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Procedure to draw a phase portrait by hand ?

  1. Apr 13, 2012 #1
    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. jcsd
  3. Apr 13, 2012 #2
    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
     
  4. Apr 13, 2012 #3
    No idea what that means...
     
  5. Apr 13, 2012 #4

    HallsofIvy

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    Okay for this problem that is
    [tex]\begin{bmatrix}x' \\ y'\end{bmatrix}= \begin{bmatrix}1 & -2 \\ 3 & -4\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex]

     
  6. Feb 24, 2013 #5
    @HallsofIvy: Thank you so much! That was such an elaborate reply! helped me a lot!
     
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