Two-dimensional systems: \[ \dot{x} = f(x, y) \] \[ \dot{y} = g(x, y) \]

• For stability, look at Jacobian
  \( J = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix} \)
• \( J \Rightarrow \lambda_1, \lambda_2 \): eigenvalues
• \( \ell_1, \ell_2 \): eigenvectors

• \( \lambda_1, \lambda_2 \) real, positive → unstable node
• \( \lambda_1, \lambda_2 \) real, negative → stable node
• \( \lambda_1 > 0, \lambda_2 < 0 \) → saddle node
• \( \lambda_1, \lambda_2 \) pure imaginary → center
• \( \lambda_1, \lambda_2 = \lambda_R \pm i \lambda_I \), \( \lambda_R > 0 \) → unstable spiral
• \( \lambda_1, \lambda_2 = \lambda_R \pm i \lambda_I \), \( \lambda_R < 0 \) → stable spiral