%matlab计算程序如下:
% Poincare_section[绘制庞加莱截面图]
[t,x]=ode45(@Poincare,[0,2800],[0,1.5,0]);
x(:,2)=mod(x(:,2),2*pi)-pi;
phi0=pi*2/3; % 选择phi=2*pi/3这个截面
for k=1:round(max(x(:,3))/2/pi);
d=x(:,3)-(k-1)*2*pi-phi0;
[P,K]=sort(abs(d));
x1l=x(K(1),1);
x1r=x(K(2),1);
x2l=x(K(1),2);
x2r=x(K(2),2);
x3l=x(K(1),3);
x3r=x(K(2),3);
if abs(P(1))+abs(P(2))<3e-16;
X1(k)=x1l;
X2(k)=x2l;
else
Q=polyfit([x3l,x3r],[x1l,x1r],1);
X1(k)=polyval(Q,(k-1)*2*pi-phi0);
Q=polyfit([x3l,x3r],[x2l,x2r],1);
X2(k)=polyval(Q,(k-1)*2*pi-phi0);
end
end
plot(X1,X2,'.');
xlabel('\theta','fontsize',14);
ylabel('d\theta/dt','fontsize',14);
function dx=Poincare(t,x);
% 单摆方程[不显含时间t的自治系统]
% 方程如下:
% dθ/dt=ω,
% dω/dt=-2*β*[dθ/dt]-ω^2*sin(θ)+F*cos(vt)
% dψ/dt=v
betaa=0.25;
F=1.093;
v=2/3;
P2=-2*betaa*x(2)-x(2).^2.*sin(x(1))+F*cos(v*t);
dx=[x(2)2;v];
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