分岔的基本概念及matlab程序实现

无水1324 · 发表于 2008-4-4 09:26

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如题，最基本的分岔概念如：saddle-node；transcritical；supercritical pitchfork；subcritical pitchfork等，如果你不是很了解，可以学习一下下面的程序。

%% Animation for a Saddle Node bifurcation
% Range of x
xmin = -4;
xmax = 4;
x = xmin:0.1:xmax
% Some auxiliary variable to plot x-axis
ax = (0.0).*x;
% Range of r
rmin = -4
rmax = 4
% number of frames in the movie
imax = 21
% Rage of dx/dt (here y=dx/dt)
ymin = rmin ;
ymax = rmax + xmax.^2;
% Some auxiliary variables to plot the y-axis
ky = ymin:0.1:ymax
ay = (0.0).*ky
%% Start of the loop for acquisition of the movie frames
for i = 1:imax
%calculate r value
r = rmin + (i-1)*(rmax-rmin)/(imax-1)
%create string for labelling r
strr = num2str(r);
strt = ['r=' strr];
%calculate dxdt
y = r + x.^2
%plot x-axis
plot(x,ax,'--');
% determine range of plot
axis([xmin xmax ymin ymax])
hold on
%plot y-axis
plot(ay,ky,'--')
%plot f(x)
plot(x,y,'r');
%plot fixed points
if (r < 0)
plot(-(-r).^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot((-r).^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k');
end
if(r == 0.0)
plot((-r).^(0.5),0,'*','MarkerSize',10,'MarkerEdgeColor','k');
end
%plot flow direction
if(r < 0)
plot(-3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(0,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r == 0)
plot(-3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r > 0)
plot(-3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k'); plot(0,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(3,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
%plot labels
xlabel('x','FontSize',20)
ylabel('dx/dt','FontSize',20)
text(-3,15,'dx/dt = r + x^2','Color','r','FontSize',20)
text(1,15,strt,'FontSize',20)
title('Saddle Node Bifurcation','FontSize',20)
%get movie frame
h = gcf;
M(i) = getframe(h,[5 5 480 380]);
hold off
% end of loop
end
%play and save movie
movie(M);
movie2avi(M,'SaddleNode','fps',1);

复制代码

无水1324 · 发表于 2008-4-4 09:27

%% Animation for a Transcritical Bifurcation (same structure as Saddle-node, limited annotations for this program)
xmin = -6;
xmax = 6;
x = xmin:0.1:xmax
ax = (0.0).*x;
rmin = -4
rmax = 4
imax = 21
ymin = -6 ;
ymax = 6;
ky = ymin:0.1:ymax
ay = (0.0).*ky
for i = 1:imax
%calculate r value
r = rmin + (i-1)*(rmax-rmin)/(imax-1)
%create title string
strr = num2str(r);
strt = ['r=' strr];
%calculate dxdt
y = r.*x - x.^2
%plot x-axis
plot(x,ax,'--');
axis([xmin xmax ymin ymax])
hold on
%plot y-axis
plot(ay,ky,'--')
%plot f(x)
plot(x,y,'r');
%plot fixed points
if (r < 0)
plot(r,0,'o','MarkerSize',10,'MarkerEdgeColor','k');
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r == 0.0)
plot((-r).^(0.5),0,'*','MarkerSize',10,'MarkerEdgeColor','k');
end
if (r > 0)
plot(r,0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k');
end
%plot flow direction
if(r < 0)
plot(-4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(0.5.*r,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r == 0)
plot(-4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r > 0)
plot(-4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k'); plot(0.5.*r,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(4.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
%plot labels
xlabel('x','FontSize',20)
ylabel('dx/dt','FontSize',20)
text(-5,5,'dx/dt = rx - x^2','Color','r','FontSize',20)
text(2,5,strt,'FontSize',20)
title('Transcritical Bifurcation','FontSize',20)
%getframe
h = gcf;
M(i) = getframe(h,[5 5 480 380]);
hold off
end
%play and save movie
movie(M);
movie2avi(M,'Transcritical','fps',1);

复制代码

无水1324 · 发表于 2008-4-4 09:27

%% Animation for a Supercritical Pitchfork bifurcation (same structure as the Saddle Node case, limited annotation)
xmin = -3;
xmax = 3;
x = xmin:0.1:xmax
ax = (0.0).*x;
rmin = -4
rmax = 4
imax = 21
ymin = -20;
ymax = 20;
ky = ymin:0.1:ymax
ay = (0.0).*ky
for i = 1:imax
%calculate r value
r = rmin + (i-1)*(rmax-rmin)/(imax-1)
%create title string
strr = num2str(r);
strt = ['r=' strr];
%calculate dxdt
y = r.*x - x.^3
%plot x-axis
plot(x,ax,'--');
axis([xmin xmax ymin ymax])
hold on
%plot y-axis
plot(ay,ky,'--')
%plot f(x)
plot(x,y,'r');
%plot fixed points
if (r < 0)
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r == 0.0)
plot(0,0,'*','MarkerSize',10,'MarkerEdgeColor','k');
end
if (r > 0)
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k');
plot(r.^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(-r.^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
%plot flow direction
if(r <= 0)
plot(-2.5,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(2.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r > 0)
plot(-2.5,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(-r.^(0.5)/2,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(r.^(0.5)/2,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(2.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
%plot labels
xlabel('x','FontSize',20)
ylabel('dx/dt','FontSize',20)
text(-2,15,'dx/dt = rx - x^3','Color','r','FontSize',20)
text(1,15,strt,'FontSize',20)
title('Supercritical Pitchfork Bifurcation','FontSize',20)
%getframe
h = gcf;
M(i) = getframe(h,[5 5 480 380]);
hold off
end
%play and save movie
movie(M);
movie2avi(M,'SuperPitchfork','fps',1);

复制代码

无水1324 · 发表于 2008-4-4 09:27

%% Animation for a Subcritical Pitchfork bifurcation
xmin = -3;
xmax = 3;
x = xmin:0.1:xmax
ax = (0.0).*x;
rmin = -4
rmax = 4
imax = 21
ymin = -20;
ymax = 20;
ky = ymin:0.1:ymax
ay = (0.0).*ky
for i = 1:imax
%calculate r value
r = rmin + (i-1)*(rmax-rmin)/(imax-1)
%create title string
strr = num2str(r);
strt = ['r=' strr];
%calculate dxdt
y = r.*x + x.^3
%plot x-axis
plot(x,ax,'--');
axis([xmin xmax ymin ymax])
hold on
%plot y-axis
plot(ay,ky,'--')
%plot f(x)
plot(x,y,'r');
%plot fixed points
if (r < 0)
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot((-r).^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k');
plot(-(-r).^(0.5),0,'o','MarkerSize',10,'MarkerEdgeColor','k');
end
if(r == 0.0)
plot(0,0,'*','MarkerSize',10,'MarkerEdgeColor','k');
end
if (r > 0)
plot(0,0,'o','MarkerSize',10,'MarkerEdgeColor','k');
end
%plot flow direction
if(r < 0)
plot(-2.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(-(-r).^(0.5)/2,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot((-r).^(0.5)/2,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(2.5,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
if(r >= 0)
plot(-2.5,0,'<','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
plot(2.5,0,'>','MarkerSize',10,'MarkerEdgeColor','k','MarkerFaceColor','k');
end
%plot labels
xlabel('x','FontSize',20)
ylabel('dx/dt','FontSize',20)
text(-2.5,15,'dx/dt = rx + x^3','Color','r','FontSize',20)
text(1,15,strt,'FontSize',20)
title('Subcritical Pitchfork Bifurcation','FontSize',20)
%getframe
h = gcf;
M(i) = getframe(h,[5 5 480 380]);
hold off
end
%play and save movie
movie(M);
movie2avi(M,'SubPitchfork','fps',1);

复制代码

xinwilliam · 发表于 2008-4-4 19:23

非常之好!
如果我能给你加分，我就加了!
:@D

无水1324 · 发表于 2008-4-4 19:54

看了这些程序希望对大家对基本的分岔过程有一个感性的认识，声明：此程序是网上找的

octopussheng · 发表于 2008-4-5 19:52

无水，如果能把运行结果的图也贴出来就更完美了！

buiesea · 发表于 2008-4-5 21:09

我不是很懂，但我用程序运行了一下，结果依次如下，供大家参考

buiesea · 发表于 2008-4-5 21:16

随着r值得增大，图形和x方向的零轴的交点也相应的变化，上面给出的只是最后r增大到4的时候的图形。

无水1324 · 发表于 2008-4-6 09:59

那你还是把基本的概念搞清楚，这个动态的过程就是一个分岔的过程

chencylx03 · 发表于 2008-4-8 16:49

很强，学习中。能不能把动态图保存了？

无水1324 · 发表于 2008-4-9 18:20

没有想过这个问题，不过可以去请matlab版的高手看看，应该可以做成moive的形式

无水1324 · 发表于 2008-4-9 20:33

还有你可以看一下程序运行之后会保存一个avi文件，那就是录像，动态变化的

whecust · 发表于 2008-5-7 15:19

原来是一个动态的过程啊
！！！！！

liyu__1004 · 发表于 2008-5-7 20:01

very good:victory:

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