Difference between revisions of "FAQ: How do you plot a 3D phase portrait?"

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<li> For a linear system, you can find the eigenvalues of matrix <math>A</math> and the corresponding eigenvectors.  Then locate the eigenvectors in the space, and correspondingly draw arrows whose tip has a direction that depends on the sign of the eigenvalue (trajectories shrink towards the origin for eigenvalues with negative real part, and vice versa).  Then try to match the behavior in the rest of the space.  The picture below shows an example of this:
 
<li> For a linear system, you can find the eigenvalues of matrix <math>A</math> and the corresponding eigenvectors.  Then locate the eigenvectors in the space, and correspondingly draw arrows whose tip has a direction that depends on the sign of the eigenvalue (trajectories shrink towards the origin for eigenvalues with negative real part, and vice versa).  Then try to match the behavior in the rest of the space.  The picture below shows an example of this:
<center>[[Image:phase3d.jpg|center|400px]]</center>
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<center>[[Image:phase3d.png|center|400px]]</center>
  
 
2) For a nonlinear system, you can have a rough idea of the phase plot near the origin as an equilibrium point, by linearizing and then proceeding as at 1).
 
2) For a nonlinear system, you can have a rough idea of the phase plot near the origin as an equilibrium point, by linearizing and then proceeding as at 1).

Latest revision as of 15:34, 4 January 2007

There are several ways to go.

  • For a linear system, you can find the eigenvalues of matrix and the corresponding eigenvectors. Then locate the eigenvectors in the space, and correspondingly draw arrows whose tip has a direction that depends on the sign of the eigenvalue (trajectories shrink towards the origin for eigenvalues with negative real part, and vice versa). Then try to match the behavior in the rest of the space. The picture below shows an example of this:
    Phase3d.png

    2) For a nonlinear system, you can have a rough idea of the phase plot near the origin as an equilibrium point, by linearizing and then proceeding as at 1).

    3) One can use dedicated software, or simulate several 3D trajectories having meaningful initial conditions (so that you would have an idea of their behavior in most of the space near the origin or the eqm you find). For Mac users, I suggest 3D-XplorMath.

    --Franco, Oct 2006