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I have code that will work for plotting the following predator prey model:
dx/dt = x − xy, dy/dt = −y + xy
from pylab import *
xvalues, yvalues = meshgrid(arange(0, 3, 0.1), arange(0, 3, 0.1))
xdot = xvalues - xvalues * yvalues
ydot = - yvalues + xvalues * yvalues
streamplot(xvalues, yvalues, xdot, ydot)
show()
But I am not sure how to use these functions to draw a phase plane (using streamplot) to model pendulum motion, defined as
d^2θ/dt^2 = (−g/L)sin(θ)
How can I implement this model to produce a phase plane using matplotlib and pylab?
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To generate the phase portrait, we need to compute the derivatives y′1 and y′2 at t=0 on a grid over the range of values for y1 and y2 we are interested in. We will plot the derivatives as a vector at each (y1, y2) which will show us the initial direction from each point.
The phase-space trajectory that represents the motion of the pendulum at the limit where the motion changes from 'back and forth' to continuous rotation is called the separatrix. The purple trajectory in figure 2 is very close in energy to the separatrix and is extremely close to it in shape.
A phase portrait represents the directional behavior of a system of ordinary differential equations (ODEs). The phase portrait can indicate the stability of the system.
You do it the same way, first transform it into a first order system
thetadot = omega
omegadot = -g/L*sin(theta)
rename theta, omega to x,y for shortness and then proceed as before:
g,L = 1,1
xvalues, yvalues = meshgrid(arange(-8, 8, 0.1), arange(-3, 3, 0.1))
xdot = yvalues
ydot = -g/L*sin(xvalues)
streamplot(xvalues, yvalues, xdot, ydot)
grid(); show()
which gives the usual phase portrait
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