Quiz Prep (Jacobian Eigensystem)

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in Problems

Consider the system:

$$
\begin{align}
x' &= x - y \\
y' &=2x(y-1).
\end{align}
$$

First, find the equilibria of the system.

Then, for each equilibrium:

  • Find the linearization (expressed as the Jacobian matrix) of the system at that equilibrium, and
  • Find the eigenvalues and eigenvectors of that Jacobian.
  • Use the eigenvalues and eigenvectors to classify each point as an
    • Unstable node,
    • Unstable spiral,
    • Stable spiral,
    • Stable node, or
    • Saddle point
  • Finally, sketch the phase plane.

Feel free to use technology (like this webpage) to help you sketch the phase plane. Be sure, though, to include eigenvectors that clearly indicate any attractive or repulsive directions that you find.

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