How to sketch phase planes by hand

[pakmingki2]

so, for the very specific cases of linear systems

i can identify what shape it will be after determining the eigenvalues, but i really do not know how to go about sketching the phase planes.

can someone give me a method?

 

[HallsofIvy]

Added in edit: I just went back and reread your post. In the "specific case of linear systems", after you have found the eigenvalues and eigenvectors, draw straight lines along the direction of the eigenvectors, including their directions as t increases. Draw the other curves close to those lines following their direction and "asymptotic" to those lines.

If your eigenvalues are complex, then, of course, you will have circles or spirals as phase lines.

I assume you have equations of the form dx/dt= f(x,y) and dy/dt= g(x,y). At each point, (x,y), then dy/dx= g(x,y)/f(x,y). Choose a number of points in the plane, and calculate dy/dx for each of them. That gives you the slope of the line through that point that is tangent to the solution curve through that point. Draw a short line through the point with that slope (you might find it easier to use f(x,y) and g(x,y) separately to fine the "run" and "rise"). Oh, and be sure to put a little "arrowhead" on the line to show the direction of "flow" as t increases- use the signs of f(x,y) and g(x,y) to get that.

The more general case, dx/dt= f(x,y,t) and dy/dt= g(x,y,t) is harder- you have a different phase plane for every value of t.

 

[AiRAVATA]

A fantastic way of sketching phase planes comes explained in

Hubbard, J. H. and West, B.H. Differential Equations: A Dynamical System Approach. Vol 2. Springer.

It basically consist in cutting the plane in "quadrants" between the isoclines. As Hall's said, you take $dy/dx=f(x,y)/g(x,y)$ and draw the curves $f(x,y)=0$, $g(x,y)=0$. This will give you the isoclines with zero slope (nullclines) and the ones with infinite slope. Then you'll have to calculate the direction of the vectors lying in those curves (i.e. when $f(x,y)=0$ and $x<0$, the vectors over the nullcline point to the East (negative direction in $x$), while if $x>0$, they point to the West (positive direction in $x$)). That way, you can tell in which direction the vectors in the regions between such isoclines point (NE, NW, SE, SW). Then you'll only have to check what kind of critical points you have, and follow the vectors. That's it!

I strongly recommend you to look at Hubbard's book. It has figures which makes it all more easier to understand. Plus is a great book for qualitative study of ODE's.