非线性可视化（3）混沌系统

clcclearclose all%% 洛伦兹吸引子h=1e-3;x0=0:h:40;[y1,~]=ODE_RK4_hyh(x0,h,[1;4;20],{'Lorenz',[10,8/3,20]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(1)subplot(1,3,1)plot(Lx,Ly);title('r=20')figure(2)subplot(1,3,1)plot3(Lx,Ly,Lz);view([51,30]);title('r=20')
[y1,~]=ODE_RK4_hyh(x0,h,[-13;-2;41],{'Lorenz',[10,8/3,28]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(1)subplot(1,3,2)plot(Lx,Ly);title('r=28')figure(2)subplot(1,3,2)plot3(Lx,Ly,Lz);view([51,30]);title('r=28')
[y1,~]=ODE_RK4_hyh(x0,h,[1;4;67],{'Lorenz',[10,8/3,99.36]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(1)subplot(1,3,3)plot(Lx,Ly);title('r=99.36')figure(2)subplot(1,3,3)plot3(Lx,Ly,Lz);view([51,30]);title('r=99.36')
%% Rossler吸引子h=2e-3;x0=0:h:180;[y1,~]=ODE_RK4_hyh(x0,h,[4.9;-5;0.07],{'Rossler',[0.1,0.1,4]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(4)subplot(2,2,1)plot3(Lx,Ly,Lz);view([51,30]);title('c=4 周期1')
[y1,~]=ODE_RK4_hyh(x0,h,[9.1;-5;0.17],{'Rossler',[0.1,0.1,6]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(4)subplot(2,2,2)plot3(Lx,Ly,Lz);view([51,30]);title('c=6 周期2')
[y1,~]=ODE_RK4_hyh(x0,h,[12.8;-5;0.277],{'Rossler',[0.1,0.1,8.5]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(4)subplot(2,2,3)plot3(Lx,Ly,Lz);view([51,30]);title('c=8.5 周期4')
[y1,~]=ODE_RK4_hyh(x0,h,[12.8;-5;0.277],{'Rossler',[0.1,0.1,9]});Lx=y1(1,:);Ly=y1(2,:);Lz=y1(3,:);
figure(4)subplot(2,2,4)plot3(Lx,Ly,Lz);view([51,30]);title('c=9 混沌')
%% Duffing吸引子h=2e-3;x0=0:h:180;[y1,~]=ODE_RK4_hyh(x0,h,[1;0.5],{'Duffing',[1.15,1,1]});Lx=y1(1,:);Ly=y1(2,:);figure(6)subplot(1,3,1)plot(Lx,Ly);title('d=1.15')
[y1,~]=ODE_RK4_hyh(x0,h,[0.8;0.75],{'Duffing',[1.35,1,1]});Lx=y1(1,:);Ly=y1(2,:);figure(6)subplot(1,3,2)plot(Lx,Ly);title('d=1.35')
[y1,~]=ODE_RK4_hyh(x0,h,[0.7;0.73],{'Duffing',[1.5,1,1]});%[0.7;0.73]Lx=y1(1,:);Ly=y1(2,:);figure(6)subplot(1,3,3)plot(Lx,Ly);title('d=1.5')
function [F,Output]=Fdydx(x,y,Input)%形式为Y'=F(x,Y)的方程，参见数值分析求解常系数微分方程相关知识%高次用列向量表示，F=[dy(1);dy(2)];y(1)为函数，y(2)为函数导数switch Input{1}    case 'Lorenz'        a=Input{2}(1);b=Input{2}(2);r=Input{2}(3);        dy(1)=a*(y(2)-y(1));        dy(2)=r*y(1)-y(2)-y(1)*y(3);        dy(3)=y(1)*y(2)-b*y(3);        F=[dy(1);dy(2);dy(3)];    case 'Rossler'        a=Input{2}(1);b=Input{2}(2);c=Input{2}(3);        dy(1)=-y(2)-y(3);        dy(2)=y(1) a*y(2);        dy(3)=b y(3)*(y(1)-c);        F=[dy(1);dy(2);dy(3)];    case 'Duffing'        d=Input{2}(1);r=Input{2}(2);w=Input{2}(3);        dy(1)=y(2);        dy(2)=-y(1)^3 y(1)-d*y(2) r*cos(w*x);        F=[dy(1);dy(2)];endOutput=[];end
function [y,Output]=ODE_RK4_hyh(x,h,y0,Input)%4阶RK方法%h间隔为常数的算法y=zeros(size(y0,1),size(x,2));y(:,1)=y0;for ii=1:length(x)-1    yn=y(:,ii);    xn=x(ii);    [K1,~]=Fdydx(xn    ,yn       ,Input);    [K2,~]=Fdydx(xn h/2,yn h/2*K1,Input);    [K3,~]=Fdydx(xn h/2,yn h/2*K2,Input);    [K4,~]=Fdydx(xn h  ,yn h*K3  ,Input);    y(:,ii 1)=yn h/6*(K1 2*K2 2*K3 K4);endOutput=[];end