Dynamics of Linear Maps

Abstract

Create a variable A initialized to be the above matrix (check the command matrix): sage: # edit here Create a variable v that contains the vector (1,1) (check the command vector): sage: # edit here To compute the image of a vector under a matrix you need to use the multiplication *. Compute the image A * v: sage: # edit here Write a function draw_orbit(A, v0, n) that draw the n-th first iterate of the orbit of v0 under A. That is the sequence of vectors v0, Av0, Av0, ..., Av0 (you can use the graphics primitives point2d or line2d): sage: # edit here On the same graphics, draw several of these orbits: sage: # edit here What is happening to them? The object AA in Sage stands for "real algebraic numbers". It is one way to consider exact numbers beyond integers and rationals.: sage: AA In the following cell, we create the algebraic number √ 5 and the golden ratio φ = 1+ √ 5 2 : sage: a = AA(5).sqrt() sage: phi = (1 + a) / 2 Check with Sage that the vectors u+ = (1,−φ) and u− = (1,φ −1) are eigenvectors of the matrix A: sage: up = vector([1, -phi]) sage: um = vector([1, phi-1]) sage: # edit here Create a graphics with several orbits together with the lines Ru+ and Ru−: sage: # edit here