What does pplane7 do? You tell me!
end3r7 said:
Basically I use pplane7, so any system of two first order differential equations I believe.
Let's back up. When you say "plot a phase plane", I think you mean "sketch a phase portrait". I have used MATLAB in the past, but I haven't used in recently and I am not familiar with "pplane7".
I have been assuming that given a second order ODE for y in terms of x, pplane7 obtains the corresponding
autonomous first order system of ODEs u=y, v=y\prime and numerically plots the
phase portrait in the u,v plane. For example, given the
van der Pol equation governing a nonlinear spring
<br />
y\prime \prime + y = \mu \, (y - y^2) y\prime<br />
the first order autonomous system is
<br />
\dot{u} = v, \; \; \dot{v} = -u + \mu \, (1-u^2) \, v<br />
and the corresponding
flow on R^2 = \left{ (u, \, v): u, \, v \in R \right} is generated by the
vector field
<br />
v \, \partial_u + \left( -u + \mu \, (1-u^2) \, v \right) \, \partial_v<br />
The
integral curves of this vector field are the
phase curves, and plotting a judicious selection of phase curves (in this case, there is a unique closed phase curve, and the other phase curves approach it as time increases, so it is a
limit cycle) gives the phase portrait. This phase portrait gives a vivid picture of the behavior of solutions to the original ODE.
Does this look familiar? (See Arnold,
Ordinary Differential Equations for many more examples.)
From your responses I am guessing you are not sure what pplane7 does either, and that this is part of the problem. I was trying to get you to realize that figuring out exactly what pplane7 is the first step in answering your own question. Do you have some on-line help which explains what is acceptable input for pplane7?
Once you understand why whatever restrictions on the acceptable input are mathematically necessary (don't forget the possibility that you don't want to consider higher dimensional phase portraits!), you will be in a better position to start thinking about whether phase portraits make sense if you start with a PDE instead of an ODE.
