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

Loading subject…

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

Question 1

Sketch the phase portrait of the system {x^{'} = y y^{'} = - 4 x - y and identify the type and stability of the origin.

[6]

Question 2

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

[7]

Question 3

Analysis of phase portraits and eigenvalues for systems of linear differential equations.

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 .

[4]

Question 4

Consider the linear system {x^{'} = 3 x + 4 y y^{'} = - 2 x - y

(a)

Find all equilibrium points for the system.

[2]

(b)

Compute the eigenvalues of the coefficient matrix of the system.

[3]

(c)

Classify the equilibrium at the origin and determine its stability.

[3]

Question 5

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

[11]

Question 6

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

[6]

Question 7

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.

[6]

Question 8

Consider the following system of linear differential equations: {x^{'} = - x + 3 y y^{'} = - 3 x - y Find the eigenvalues of the system and classify the equilibrium point at the origin.

[6]

Question 9

Stability and phase portraits for two-dimensional systems, including non-hyperbolic equilibria characterized by one zero eigenvalue and one negative eigenvalue.

Consider a two-dimensional autonomous system whose equilibrium at the origin is a non-hyperbolic node with eigenvalues λ_{1} < 0 and λ_{2} = 0 . Describe the qualitative behavior of the phase portrait in the neighborhood of the origin and provide a sketch showing representative trajectories.

[5]

Question 10

Consider a system with a Jacobian matrix at the equilibrium (0, 0) given by J = (0 - 1 10) Determine the type and stability of the origin.

[4]

Question 11

A nonlinear predator-prey model represents the interaction between two species.

A nonlinear predator-prey model is given by the following system of differential equations: {x^{'} = x (2 - x - y) y^{'} = y (- 1 + x) Locate the nontrivial equilibrium and classify it by computing the Jacobian and its eigenvalues.

[9]

IB Mathematics Applications & Interpretation Question Type 5: Interpreting Key Informative Points Using Phase Portraits Exercises