The PhasePlane for a Linear system

In this brief set of exercises, we will look at the phaseplane for the two-dimensional linear system of differential equations:
x' = ax + b y
y'= cx + dy

We know that the behavior of this system is completely determined by the eigenvalues of the matrix A whose entries are a,b,c,d. These are the normal possibilities:

In addition there are a number of degenerate cases: In the last degenerate case there are infinitely many fixed points corresponding to the eigenvector for the zero eigenvalue. Thus, the fixed points form a line in the plane. There are several concepts that are important and are nicely illustrated by the simple linear system.


Invariant sets for linear planar systems

An positively (negatively) invariant set is a set of points in the plane such that if we choose initial conditions from the set, then the solution to the differential equation lies in the set for all positive (negative) time. Two obvious invariant sets are fixed points which are unchanged by the differential equation and the whole plane.

Two important invariant sets are related to fixed points. The stable (unstable) manifold for a fixed point is the set of all points in the plane which tend to the fixed point as time goes to positive (negative) infinity. For example, a stable node has the whole plane as its stable manifold and just the node itself as its unstable manifold. What are the stable and unstable manifolds for an unstable vortex? How about a center ?

In a linear system, 0 is the only fixed point. We have figured out the invariant sets for nodes, centers, and vortices. All that remains is the saddle point. This has a positive eigenvalue and a negative eigenvalue. The unstable manifold for a saddle point is the eigenvector corresponding to the positive eigenvalue. This is a one-dimensional set unlike the case of a node or vortex where it is either the whole plane (two-dimensional) or a single point ("zero" dimensional.) Similarly, the stable manifold for a saddle point is the eigenvector corresponding to the negative eigenvalue and is also one-dimensional. Except for degenerate cases, the sum of the dimensions of the unstable and stable manifolds is equal to 2, the dimension of the plane.


XPP stuff

XPP is able to classify fixed points and for saddle points, it is able to find the stable and unstable manifolds. Click on the problem above to fire up XPP. Click on (Dir.Field/flow), then choose (Dir. field) and accept the default of 10. You will see a bunch of lines indicating the direction fields. It looks like some sort of vortex. Click on (Initial conds) (Mouse) and choose initial data. Note how they spiral into the origin. Choose some other initial data. (Hint: A new option has been added that lets you automatically freeze every trajectory -- click on (Graphic stuff) (Freeze) (On Freeze) to turn this on. Repeat the procedure to turn it off. Also, click on (Graphic stuff) (Freeze) (Remove all) to clear the curves from the screen.)

Now click on (Sing Pts) (Go) to compute the "singular points" or as we know them, fixed points. Choose (no) when asked to print eigenvalues (since you are running Netscape -- terminal output is lost in the ozone.) A new window appears that tells you that the fixed point is stable and that there are two complex eigenvalues with negative real parts (that's what the c- = 2 means. Also, you are informed that X=0, Y=0 is the fixed point. Since there are two complex eigenvalues, it is a stable node.

Change the parameter b from 3 to -3. Click on (Graphic stuff) (Freeze) (Remove all). Redraw the direction field. Draw a bunch of trajectories with the mouse initial condition. (You will get a lot of "Out of Bounds Errors" -- just hit any key to ignore them!) This is a saddle point. Lets get the invariant sets. Click on (Sing pts) (Go) and answer (No) for printing eigenvalues but answer (Yes) to invariant sets. As before, you get a lot of out of bounds errors -- but nevermind them. You will see 4 trajectories that are straight lines emanating from the origin. One pair is the stable manifold and the other is the unstable manifold. They are also the eigenvectors corresponding to the negative and positive eigenvalues. XPP does not keep these in its memory, but does keep the initial conditions corresponding to them in memory. (You can access these initial conditions by clicking (Initial conds) (sHoot) and choosing 1-4 corresponding to the 4 different trajectories. NOTE that for the stable manifolds, you must integrate backward in time to move away from the origin -- use the (nUmerics) (Dt) menu to change the sign of the integration.)

HOMEWORK
In the following problems (i) sketch the direction field (ii) draw a few sample trajectories (iii) classify the behavior of the origin (iv) in the case of saddle points, find the stable and unstable manifolds. Verify the numerical calculations by explicitly calculating the eigenvalues and eigenvectors. Make a print out of your phase portrait. (Hint -- you can label your picture by using the (Text,etc) (Text) menu item and typing text in whatever font and size you want and using the mouse to position it.)


I am ready for nonlinear two-dimensional systems!