Topic 5.17: Phase portraits (HL only) Questions

Which statement guarantees the origin is STABLE (all trajectories approach it)?

The eigenvalues of a system's matrix M are −2 and −5. The origin is:

A coupled system is dx/dt = 2x + y, dy/dt = x + 2y (per year). Find the eigenvalues of M and classify the equilibrium at the origin.

A predator (x) and prey (y) model is dx/dt = 0.2x − 0.1y, dy/dt = 0.1x + 0.05y (hundreds of animals, t in years). Starting at x = 40, y = 20 (hundreds), use Euler's method with step h = 1 year to estimate the populations after 1 year.

The matrix M = (a, b; c, d) of a coupled system has det M = 6 and tr M = −5. The eigenvalues satisfy λ² + 5λ + 6 = 0, giving λ = −2, −3. The equilibrium is:

A coupled system has eigenvalues λ = ±3i (purely imaginary). The trajectories are:

For a saddle point, a trajectory generally:

For the system with M = (−3, 0; 0, −1), find the eigenvalues and eigenvectors, then write the general solution.

Show that the origin is a saddle point.

Sketch the phase portrait near the origin.

Interpret the long-term behaviour of the two populations.

A predator–prey deviation model has M = (0, −2; 2, 0). Find the eigenvalues and describe the long-term behaviour of the trajectories.

Two competing firms' market shares (deviations from equilibrium) follow dx/dt = x + y, dy/dt = 4x + y. Find the eigenvalues, an eigenvector for each, and describe the trajectory shape.

A system has M = (1, −2; 2, 1). Find the eigenvalues and classify the equilibrium, stating whether trajectories spiral inward or outward.

Find its eigenvalues and eigenvectors.