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Taking a four-dimensional energy resources demand-supply system between the East and West of China, this paper discusses its chaotic behavior, and via the unilateral coupling method we lead the system to synchronization successfully. But not all the values of coupling coefficient can lead to synchronization. The values of coupling coefficient have a range. By calculating the maximal relative Lyapunov exponents’ spectrum, we gained the value range of coupling coefficients. Within the value range, the two coupling systems can achieve synchronization, otherwise can’t. Further more, the values of coupling coefficient are in connection with the chaos synchronizing time. At last, we get the relationship of coupling coefficients and chaos synchronizing time.

Generally, designing a controller to force a system to imitate the behavior of another chaotic system is called synchronization [

Nowadays, many countries attach importance to the exploitation and utilization of clear energies, and develop low carbon economy. Considering renewable energy resources, based on a three dimensional system, gained a four-dimensional energy resources demand-supply system between east and west in China by adding a new variable [

In 1993, K. Pyragas proposed a way to control a nonlinear system, which is called the error variable negative feedback control also the unilateral coupling method [

Supposing there are two chaotic systems: , , which are defined by following dynamic functions:

where K(X−Y) is a coupling item, is coupling coefficient, which are equal upon each variable and take positive values commonly, i.e. . When the parameters of the two systems are matched, as long as take an appropriate coupling coefficient K, the two systems can achieve synchronization. This method won’t change the system’s initial dynamic characteristic, because the coupling item K(X – Y) = 0 after synchronization. The characteristic of the method is we needn’t analyze the system in advance. Further more, we can confirm the domain of the coupling coefficient by calculating.

Adding a new variable to a three dimensional energy resource system, gained a four dimensional energy resources demand-supply system between east and west in China, which is defined by the functions below [

where x(t) is the energy resource shortage in A region, y(t) expresses the energy resources supply increment in B region, z(t) the energy resources import in A region, w(t) is the amount of renewable energy resources in A region; a_{i}, b_{i}, c_{i}, d_{i} and M, N are positive real constants [_{1} = 0.1, a_{2} = 0.15, b_{1} = 0.06, b_{2} = 0.082, b_{3} = 0.07, c_{1} = 0.2, c_{2} = 0.5, d_{1} = 0.1, d_{2} = 0.06, d_{3} = 0.07, and the initial condition , the system can generate complex chaotic attractor, the numerical simulation is shown in

According to the unilateral coupling method, we define the energy resource system (3), which is described as follows:

whose initial condition x_{1}(0), x_{2}(0), x_{3}(0), x_{4}(0) takes a random real constant from (0, 1) respectively.

Copying a system and adding coupling items, we obtain the system (4) defined below:

(4)

where c is the coupling coefficient, and the initial condi-

tion y_{1}(0), y_{2}(0), y_{3}(0), y_{4}(0) also takes a random real constant from (0, 1) respectively.

The errors of corresponding variables between system (3) and system (4) are denoted as

and

which express the situation of chaos synchronization. When E_{i}(t) and E equals to zero and then stable over time, we can say systems (3) and (4) have achieved synchronization, otherwise not. Randomly, chose c = 2, the errors E_{i}(t) is shown as

_{i}(t) (or E) tends to zero immediately and then stable, in other words, systems (3) and (4) achieved synchronization quickly. Thus, via the unilateral coupling method, we have made two systems achieve synchronization successfully.

But not all the values of coupling coefficient can lead to synchronization according to the following discussion.

Then let c = 0.01, the numerical simulation of the errors E_{i}(t) is shown as

From _{i}(t) (or E) vibrates desultorily all the time, namely this value of coupling coefficient c won’t

lead the two systems to synchronization.

So, we conclude not all the values of coupling coefficient can lead the two systems to synchronization; Values of c have a domain, values in which can make the two systems achieve synchronization, values outside it are not appropriate. By calculating the maximal relative Lyapunov exponents’ spectrum, we can get the value range of c.

In a chaotic system, the maximal Lyapunov exponent is a quantity characterizes the rate of separation of infinitesimally close trajectories, also the quantity indicates the strength of butterfly effect. In coupled systems, when maximal relative Lyapunov exponent is less than zero, the two systems will achieve synchronization, otherwise won’t [

where D_{t} is the distance of the two system’s trajectories at the time of t, D_{0} is the distance of the two system’s trajectories at the initial time, E_{i}(t) the error of corresponding variables between systems at t moment, E_{i}(0) the error of corresponding variables between systems initially, N is dimension. The numerical relation between coupling coefficients and the maximal relative Lyapunov exponents is shown in

As shown in

When c = 0.2, the errors E_{i}(t) is shown as

Comparing

In fact, sometime we need achieve synchronization immediately, while sometime we need transit a certain time and then achieve synchronization. So, the choice of a coupling coefficient’s value is crucial, and it’s neces-

sary to discuss the relation between coupling coefficients and the time achieving synchronization.