Figure: A phase portrait on a Cartesian coordinate system with axes \(x_1\) and \(x_2\). The origin is a nodal sink. Two straight-line trajectories correspond to the eigenvectors: one along the vector \(\begin{bmatrix} 1 \\ \sqrt{2} \end{bmatrix}\) with eigenvalue \(\lambda = -1\), and another along \(\begin{bmatrix} -\sqrt{2} \\ 1 \end{bmatrix}\) with eigenvalue \(\lambda = -4\). All other trajectories are curved and approach the origin. Arrows on all trajectories point toward the origin, indicating that as time \(t\) increases, solutions decay to zero. The curves are tangent to the eigenvector associated with the dominant eigenvalue \(\lambda = -1\) as they approach the origin.
Phase portrait showing a nodal sink at the origin.
Which one is asymptote into origin?
Origin is \(t = \infty\) because \(e^{\lambda t} \to 0\) as \(t \to \infty\).
As \(t \to \infty\), \(e^{-t} > e^{-4t}\).
So, the eigenvector w/ \(\lambda = -1\) is more important \(\to\) asymptote.
Figure: A phase portrait on a Cartesian coordinate system with axes \(x_1\) and \(x_2\). The origin is an improper nodal source. Multiple green solution curves originate from the origin and move outward into all four quadrants. Two straight-line trajectories are visible corresponding to the eigenvectors: one along the line defined by the vector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\) with eigenvalue \(\lambda = 2\), and another along the line defined by the vector \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\) with eigenvalue \(\lambda = 4\). Arrows on the curves indicate movement away from the origin as time \(t\) increases.
Phase portrait showing trajectories leaving the origin.
Figure: A phase portrait on a Cartesian coordinate system with axes x1 and x2. The origin is a saddle point. Two straight-line trajectories (eigenvectors) pass through the origin. One line corresponds to the eigenvector \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\) with eigenvalue \(\lambda = 3\); arrows on this line point away from the origin, indicating instability. The other line corresponds to the eigenvector \(\begin{bmatrix} 1 \\ -2 \end{bmatrix}\) with eigenvalue \(\lambda = -1\); arrows on this line point toward the origin, indicating stability in that direction. Hyperbolic trajectories fill the regions between these lines, curving toward the unstable eigenvector as time increases.
Conclusion:
The origin is a saddle point.
For \(\lambda = 3\), the eigenvector is \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\).
For \(\lambda = -1\), the eigenvector is \(\begin{bmatrix} 1 \\ -2 \end{bmatrix}\).
Figure: A phase portrait of a proper nodal source in a 2D Cartesian coordinate system with axes \(x_1\) and \(x_2\). Numerous straight-line trajectories radiate outward from the origin in all directions, indicated by arrows pointing away from the center. Two specific vectors are labeled: the horizontal vector is labeled with the eigenvector \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and eigenvalue \(\lambda = 1\), and the vertical vector is labeled with the eigenvector \(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\) and eigenvalue \(\lambda = 1\). The symmetry of the outward flow shows no preferred direction or asymptote.
Phase portrait showing a proper nodal source where all trajectories radiate from the origin.
both \(\lambda\)'s are the same, so both eigenvectors are equally important \(\rightarrow\) solutions are along linear combos of them
Figure: A phase portrait on a Cartesian coordinate system with axes x1 and x2. A single straight-line solution is shown along the eigenvector \( \vec{v} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \) for \( \lambda = -3 \). All other trajectories are curved in an 'S' shape, resembling an incomplete spiral, and all arrows point toward the origin (0,0). This represents an improper nodal sink.
Figure 1: Phase portrait for a system with a repeated negative eigenvalue and a defect of one.
Figure: Two small sketches of ellipses on Cartesian axes. The first ellipse is oriented with its major axis along the line \(y = x\), and the second is oriented with its major axis along the line \(y = -x\). They are separated by the word 'or'.
The vectors \(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\) and \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) tell us the direction of the major axis.
* incomplete see next page
Figure: A phase portrait showing concentric elliptical trajectories centered at the origin in the \(x_1, x_2\) plane. The ellipses are elongated along an axis in the first and third quadrants. Green arrows on the trajectories indicate a counter-clockwise (CCW) direction. Two vectors are drawn from the origin: one labeled \(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\) and another labeled \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\). A red vector labeled \(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\) points from the tip of the horizontal vector.
Figure: A Cartesian coordinate system with axes labeled x1 and x2. A red parallelogram is centered at the origin. The vertices and sides are defined by the vectors \( \begin{bmatrix} 2 \\ 1 \end{bmatrix} \) and \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \). Specifically, the vector \( \begin{bmatrix} 2 \\ 1 \end{bmatrix} \) points to a vertex in the first quadrant, and \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and its negative \( -\begin{bmatrix} 1 \\ 0 \end{bmatrix} \) define horizontal segments. A small black ellipse is sketched inside this parallelogram.
Figure 1: Parallelogram formed by the real and imaginary parts of the eigenvectors on the \(x_1-x_2\) plane.
The ellipse is inside the parallelogram.
Figure: A conceptual diagram showing a blue ellipse perfectly inscribed within a red tilted parallelogram, illustrating the geometric relationship between the eigenvector components and the resulting trajectory shape.
Figure 2: Detail of the ellipse inscribed within the parallelogram.
PAGE 8
Linear Systems: Complex Eigenvalues and Spiral Sinks
\[ \vec{v} = \begin{bmatrix} 2 + i \\ 5 \end{bmatrix}, \quad \begin{bmatrix} 2 - i \\ 5 \end{bmatrix} \]
Orient the "ovals" the same way as in the previous case.
This time, the real part of \( \lambda \) is \( -1 \), so the \( e^{-t} \) factor drives solutions into the origin. The ovals shrink as they go.
Figure: A phase portrait on a Cartesian coordinate system with axes labeled \( x_1 \) and \( x_2 \). A green spiral curve starts from the outer regions and spirals inward toward the origin in a clockwise direction. Arrows on the spiral indicate the direction of motion toward the center, representing a spiral sink or stable spiral.
Spiral source/sink
The trajectory shown is a spiral sink because the real part of the eigenvalue is negative, causing the magnitude to decay over time.
Everything on this line is a solution to \( \vec{x}' = A\vec{x} \) (a whole line of equilibria).
Phase Portrait
Figure: A phase portrait on a Cartesian coordinate system with axes x1 and x2. A line of equilibria is shown along the eigenvector \( \begin{bmatrix} -2 \\ 1 \end{bmatrix} \) corresponding to \( \lambda = 0 \). Other trajectories are straight lines parallel to the eigenvector \( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) corresponding to \( \lambda = 5 \). Since \( \lambda = 5 \) is positive, arrows on these trajectories point away from the line of equilibria, indicating an unstable growth direction.
Figure 1: Phase portrait showing a line of equilibria for the case where one eigenvalue is zero.
