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7.4 Solution Curves of Linear Systems (continued)

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

\( \lambda = 1, 1 \)

\( \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) only need a generalized eigenvector

let \( \vec{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)

\[ (A - \lambda I) \vec{v}_2 = \vec{v}_1 \quad \begin{bmatrix} 0 & 1 & : & 1 \\ 0 & 0 & : & 0 \end{bmatrix} \quad \vec{v}_2 = \begin{bmatrix} r \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad (r=0) \]

Solution:

\[ \vec{x}(t) = c_1 e^t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 e^t \left( t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) \]

as \( t \to \infty \), both \( e^t \) and \( t e^t \) are important

both \( e^t \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( t e^t \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) follow the same vector: \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) → ordinary eigenvector

the generalized one: \( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) affects orientation but is not really visible

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Only one asymptote: \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) (ordinary eigenvector)

Phase portrait on x1-x2 axes showing curved trajectories moving away from the origin, labeled with c2 signs.

Origin is an improper nodal source (defective matrix)

if matrix is complete then proper

Small diagram of a proper nodal source with straight lines radiating from the origin in all directions.

if \( c_2 > 0 \), \( c_2 e^t \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) gives the solutions an "up" direction as \( t \) increases

\( c_2 < 0 \), "down" direction as \( t \) increases

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Phase Portrait Analysis: Case Study

System Definition

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

The eigenvalues and eigenvectors for this system are:

  • \( \lambda = 0, 1 \)
  • \( \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)

General Solution

\[ \vec{x}(t) = c_1 e^{0 \cdot t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \\ 1 \end{bmatrix} \] \[ = c_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} \]

Component-wise solutions:

\( x_1(t) = c_1 \)

\( x_2(t) = c_2 e^t \)

Solutions are up/down

Phase Portrait Visualization

Phase portrait with vertical trajectories moving away from the horizontal x1-axis.

Parallel lines

\( c_2 > 0 \) (above axis) \( c_2 < 0 \) (below axis)

Analysis on the \( x_1 \)-axis

On the \( x_1 \)-axis, all points are of the form \( \begin{bmatrix} k \\ 0 \end{bmatrix} \).

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

No tangent vectors on the \( x_1 \)-axis.

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Gallery of Typical Phase Portraits for the System \( x' = Ax \): Nodes

Vector field showing a proper nodal source with all vectors pointing away from the origin.

Proper Nodal Source

A repeated positive real eigenvalue with two linearly independent eigenvectors.

Vector field showing a proper nodal sink with all vectors pointing toward the origin.

Proper Nodal Sink

A repeated negative real eigenvalue with two linearly independent eigenvectors.

Vector field showing an improper nodal source with curved trajectories moving away from the origin.

Improper Nodal Source

Distinct positive real eigenvalues (left) or a repeated positive real eigenvalue without two linearly independent eigenvectors (right).

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Gallery of Typical Phase Portraits for the System \(x' = Ax\): Nodes

Phase portrait of an improper nodal sink with trajectories curving toward the origin.

Improper Nodal Sink

Distinct negative real eigenvalues (left) or a repeated negative real eigenvalue without two linearly independent eigenvectors (right).

Phase portrait of a center with concentric elliptical trajectories around the origin.

Center

Pure imaginary eigenvalues.

Phase portrait of an improper nodal sink showing trajectories converging to the origin along a specific axis.
Phase portrait of a saddle point with trajectories approaching and then diverging from the origin.

Saddle Point

Real eigenvalues of opposite sign.

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Gallery of Typical Phase Portraits for the System \(x' = Ax\): Nodes

Phase portrait of a spiral source with trajectories spiraling outward from the origin.

Spiral Source

Complex conjugate eigenvalues with positive real part.

Phase portrait of a spiral sink with trajectories spiraling inward toward the origin.

Spiral Sink

Complex conjugate eigenvalues with negative real part.

Phase portrait showing parallel lines of flow with one zero and one negative real eigenvalue.

Parallel Lines

One zero and one negative real eigenvalue. (If the nonzero eigenvalue is positive, then the trajectories flow away from the dotted line.)

Phase portrait showing parallel lines of flow for a repeated zero eigenvalue.

Parallel Lines

A repeated zero eigenvalue without two linearly independent eigenvectors.