本文基于Lypunov稳定性定理，通过构造Lyapunov函数以及设计提出一非线性反馈控制输入器
u
u
u，进而实现一类六维Lorenz型混沌系统的有限时间同步。
本文所基于的一类六维Lorenz型超混沌系统可表示如下：
{
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\left\{ \begin{array}{l} \dot{x}_1=h\left( x_2-x_1 \right) +x_4\\ \dot{x}_2=-fx_2-x_1x_3+x_6\\ \dot{x}_3=-l+x_1x_2\\ \dot{x}_4=-x_2-x_5\\ \dot{x}_5=kx_2+x_4\\ \dot{x}_6=gx_1+mx_2\\ \end{array} \right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)
⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧x˙1=h(x2−x1)+x4x˙2=−fx2−x1x3+x6x˙3=−l+x1x2x˙4=−x2−x5x˙5=kx2+x4x˙6=gx1+mx2 (1)
将
(
1
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(1)
(1)作为驱动系统，对应的可以得到响应系统如下：
{
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\left\{ \begin{array}{l} \dot{y}_1=h\left( y_2-y_1 \right) +y_4\\ \dot{y}_2=-fy_2-y_1y_3+y_6\\ \dot{y}_3=-l+y_1y_2\\ \dot{y}_4=-y_2-y_5\\ \dot{y}_5=ky_2+y_4\\ \dot{y}_6=gy_1+my_2\\ \end{array} \right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 2 \right)
⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧y˙1=h(y2−y1)+y4y˙2=−fy2−y1y3+y6y˙3=−l+y1y2y˙4=−y2−y5y˙5=ky2+y4y˙6=gy1+my2 (2)
我们令
e
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e_i=y_i-x_i,(i=1,2,3,4,5,6)
ei=yi−xi,(i=1,2,3,4,5,6),从而能够得到误差同步系统如下：
{
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\left\{ \begin{array}{l} \dot{e}_1=h\left( e_2-e_1 \right) +e_4\\ \dot{e}_2=-fe_2-y_1e_3-y_3e_1+e_1e_3+e_6\\ \dot{e}_3=y_1e_2+y_2e_1-e_1e_2\\ \dot{e}_4=-e_2-e_5\\ \dot{e}_5=ke_2+e_4\\ \dot{e}_6=ge_1+me_2\\ \end{array} \right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 3\right)
⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧e˙1=h(e2−e1)+e4e˙2=−fe2−y1e3−y3e1+e1e3+e6e˙3=y1e2+y2e1−e1e2e˙4=−e2−e5e˙5=ke2+e4e˙6=ge1+me2 (3)
通过构造非线性反馈控制输入器
u
=
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T
u=\left( \begin{matrix} u_1& u_2& u_3& u_4& u_5& u_6\\ \end{matrix} \right) ^T
u=(u1u2u3u4u5u6)T
即可使得该系统在有限时间内达到同步。本文所提出的控制器如下：
u
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\tilde{u}\left( x_1,y_1,z_1,e \right) =\left\{ \begin{array}{l} -\left( L_BV \right) ^T\frac{L_fV+\beta V^m}{\lVert L_BV \rVert ^2},L_BV\ne 0\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_BV=0\\ \end{array} \right.
u~(x1,y1,z1,e)={−(LBV)T∥LBV∥2LfV+βVm,LBV=00, LBV=0
其中，
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V=\frac{1}{2}\left( e_{1}^{2}+e_{2}^{2}+e_{3}^{2}+e_{4}^{2}+e_{5}^{2}+e_{6}^{2} \right)
V=21(e12+e22+e32+e42+e52+e62)
B
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B=\left( \begin{matrix} 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ \end{matrix} \right)
B=⎝⎜⎜⎜⎜⎜⎜⎛100000000000001000000100000010000001⎠⎟⎟⎟⎟⎟⎟⎞
该六维混沌系统的相图刻画如下：
对应的MATLAB实现代码如下：
y0 = [0.5;0.35;0.15;0.05;0.1;0.1];
tspan = 0:0.001:200;
[T,X] = ode45(@dydt,tspan,y0);
figure(1);
plot(X(30:end,1),X(30:end,2),'Color',[0.10,0.05,0.70]);%grid;
xlabel('x1','FontName','Times New Roman','FontSize',24);
ylabel('x2','FontName','Times New Roman','FontSize',24);
figure(2);
plot(X(30:end,2),X(30:end,3),'Color',[0.10,0.05,0.70]);%grid;
xlabel('x2','FontName','Times New Roman','FontSize',24);
ylabel('x3','FontName','Times New Roman','FontSize',24);
figure(3);
plot(X(30:end,3),X(30:end,4),'Color',[0.10,0.05,0.70]);%grid;
xlabel('x3','FontName','Times New Roman','FontSize',24);
ylabel('x4','FontName','Times New Roman','FontSize',24);
figure(4);
plot(X(30:end,4),X(30:end,5),'Color',[0.10,0.05,0.70]);%grid;
xlabel('x4','FontName','Times New Roman','FontSize',24);
ylabel('x5','FontName','Times New Roman','FontSize',24);
figure(5);
plot(X(30:end,5),X(30:end,6),'Color',[0.10,0.05,0.70]);%grid;
xlabel('x5','FontName','Times New Roman','FontSize',24);
ylabel('x6','FontName','Times New Roman','FontSize',24);
function dy = dydt(~,y)
h = 10;l =100;f = 2.7;k = 2;g = -3;m = 1;
dy = zeros(6,1);
dy(1) = h*(y(2)-y(1))+y(4);
dy(2) = -f*y(2)-y(1)*y(3)+y(6);
dy(3) = -l+y(1)*y(2);
dy(4) = -y(2)-y(5);
dy(5) = k*y(2)+y(4);
dy(6) = g*y(1)+m*y(2);
end
该混沌系统实现同步的一维时间序列图如下：
其对应的MATLAB实现代码如下：
x0=[61;68;58;-40;63;65;59;35;34;22;42;46];
tspan = [0 500];
[t,x]=ode45(@dydt,tspan,x0);
figure(1)
subplot(2,1,1)
plot(t,x(:,7)-x(:,1),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_1','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)
subplot(2,1,2)
plot(t,x(:,8)-x(:,2),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_2','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)
figure(2)
subplot(2,1,1)
plot(t,x(:,9)-x(:,3),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_3','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)
subplot(2,1,2)
plot(t,x(:,10)-x(:,4),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_4','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)
figure(3)
subplot(2,1,1)
plot(t,x(:,11)-x(:,5),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_5','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)
subplot(2,1,2)
plot(t,x(:,12)-x(:,6),'Color',[0.30,0.10,0.70])
xlabel('t','FontName','Times New Roman','FontSize',20)
ylabel('e_6','FontName','Times New Roman','FontSize',20)
set(gca,'FontName','Times New Roman','FontSize',20,'LineWidth',2)