5.3 Solution Curves / Phase Portraits

Summary of \(\vec{x}' = A\vec{x}\)

1. Real and Distinct (Nodal Sink)

\vec{x}' = \begin{bmatrix} -3 & \sqrt{2} \\ \sqrt{2} & -2 \end{bmatrix} \vec{x}

\(\lambda = -1, -4\); \(\vec{v} = \begin{bmatrix} 1 \\ \sqrt{2} \end{bmatrix}, \begin{bmatrix} -\sqrt{2} \\ 1 \end{bmatrix}\)

2. Improper Nodal Source

\vec{x}' = \begin{bmatrix} 5 & -1 \\ 3 & 1 \end{bmatrix} \vec{x}

\(\lambda = 2, 4\); \(\vec{v} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}\)

Trajectories flow away from origin, leaving tangent to the eigenvector [1, 3].

3. Opposite Signs (Saddle Point)

\vec{x}' = \begin{bmatrix} 1 & 1 \\ 4 & 1 \end{bmatrix} \vec{x}

Saddle point: trajectories flow in along vector [1, -2] and out along vector [1, 2].

4. Repeated - Proper (Star Node)

\vec{x}' = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \vec{x}

All solutions are straight lines emanating directly away from the origin in all directions.

5. Repeated - Defective

\vec{x}' = \begin{bmatrix} 1 & -4 \\ 4 & -7 \end{bmatrix} \vec{x}

Single straight line solution. Curved trajectories form an S-shape into the origin.

6. Complex - Purely Imaginary (Center)

\vec{x}' = \begin{bmatrix} 2 & -5 \\ 1 & -2 \end{bmatrix} \vec{x}

Elliptical orbit inscribed in a parallelogram defined by real and imaginary eigenvector parts.

Solution curves are concentric ovals

Direction: Counter-Clockwise (CCW). Origin is a center.

7. Complex - Spiral Sink

\vec{x}' = \begin{bmatrix} 1 & -1 \\ 5 & -3 \end{bmatrix} \vec{x}

Solution curves spiral inward toward the origin because the real part of the eigenvalue is negative.

8. Zero Eigenvalue

\vec{x}' = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \vec{x}

A line of equilibrium points exists along the eigenvector [-2, 1]. All other paths are parallel.