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Finding the spiral sinks and spiral sources of a linear system

  1. Feb 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Basically, the problem involves a linear system dx/dt = ax + by and dy/dt = -x - y, with a and b being parameters that can take on any real value. Basically, you go through this system for several values of a and b (I did -12 to 12) to find the state at various points. That is, whether or not the point is a saddle, source, sink, perodic, or whatever else. I've found all of that. The only thing I haven't found yet is which sources and sinks are spirals. I know that it's a spiral if a complex number is involved. I've been told two different ways to find this.

    3. The attempt at a solution

    A: Graph the trace of the system (a + d)*x and the determinant (ad - bc) and the area inside the parabola has the spirals. I'm having a hard time doing this on the graphing tools I've found on the internet.

    B: Using the quadratic x^2 - (trace*x) + determinant = 0, find what values of the trace and determinant make is such that x = sqrt(negative number). This equation simplifies to x = sqrt(trace*x - determinant). An x on both sides, so that complicates things...

    What would be the easiest way to go about this?
  2. jcsd
  3. Feb 12, 2009 #2


    Staff: Mentor

    The behavior of a linear system like you describe is pretty much determined by the eigenvalues of the matrix. Here's a link to an article on Equilibrium in dynamical systems, as these are often called. The section titled Two-Dimensional Space might answer some of your questions.
  4. Feb 12, 2009 #3
    For the most part, I understand the relationship between behavior and eigenvalues. When you have two positive eigenvalues it's a source, two negative it's a sink, so on and so forth...

    My issue is, what's the easiest way to find for which values of A and B are there complex values involved.
  5. Feb 12, 2009 #4


    Staff: Mentor

    This is pretty hazy in my memory since it was a long time ago that I studied this stuff, but your quadratic (B) brings up some memories. It could be that this equation is the characteristic polynomial for the system, which gives you roots r1 and r2, and from which you get solutions e^(r1*t) and e^(r2 * t). If the solutions to the quadratic are complex, they will be conjugates, so you'll get r = a +/-bi. The imaginary parts will lead to solutions involving sin() and cos(), and that's why you get spiral behavior, either toward a source (if a < 0) or away to a sink (if a > 0).

    Again, this is pretty fuzzy in my mind, but maybe I've given you some directions to go in.
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