| 原始碼 InfoField |
Python code
| Python source code
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from numpy import *
from scipy import *
from scipy.integrate import odeint
from matplotlib.pyplot import *
from mpl_toolkits.axes_grid.axislines import SubplotZero
def myFun(u,t=0.,mu=.5):
x = u[0]
v = u[1]
dx = v
dv = -sin(x)
return (dx,dv)
if __name__ == "__main__":
fig = figure(figsize=(5.5,7))
ax = SubplotZero(fig,211)
x = linspace(-3*pi,3*pi,100)
ax.plot(x,-cos(x),'b',lw=1.5)
fig.add_subplot(ax)
ax.grid(True,which='major')
ax.minorticks_on()
ax.axis('tight')
ax.axis([-3*pi,3*pi, -1,1])
ax.set_xticks(arange(-3*pi,3.1*pi,pi))
ax.set_xticklabels(
[r'$-3\pi$',r'$-2\pi$',
r'$-\pi$',r'$0$',r'$\pi$',
r'$2\pi$',r'$3\pi$'])
ax.set_xlabel(r'$\theta$')
ax.set_ylabel(r'$V(\theta)$')
ax = SubplotZero(fig,212)
fig.add_subplot(ax)
t = linspace(0,50,200)
for m in range(0,60,5):
u = odeint(myFun,[m/10.,0.],t)
ax.plot(u[:,0],u[:,1],'b',lw=1.5)
ax.plot(-u[:,0],u[:,1],'b',lw=1.5)
u = odeint(myFun,[0,m/10.],t)
ax.plot(ma.masked_outside(u[:,0],-3*pi,3*pi),
ma.masked_outside(u[:,1],-3,3),'b',lw=1.5)
ax.plot(ma.masked_outside(-u[:,0],-3*pi,3*pi),
ma.masked_outside(u[:,1],-3,3),'b',lw=1.5)
ax.plot(ma.masked_outside(u[:,0],-3*pi,3*pi),
ma.masked_outside(-u[:,1],-3,3),'b',lw=1.5)
ax.plot(ma.masked_outside(-u[:,0],-3*pi,3*pi),
ma.masked_outside(-u[:,1],-3,3),'b',lw=1.5)
x = linspace(-3*pi,3*pi,20)
y = linspace(-3,3,15)
x,y = meshgrid(x,y)
X,Y = myFun([x,y])
M = (hypot(X,Y))
M[M==0]=1.
X,Y = X/M, Y/M
ax.quiver(x,y,ma.masked_outside(X,-3*pi+.1,3*pi-.1),Y,M,pivot='mid',color='r')
ax.minorticks_on()
ax.axis('scaled')
ax.axis([-3*pi,3*pi, -3,3])
ax.set_yticks(arange(-3,3.1,1.5))
ax.set_xticks(arange(-3*pi,3.1*pi,pi))
ax.set_xticklabels(
[r'$-3\pi$',r'$-2\pi$',
r'$-\pi$',r'$0$',r'$\pi$',
r'$2\pi$',r'$3\pi$'])
ax.set_xlabel(r'$\theta$')
ax.set_ylabel(r'$\frac{\mathrm{d}\theta}{\mathrm{d}t}$')
ax.grid(True)
subplots_adjust(wspace=.1,hspace=-.1)
fig.show()
fig.savefig("pendulum.svg", bbox_inches="tight",\
pad_inches=.15, transparent=False)
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Data
| Matlab source code
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function pendulumOde
% main function to numerically solve the pendulum ODE and plot the phase portrait
figure;
subplot(211);
x = -pi:.1:3*pi;
h = plot(x,-cos(x),'linewidth',2);
set(gca,'yminortick','on','xtick',[-pi:pi/2:3*pi],'xticklabel',
{'$-\\pi$';'$-\\frac{\\pi}{2}$';'$0$';'$\\frac{\\pi}{2}$';'$\\pi$';
'$\\frac{3}{2}\\pi$';'$2\\pi$';'$\\frac{5}{2}\\pi$';'$3\\pi$'});
xlim([-pi 3*pi])
xlabel('$\theta$');
ylabel('$V(\theta)$');
grid on;
subplot(212);
[x,y] = meshgrid(-pi:.4:3*pi,-3:.2:3);
u = zeros(size(x));
v = zeros(size(y));
for i = 1:numel(x)
yy = ode_eq(0, [x(i),y(i)]);
u(i) = yy(1);
v(i) = yy(2);
vmod = sqrt(u(i).^2 + v(i).^2);
u(i) = u(i) / vmod;
v(i) = v(i) / vmod;
end
quiver(x,y,u,v,'r');
xlabel('$\theta$');
ylabel('$\frac{\mathrm{d}\theta}{\mathrm{d}t}$');
xlim([-pi 3*pi])
ylim([-pi pi])
grid on;
set(gca,'yminortick','on','xtick',[-pi:pi/2:3*pi],'xticklabel',
{'$-\\pi$';'$-\\frac{\\pi}{2}$';'$0$';'$\\frac{\\pi}{2}$';'$\\pi$';
'$\\frac{3}{2}\\pi$';'$2\\pi$';'$\\frac{5}{2}\\pi$';'$3\\pi$'});
hold all;
dT = .01;
T = 40;
for c = 0:.5:5
[x,y] = rungeKutta([c;0],dT,T,@ode_eq);
plot(y(1,:),y(2,:),'b','linewidth',2);
plot(y(1,:),-y(2,:),'b','linewidth',2);
[x,y] = rungeKutta([0;c],dT,T,@ode_eq);
plot(y(1,:),y(2,:),'b','linewidth',2);
plot(-y(1,:),y(2,:),'b','linewidth',2);
plot(y(1,:),-y(2,:),'b','linewidth',2);
plot(-y(1,:),-y(2,:),'b','linewidth',2);
[x,y] = rungeKutta([c;pi*2],dT,T,@ode_eq);
plot(y(1,:),y(2,:),'b','linewidth',2);
plot(y(1,:),-y(2,:),'b','linewidth',2);
[x,y] = rungeKutta([pi*2;c],dT,T,@ode_eq);
plot(y(1,:),y(2,:),'b','linewidth',2);
plot(-y(1,:),y(2,:),'b','linewidth',2);
plot(y(1,:),-y(2,:),'b','linewidth',2);
plot(-y(1,:),-y(2,:),'b','linewidth',2);
end
print -depslatexstandalone "-S512,512" "pendulum.tex";
end
function dy = ode_eq(x,y)
% function that defines an n-dimensional ODE.
% In this case, the two linear ODEs of pendulum
dy = [0;0];
dy(1) = y(2);
dy(2) = -sin(y(1));
end
function [x, y] = rungeKutta(y0, dT, T, dyFun, x0)
% A generalized Runge-Kutta algorithm to solve 'n' number of linear ODE
% obtained from an 'n'th degree ODE
n = length(y0);
if n > 1 && size(y0,2) == n
y0 = y0';
end
if nargin < 5
x0 = 0;
end
N = round(T/dT);
x = zeros(1,N);
y = zeros(n,N);
x(1) = x0;
y(:,1) = y0;
for nn = 1:N-1
k1 = feval(dyFun, x(nn), y(:,nn));
k2 = feval(dyFun, x(nn)+.5*dT, y(:,nn)+.5*k1*dT);
k3 = feval(dyFun, x(nn)+.5*dT, y(:,nn)+.5*k2*dT);
k4 = feval(dyFun, x(nn)+dT, y(:,nn)+k3*dT);
y(:,nn+1) = y(:,nn) + (dT/6) * (k1 + 2*k2 + 2*k3 + k4);
x(nn+1) = x(nn) + dT;
end
end
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