Divergence at a Point

In this lab we will consider the vector field given by the equations:

dx/dt = y
dy/dt = -x^3 + x + (3/2)x^2 + y - x^2 y

The phase portrait of this vector field looks like Figure 1. (Follow this link to enter this dynamical system and plot more trajectories -- you will need a Java-enhanced browser).


Figure 1: Phase portrait and equilibria.

Let f denote the right-hand side of this system,
f(x,y)=(y, -x^3 + x + (3/2)x^2 + y - x^2 y),
and recall that

.

Question #1

  1. Compute the location of equilibria for this vector field.
  2. Compute div(f).
  3. Evaluate div(f) at each equilibrium.
  4. Based on the previous computation, what do you think is the stability type of each equilibrium? How confident are you?
Next: Divergence over a Region
Previous: Introduction
Robert E. Thurman <thurman@geom.umn.edu>

Document Created: Wed Jan 25 CST
Last modified: Thu Mar 27 15:09:19 1997