Hyperchaotic Lü system [17] is described as

in this paper choose, , and so that system (4) exhibits a hyperchaotic behavior [17] , Figure 1 shows the projections of hyperchaotic Lü system’s attractor.

Choose hyperchaotic Lü system

as drive system. System (5) can be described as

where

The response system is described as

where.

The error system is

where.

Next the sufficient conditions on feedback strength and impulsive distances will be established from two different angles to guarantee the synchronization.

Theorem 1 Suppose, is the largest eigenvalue of , is the largest eigenvalue of, the constant, is impulsive distance, if choose suitable and such that

then system (6) and system (7) can be synchronized.

Proof: Choose Lyapunov function as

Calculate the derivative of Equation (10), yield

When, we have

When, system (8) is a discrete system, according to Equation (8), we get

Let, according to Equation (12), when,

When, we obtain

From Equation (13) and Equation (15),

According to Equation (12), when,

In the same way, when,

From Equation (9), yield

hence

Substitute Equation (20) into Equation (18), we can obtain when,

Since,．From the assumption given in Theorem 1, we have, Thus obtain, i.e. system (6) and system (7) will be synchronized when Equation (9) is satisfied.

From Figure 1, we can choose. We calculate the eigenvalues of, obtain.

Suppose Equation (9) can be satisfied, then, i.e.,. It is obvious that and is near zero. Choose and, we obtain by solving Equation (9). That is to say, when, (sec.), System (6) and system (7) can be synchronized.

In this numerical simulation, let, impulsive distance. A time step of size 0.0001(sec.) is employed and fourth-order Runge-Kutta method is used to solve Equation (6) and

Equation (7). The initial states of the drive system (6) and the response system (7) are taken as and. The error system (8) has the initial state. Figure 2 shows the history of, , , in the error system (8). From Figure 2, we can see that, , , are steady near zero at last, i.e., system (6) and system (7) can be synchronized when and.

Theorem 1 provides sufficient condition for the synchronization of system (6) and system (7), but Equation (9) is not necessary condition. Through simulations we find that the above condition is too rigorous. In fact the qualified impulsive distance can be much larger than solved from Equation (9). Next we will present a new condition based on the largest Lyapunov exponent, which is much looser than Equation (9).

Suppose the initial distance between system (6) and system (7) is, the largest Lyapunov exponent of

system(6) is. Without control, the largest distance between system (6) and system (7) will not go beyond

after a short time (considering the average case). Generally speaking, the longest predicted

time of the chaotic system is [18] . Based on the above theory, we can obtain the following conclusions.

Theorem 2 Suppose is the largest eigenvalue of , is the largest Lyapunov exponent of system (6), the constant, is impulsive distance. Let, if choose suitable and such that

System (6) and system (7) can be synchronized

Proof: Suppose

then

hence

In the same way, we have

According to the condition of Theorem 2:, we have

According to Equation (29) and Equation (30), we obtain, then, i.e. system (6) and system (7) can be synchronized if the condition of Theorem 2 is satisfied.

Here, we still choose, , , so that system (5) exhibits hyperchaotic behavior [17] . For this system, the largest Lyapunov exponent. Suppose system (5) is described as system (6), choose system (6) as drive system, system (7) is the relevant response system, system (8) is the error system, we

have, ,. Choose, , substitute them into Equation (22), we obtain. Considering

, we have the following results: when, , system (6) and system (7) can achieve impulsive synchronization.

In this numerical simulation, let, impulsive distance. A time step of size 0.0001(sec.) is employed and fourth-order Runge-Kutta method is used to solve Equation (6) and

Equation (7). The initial states of the drive system (6) and the response system (7) are taken as and. The error system (8) has the initial state. Figure 3

shows the history of, , , in the error system (8). From Figure 3, we can see that

, , , are steady near zero at last, i.e., system (6) and system (7) can be synchronized when and.

If system (6) and system (7) achieve synchronization, system (8) will be steady at zero. Suppose, stands for impulsive distance, Figure 4 shows the boundaries of the stable region for Theorem 1, Figure 5 shows the boundaries of the stable region for Theorem 2. (Taking as examples, the region below the boundary is stable, not considering in Figure 5).

From Figure 4 and Figure 5, we can see that the requirement of impulsive distance in Theorem 1 is more rigorous than Theorem 2. Comparing the methods of Theorem 1 and Theorem 2, the former is based on the boundedness of chaotic system, it considers the extreme case all the time, while the latter is based on the largest Lyapunov exponent of chaotic system, it represents the average case. Therefore, the sufficient condition in Theorem 1 is a very small part of the condition in Theorem 2. Of course, in view of the requirements of synchronous time and quality, it is not suitable to choose very large impulsive distance in practical applications.