According??matrix for synchronization can be given byP(h01)=[?�̦̦�?1], to the projective system approach, two-variable partially projective synchronization occurs in system (15) provided that any of matrices P(h01) and P(h02, h03) is stable. The condition for synchronization also is �� > ?1 based on the projective system approach, which shows that a necessary and sufficient condition for modified projective synchronization may be found by using the approach.

5. ConclusionIn this paper, the projective system approach is proposed to realize modified projective synchronization of two different chaotic systems up to a desired scaling matrix. It is found that a projective system can be obtained GSK-3 from the original system to judge the occurrence of modified projective synchronization. A numerical example is given to illustrate the effectiveness of the projective system approach. Furthermore, another example of two-variable partially projective synchronization in two coupled Lorenz systems shows that a necessary and sufficient synchroniz