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Math Help - Generating vector field with critical points

  1. #1
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    Sep 2009
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    Generating vector field with critical points

    Hello,
    For a school project I am asked to analyze a vector field. To analyze it, we need to generate a vector field with several critical points(focus, spiral, saddle..)

    I have two questions about it:

    1. I can generate most of them by manipulating functions (such as the function of a circle at (x,-y) has a vector field that is classified as focus) but how can I create one with all of them at different points? Such as a focus at around -2,2 and a spiral at 1,1 and so on?

    2. If I was given vector field, then I could analyze the critical points by the Jacobian method but the other way around is quite difficult for some classes of critical points. For instance a spiral is formed if the eigenvalues of the Jacobian is complex conjugates (with nonzero imaginary values), so how do I go about coming up with a function of sort?
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  2. #2
    Super Member
    Joined
    Aug 2008
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    Would be quite a challenge to come up with a system that has all eight styles. But as far as generating each one separately, do you know about the Trace-Determinant? If not, look it up. Now draw a circle around the origin in the T-D coordinate system. As the point (t,d) moves over this circle, it will pass through regions corresponding to each of the dynamic types. Now, calculate the coefficients corresponding to (t,d) for each region for the system:

    x'=ax+by

    y'=cx+ky

    and the phase-space diagram for that system will pass through all eight of the dynamic styles as the point (t,d) travels around the circle.
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