Question Type 6: Determining the type of eigenvalue given a phase portrait diagram Exercises

The following phase portrait is given:

![Trajectories repel from the origin along two distinct straight lines].

Classify the equilibrium at the origin and state the nature of its eigenvalues.

Consider the phase portrait shown below:

![Trajectories approach the origin along two distinct straight lines and no spirals].

Determine the type of equilibrium at the origin and describe the eigenvalues (real or complex, distinct or repeated, and sign).

Analyze this phase portrait:

![Trajectories approach the origin in a spiral fashion].

Identify the equilibrium type and characterize the eigenvalues.

Consider this phase portrait:

![Trajectories approach along one axis and repel along the perpendicular axis].

Classify the equilibrium at the origin and specify the eigenvalues.

Examine the phase portrait below:

![Trajectories spiral outward away from the origin].

Determine the type of equilibrium and describe the eigenvalues.

The diagram shows:

![Closed circular orbits around the origin].

Identify the equilibrium type and nature of its eigenvalues.

Look at this phase portrait:

![Trajectories repel from the origin tangent to a single line and then diverge].

Classify the equilibrium and eigenvalues.

Study the phase portrait:

![Trajectories approach the origin tangent to a single line and then merge].

Determine the equilibrium type and eigenvalue nature.

Examine this phase portrait:

![Trajectories spiral inward but slow down, approaching axes more slowly].

Determine the equilibrium type and characterize the eigenvalues.

Consider this phase portrait:

![Trajectories spiral outward and wrap more tightly near the origin].

Identify the equilibrium type and describe the eigenvalues.

Analyze the portrait below:

![Trajectories approach from one direction in a curved path and leave on the same direction].

Classify the equilibrium and specify eigenvalues.

Observe the phase portrait:

![Trajectories spiral outward in one quadrant and inward in the opposite quadrant].

Determine the equilibrium type and eigenvalues.