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7.4 Solution Curves of Linear Systems (part 1)

\[ \vec{x}' = \begin{bmatrix} 4 & 7 \\ 7 & 4 \end{bmatrix} \vec{x} \quad \lambda = -3, 11 \]\[ \vec{v} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]

solution: \( \vec{x}(t) = c_1 e^{-3t} \begin{bmatrix} -1 \\ 1 \end{bmatrix} + c_2 e^{11t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \)

\[ x_1(t) = -c_1 e^{-3t} + c_2 e^{11t} \]\[ x_2(t) = c_1 e^{-3t} + c_2 e^{11t} \]

graph of \( x_1(t) \) vs. \( x_2(t) \) is called a phase portrait

gives relationship (qualitatively) between \( x_1, x_2 \) as \( t \) changes

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\[ \vec{x} = c_1 e^{-3t} \begin{bmatrix} -1 \\ 1 \end{bmatrix} + c_2 e^{11t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
  • \( c_1 e^{-3t} \begin{bmatrix} -1 \\ 1 \end{bmatrix} \) dominates as \( t \to -\infty \)
  • \( c_2 e^{11t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) dominates as \( t \to \infty \)

as \( t \to \infty \), all solutions approach the vector \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)

as \( t \to -\infty \), all solutions approach the vector \( \begin{bmatrix} -1 \\ 1 \end{bmatrix} \)

these vectors behave like asymptotes

Phase portrait on x1-x2 axes showing hyperbolic trajectories and a saddle point at the origin.

\( \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \lambda > 0 \) if initial conditions are such that \( c_1 = 0 \), then \( \vec{x} = c_2 e^{11t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)

then \( x_1 \to \infty, x_2 \to \infty \) as \( t \to \infty \) (because \( \lambda > 0 \)) if \( c_2 > 0 \)

if \( c_2 < 0 \), then \( \to -\infty \)

for the other, following similar reasoning, solutions go toward origin because \( \lambda < 0 \)

the origin here is called a saddle point

some solutions go toward origin and some go away \( \to \lambda \)'s are opposite in signs

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Phase Portrait of a Linear System

The following figure illustrates a phase portrait for a system of linear differential equations. The plot shows the vector field and trajectories in the xy-plane, centered at the origin.

Phase portrait showing trajectories flowing away from the origin in a nodal source pattern.
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Example

\[ \vec{x}' = \begin{bmatrix} 6 & 5 \\ -3 & -2 \end{bmatrix} \vec{x} \]

Eigenvalues and Eigenvectors:

\[ \lambda = 3, \quad 1 \]\[ \vec{v} = \begin{bmatrix} -5 \\ 3 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 1 \end{bmatrix} \]

General Solution:

\[ \vec{x} = c_1 e^{3t} \begin{bmatrix} -5 \\ 3 \end{bmatrix} + c_2 e^t \begin{bmatrix} -1 \\ 1 \end{bmatrix} \]
  • The term \( c_1 e^{3t} \begin{bmatrix} -5 \\ 3 \end{bmatrix} \) dominates as \( t \to \infty \).
  • The term \( c_2 e^t \begin{bmatrix} -1 \\ 1 \end{bmatrix} \) dominates for smaller \( t \).

So, solutions will leave the origin, first following \( \begin{bmatrix} -1 \\ 1 \end{bmatrix} \), then as \( t \to \infty \), following \( \begin{bmatrix} -5 \\ 3 \end{bmatrix} \).

The origin is called a source because things flow out of it. It is an improper nodal source.

Solutions all follow an asymptote near the origin.
Phase portrait sketch showing trajectories emerging from the origin along eigenvectors.
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Phase Portrait Analysis

The following figure illustrates a phase portrait for a system of differential equations. The vector field shows trajectories curving and flowing in a specific direction, indicating the behavior of the system near the origin.

A phase portrait on a Cartesian grid from -4 to 4 on both axes, showing blue vector flow lines curving toward the origin.
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If both \(\lambda\)'s are negative, the same (or similar) picture but all arrows go toward origin \(\rightarrow\) improper nodal sink.

Example

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

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

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

\(x_1 = c_1 e^{-t}\)    \(x_2 = c_2 e^{-t}\)

\[ \frac{x_2}{x_1} = \frac{c_2}{c_1} = m \rightarrow x_2 = m x_1 \]

line thru origin w/ slope m
A proper nodal sink diagram showing multiple straight-line trajectories with arrows pointing directly toward the origin.

Origin is a proper nodal sink

  • \(\lambda < 0\)
  • No asymptotes
  • Repeated \(\lambda\)'s w/ enough eigenvectors (or matrix is complete)
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Example: Linear Systems with Purely Imaginary Eigenvalues

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

The eigenvalues and eigenvectors for this system are:

\[ \lambda = i, -i \]\[ \vec{v} = \begin{bmatrix} 1+i \\ 2 \end{bmatrix}, \begin{bmatrix} 1-i \\ 2 \end{bmatrix} \]

The general solution is given by:

\[ \vec{x} = c_1 \begin{bmatrix} \sin t + \cos t \\ 2 \sin t \end{bmatrix} + c_2 \begin{bmatrix} -\sin t \\ \cos t - \sin t \end{bmatrix} \]

Since the solution involves sine and cosine, it is periodic, which corresponds to rotation. The solutions are ovals that go around the origin.

Phase portrait showing concentric elliptical trajectories centered at the origin in the x1-x2 plane.
The origin is called a center.

Determining Direction

The real part of \( \vec{v} \) is \( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \), which represents the major axis.

To determine the direction (counter-clockwise or clockwise), pick a location. For example, let:

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

Then the tangent vector is:

\[ \vec{x}' = A\vec{x} = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]

This tangent vector points one unit right and two units up, indicating the direction of flow.

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Example: Linear Systems with Complex Eigenvalues (Real Part > 0)

\[ \vec{x}' = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \vec{x} \]
\[ \lambda = 1-i, 1+i \]\[ \vec{v} = \begin{bmatrix} 1 \\ -i \end{bmatrix}, \begin{bmatrix} 1 \\ i \end{bmatrix} \]

The general solution is:

\[ \vec{x} = c_1 e^t \begin{bmatrix} \cos t \\ -\sin t \end{bmatrix} + c_2 e^t \begin{bmatrix} \sin t \\ \cos t \end{bmatrix} \]

The direction is determined as in the last example. Due to the positive real part of \( \lambda \) (the \( e^t \) term), the magnitude increases as \( t \to \infty \). So the curves get farther from the origin as they spiral outward.

Phase portrait showing a trajectory spiraling outward from the origin in the x1-x2 plane.
The origin here is a spiral source.

If the real part of \( \lambda \) is less than 0, it is a spiral sink.