# Procedure to draw a phase portrait by hand ?

1. Apr 13, 2012

### sid9221

So I want to be able to draw the phase portrait for linear systems such as:

x'=x-2y
y'=3x-4y

I am completely confused, but this is what I have come up with so far:

Step 1: Write out the system in the form of a matrix.
Step 2: Find the eigenvalues and eigenvectors for the matrix.
Step 3: Using the eigenvectors draw the eigenlines.
Step 4: Using the the eigenvalues label the direction of the eigenlines[(+) = away, (-)= towards]
Step 5: Using the eigenvalues determine the type of the system. Eg: node, star, spiral etc.
Step 6: Fill in a few trajectories.

My issues appear at steps 5 and 6, I can't figure out how to draw the trajectories. Which eigenline should they be based around ?
Also with complex eigenvalues or when there is 1 eigenvector what do I have to do ?

Any help or link to a webpage would be greatly appreciated.

2. Apr 13, 2012

### wisvuze

since you have a linear system, as with all linear systems, you want to put it in a "cannonical form", and draw your phase portrait as a linear shift from the cannonical form phase portrait ( with the change of basis operator as your shift )
the cannonical forms are exactly the situations where you have 2 different eigenvalues, 1 eigenvector... et c if you haven't learnt about these, I will elaborate

3. Apr 13, 2012

### sid9221

No idea what that means...

4. Apr 13, 2012

### HallsofIvy

Staff Emeritus
Okay for this problem that is
$$\begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix}1 & -2 \\ 3 & -4\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$$

5. Feb 24, 2013