Quiz Prep (Tr/Det 2)

mark

Consider the system:

$$
\begin{align}
x' &= x-y+x^2 \\
y' &=x+y.
\end{align}
$$

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
• Use the determinant and trace of the Jacobian to classify as an
  • Unstable node,
  • Unstable spiral,
  • Stable spiral,
  • Stable node, or
  • Saddle point

Note that the Det/Tr analysis of a linear system is described in detail in section 4.5 of our text, as well as on this webpage. Linearization is described in detail in section 5.1 of our text.

sara

How do we set up the matrix for this?

mark

The Jacobian Matrix for a 2D system is $2\times2$. The first row lists the partial derivatives of $x'$ and the second row lists the partial derivatives of $y'$. Thus, we get

$$
J = \begin{pmatrix}
1 +2x & -1 \\
1 & 1
\end{pmatrix}.
$$