Question Type 5: Interpreting key informative points using phase portraits Exercises

Sketch the phase portrait of the system $\begin{cases} x' = y \\ y' = -4x - y \end{cases}$ and identify the type and stability of the origin.

The system $\begin{cases}x' = 4x - y \\ y' = 2x + y\end{cases}$ has an equilibrium at the origin. Determine the eigenvalues and eigenvectors of the linearization, then classify the equilibrium.

A phase portrait exhibits spiral trajectories approaching an equilibrium inwards, making two rotations before settling. What does this indicate about the eigenvalues’ real and imaginary parts? Explain.

Given the nonlinear system $\begin{cases}x' = x(1 - x) - xy\\y' = y(-2 + x)\end{cases}$ draw the nullclines, find all equilibrium points, and determine the stability of each using linearization.

For the system $\begin{cases} x' = x - y^2 \\ y' = -y + x^2 \end{cases}$ determine the type of equilibrium at the origin $(0,0)$ using linearization and comment on the effect of the nonlinear terms.

A phase portrait shows a saddle at the origin with unstable manifold along the $x$-axis and stable manifold along the $y$-axis. Propose a linear system with these properties and justify your choice.

Consider a two-dimensional autonomous system whose equilibrium at the origin is a non-hyperbolic node with eigenvalues $\lambda_1 < 0$ and $\lambda_2 = 0$.

Describe the qualitative behavior of the phase portrait in the neighborhood of the origin and provide a sketch showing representative trajectories.

Consider a system with a Jacobian matrix at the equilibrium $(0,0)$ given by

$J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}$

Determine the type and stability of the origin.

A nonlinear predator–prey model is given by the following system of differential equations:

Locate the nontrivial equilibrium and classify it by computing the Jacobian and its eigenvalues.