分岔圖的總結(jié)

1、混沌動力學(xué)行為研究的程序說明1.分岔圖:x1二-Jx1%(x3a)一個耦合發(fā)電機(jī)系統(tǒng)：x2二-x2x1(x3-a)x3=x3-xix2耦合系統(tǒng)函數(shù)：functiondx=ouhe1(t,x)dx(1,1)=-x(4)*x(1)+x(2)*(x(3)+x(5);dx(2,1)=-x(4)*x(2)+x(1)*(x(3)-x(5);dx(3,1)=x(3)-x(1)*x(2);dx(4,1)=0;dx(5,1)=0;分岔圖程序：clear;clc;z=;fora=linspace(0.5,10.5,500);%舍棄前面迭帶的結(jié)果，用后面的結(jié)果畫圖：即0.5-10.5分為500點(diǎn)T,Y=ode45(

2、'ouhe1',1,1;1;1;2;a);T,Y=ode45('ouhe1',20,丫(length(Y),:);Y(:,1)=丫(:,2)-Y(:,1);fork=2:length(Y)f=k-1;ifY(k,1)<0ifY(f,1)>0y=Y(k,2)-Y(k,1)*(Y(f,2)-Y(k,2)/(Y(f,1)-Y(k,1);Z=Za+abs(y)*i;endelseifY(f,1)<0y=Y(k,2)-Y(k,1)*(Y(f,2)-Y(k,2)/(Y(f,1)-Y(k,1);Z=Za+abs(y)*i;endendendendplot(Z

3、,'.','markersize',1);title('ouhe映射分岔圖,);xlabel('a'),ylabel('|y|')121086420-2-4-6246810a2 .功率譜：對上面耦合系統(tǒng)的功率譜的研究clearallx=zeros(1,10001);y=zeros(1,10001);z=zeros(1,10001);%數(shù)組置零x(1)=1;y(1)=1;z(1)=1;h=0.001;k=10000;a=3;u=2;fori=1:kx(i+1)=x(i)+h*(-u*x(i)+y(i)*(z(i)+a);%歐

4、拉離散y(i+1)=y(i)+h*(-u*y(i)+x(i)*(z(i)-a);z(i+1)=z(i)+h*(z(i)-x(i)*y(i);end%對x做傅里葉變換，取8192個點(diǎn)%求x的模及功率譜密度，單位：dB同樣可求y或z%取雙邊，也可取單邊c=0:4095/0.8192;%頂端題目X1=fft(z,16384);p=X1.*conj(X1)/16384;c=100*0:8191/16384;%title('題目)plot(c,log10(p(1:8192),'k')%畫出左半部分axis(04-36);%限制橫、縱坐標(biāo)范圍%plot(c,abs(X1(1:409

5、6)有時候也可用x的絕對值表示功率大小，沒求對數(shù)xlabel('itfrm/HZ','fontsize',18,'fontName','timesnewRoman','fontweight','bold','color','k');%加橫坐標(biāo)，it表傾斜，rm表復(fù)正ylabel('powerspectrum/dB','fontsize',18,'fontName','timesnewRoman','

6、;fontweight','bold','color','k');%縱坐標(biāo)標(biāo)示-505101520253035404550f/HZBdrmurtceoorewop3 .最大李雅譜指數(shù)程序：仍對上述耦合系統(tǒng)clear;%與ouhe系統(tǒng)的分岔圖（fotran）符合的很好clc;d0=1e-8;le=0;lsum=0;x=1;y=i;z=1;x1=1;y1=1;z1=1+d0;fori=1:500T1,Y1=ode45('ouhe1',0,1,x;y;z;4.05;7.0);T2,Y2=ode45('ouhe1'

7、,0,1,x1;y1;z1;4.05;7.0);n1=length(Y1);n2=length(丫2);x=Y1(n1,1);y=Y1(n1,2);z=Y1(n1,3);x1=Y2(n2,1);y1=Y2(n2,2);z1=Y2(n2,3);d1=sqrt(x-x1)A2+(y-y1)A2+(z-z1)A2);x1=x+(d0/d1)*(x1-x);y1=y+(d0/d1)*(y1-y);z1=z+(d0/d1)*(z1-z);ifi>100lsum=lsum+log(d1/d0);endendle=lsum/(i-100)最大李雅譜指數(shù)譜程序：仍對上述耦合系統(tǒng)clear;%可調(diào)參數(shù)區(qū)間

8、和步長，如a的第5和36行clc;LE1=;d0=1e-8;fora=linspace(0.5,10.5,300);le=0;lsum=0;x=1;y=1;z=1;x1=1;y1=1;z1=1+d0;fori=1:150T1,Y1=ode45('ouhe1',0,1,x;y;z;2;a);T2,丫2=ode45('ouhe1',0,1,x1;y1;z1;2;a);n1=length(Y1);n2=length(Y2);x=Y1(n1,1);y=Y1(n1,2);z=Y1(n1,3);x1=Y2(n2,1);y1=Y2(n2,2);z1=Y2(n2,3);d1=s

9、qrt(x-x1)A2+(y-y1)A2+(z-z1)A2);x1=x+(d0/d1)*(x1-x);y1=y+(d0/d1)*(y1-y);z1=z+(d0/d1)*(z1-z);ifi>50lsum=lsum+log(d1/d0);endendle=lsum/(i-50);LE1=LE1le;enda=linspace(0.5,10.5,300);plot(a,LE1,'-');title('largestLyapunovexponentsofouhe1');xlabel('parametera'),ylabel('larges

10、tLyapunovexponents');gridsinenoDxeVonupavfrsenpa-0.20246parametera81012largestLyapunovexponentsofouhe1雙參數(shù)空間的最大李雅譜指數(shù)譜程序：仍對以上耦合系統(tǒng)clear;%可調(diào)參數(shù)區(qū)間和步長clc;globaluaN1=linspace(0,0,200);N2=linspace(0,0,400);for1=1:200u=1.5+I*0.025;d0=1e-8;forL=1:400a=0.5+L*0.025;le=0;lsum=0;x=1;y=1;z=1;x1=1;y1=1;z1=1+d0;f

11、ori=1:150T1,Y1=ode45('ouhe1',0,1,x;y;z;u;a);T2,Y2=ode45('ouhe1',0,1,x1;y1;z1;u;a);n1=length(Y1);n2=length(Y2);x=Y1(n1,1);y=Y1(n1,2);z=Y1(n1,3);x1=Y2(n2,1);y1=Y2(n2,2);z1=Y2(n2,3);d1=sqrt(x-x1)A2+(y-y1)A2+(z-z1)A2);x1=x+(d0/d1)*(x1-x);y1=y+(d0/d1)*(y1-y);z1=z+(d0/d1)*(z1-z);ifi>50l

12、sum=lsum+log(d1/d0);endendle=lsum/(i-50);LE1(I,L)=le;N2(L)=a;endN1(I)=u;endX,Y=meshgrid(N1,N2);Z=LE1;pcolor(X,Y,Z);%畫偽彩圖%colormapjet,shadinginterp%連續(xù)變化的變異飽和色圖，表面畫偽彩圖%contourf(X,Y,Z)%畫等高線title('largestLyapunovexponentsofouhe1')xlabel('parameteritu')ylabel('parameterita')zlabel

13、('最大李雅普指數(shù)delta','FontSize',12)1086a4201E-30.20000.30000.40000.50000.60000.70000.80000.90001.0001.1001.2004 .李雅譜指數(shù)程序：仍對以上耦合系統(tǒng)clear;clc;x=1;y=1;z=1;h=0.003;a=3.5;u=2;V=eye(3);S=V;b1=0;k=50000;fori=1:kx1=x+h*(-u*x+y*(z+a);y1=y+h*(-u*y+x*(z-a);z1=z+h*(z-x*y);x=x1;y=y1;z=z1;J=-ua+zyz-a-u-

14、y-x1;J=eye(3)+h*J;B=J*V*S;V,S,U=svd(B);a_max=max(diag(S);S=(1/a_max)*S;b1=b1+log(a_max);endLyapunov=(log(diag(S)+b1)/(k*h)李雅譜指數(shù)譜程序：仍對以上耦合系統(tǒng)clear;%奇異值分解法計(jì)算ouhe系統(tǒng)的李雅普諾夫指數(shù)譜clc;zi=;Z2=;Z3=;x=1;y=1;z=1;h=0.002;u=2;%a=3;k=10000;fora=linspace(0.5,10.5,1000);V=eye(3);S=V;b1=0;lp=0;fori=1:kx1=x+h*(-u*x+y*(z+

15、a);y1=y+h*(-u*y+x*(z-a);z1=z+h*(z-x*y);x=x1;y=y1;z=z1;J=-ua+zyz-a-ux-y-x1;J=eye(3)+h*J;B=J*V*S;V,S,U=svd(B);a_max=max(diag(S);S=(1/a_max)*S;b1=b1+log(a_max);endlp=(log(diag(S)+b1)/(k*h);Z1=Z1Ip;Z2=Z2lp(2);Z3=Z3lp(3);enda=linspace(0.5,10.5,1000);plot(a,Z1,'-',a,Z2,'-',a,Z3,'-'

16、);title('Lyapunovexponentsofouhe');xlabel('parametera'),ylabel('lyapunovexponents');gridonLyapunovexponentsofouheparameteraRrnenoDxevonuDav6.三維、二維以及時間序列圖：耦合系統(tǒng)的函數(shù)程序：functiondx=ouhe(t,x)dx=zeros(3,1);a=3;u=2;dx(1)=-u*x(1)+x(2)*(x(3)+a);dx(2)=-u*x(2)+x(1)*(x(3)-a);dx(3)=x(3)-x(1

17、)*x(2);圖程序：clc;clear;t,x=ode45('ouhe',01000,1-11);subplot(2,2,1);plot(x(:,1),'k','markersize',0.5);xlabel('t/ms','fontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k');ylabel('x','f

18、ontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k');subplot(2,2,2);plot(x(:,2),'k','markersize',0.5);xlabel('t/ms','fontsize',12,'fontName','timesnewRoman','fontweight',

19、9;bold','color','k');ylabel('y','fontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k');subplot(2,2,3);plot(x(:,3),'k','markersize',0.5);xlabel('t/ms','fontsize',12,&

20、#39;fontName','timesnewRoman','fontweight','bold','color','k');ylabel('z','fontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k');subplot(2,2,4);plot3(x(:,1),x(:,2),x(:,3),&#

21、39;k','markersize',0.5);gridon;xlabel('x','fontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k');ylabel('y','fontsize',12,'fontName','timesnewRoman','fontweight','bold','color','k