Is the origin an attractor a repeller or a saddle point of

Question

Question: Let A have the properties described in Exercise.

a. Is the origin an attractor, a repeller, or a saddle point of the dynamical system xk+1 = Axk?

b. Find the directions of greatest attraction and/or repulsion for this dynamical system.

c. Make a graphical description of the system, showing the directions of greatest attraction or repulsion. Include a rough sketch of several typical trajectories (without computing specific points).

Exercise: Let A be a 2 X 2 matrix with eigenvalues 3 and 1/3 and corresponding eigenvectors v₁ = {xk} be a solution of the difference equation xk+1 = Axk, x0 =

a. Compute x1 = Ax0

b. Find a formula for xk involving k and the eigenvectors v₁ and v₂.