Problem Set 16 – Introduction to Discrete Dynamical Systems and Eigenanalysis

Problem Set 16 – Introduction to Discrete Dynamical Systems and Eigenanalysis

Learning Objectives:

• You should know and be able to use the definition of eigenvalues and eigenvectors of a matrix A.

• You should be comfortable writing a discrete dynamical system that models a given scenario. You should also be comfortable interpreting a given discrete dynamical system in terms of a situation.

• If A is an n×n matrix and you are given a basis of Rn consisting of eigenvectors of A, you should be able to solve the discrete dynamical system ~x(t + 1) = A~x(t) with any given initial condition ~x(0). (Remember that solving a discrete dynamical system ~x(t+ 1) = A~x(t) means finding a closed formula for ~x(t).)

• You should feel comfortable sketching and interpreting solution trajectories of a discrete dynamical system.

1. Diabetes is often diagnosed using a glucose tolerance test, in which a person fasts before taking a large dose of glucose. The concentration of glucose in the person’s blood is measured at regular intervals after the dose, and these measurements are compared with the person’s fasting glucose level. [?] described a model of the human glucose regulatory system that could be used to interpret the results of a glucose tolerance test. Here, you’ll look at a simplified version of the model focusing on two quantities:

g(t) = glucose concentration in the body at time t minus fasting glucose level

h(t) = hormone concentration in the body at time t minus fasting hormone level

If t is measured in minutes, the model for a particular person might be∣∣∣∣∣ g(t + 1) = 0.89g(t) − 0.03h(t)h(t + 1) = 0.02g(t) + 0.96h(t) ∣∣∣∣∣

(a) Let ~x(t) =

[ g(t) h(t)

] . Find a matrix A such that ~x(t + 1) = A~x(t).

(b) ~v1 =

[ −1

2

] and ~v2 =

[ 3 −1

] are eigenvectors of A. Find the associated eigenvalues.

In the rest of this problem, you’ll find some solutions of the discrete dynamical system ~x(t+ 1) = A~x(t) and sketch the corresponding solution trajectories.

(c) If ~x(0) is a scalar multiple of ~v1 =

[ −1

2

] , say ~x(0) = c1~v1, what is ~x(t)? What is lim

t→∞ ~x(t)? Sketch

the corresponding trajectories (be sure to consider both the case where c1 is positive and where c1 is negative).

(d) If ~x(0) is a scalar multiple of ~v2 =

[ 3 −1

] , say ~x(0) = c2~v2, what is ~x(t)? Sketch the corresponding

trajectories on your sketch from (c).

1

(e) A person who takes a large dose of glucose at time t = 0 could have g(0) = 70 and h(0) = 0; that

is, ~x(0) =

[ 70 0

] . Find ~x(t) in this case; express your answer as a linear combination of ~v1 =

[ −1

2

] and ~v2 =

[ 3 −1

] .

(f) In (e), you expressed your solution ~x(t) as a linear combination c1(t)~v1 + c2(t)~v2. (1) When t is

very large, is c1(t) or c2(t) larger? Based on that, is ~x(t) closer to the line span(~v1) or to the line span(~v2) when t is large?

(g) Graph the trajectory you found in (e); add this trajectory to your sketch in (c), and make sure your answer to (f) agrees with the graph.

2. (a) Bretscher #7.1.2

(b) Bretscher #7.1.6

(c) Bretscher #7.1.34

Hint: If you have trouble getting started, a good strategy is to use the definitions of eigenvalues and eigenvectors. For example, how can you use the definitions to rewrite the fact “~v is an eigenvector of A with eigenvalue λ” as an equation?

3. (a) Bretscher #7.1.16

(b) Bretscher #7.1.18

(c) Arguing geometrically, find all eigenvectors and eigenvalues of the shear

[ 1 1 0 1

] . If possible,

find a basis of R2 consisting of eigenvectors of this shear.

4. Bretscher #7.1.36

5. Let A be an n×n matrix and λ be any number. Let Eλ = {~v ∈ Rn : A~v = λ~v}. (So, Eλ consists of the eigenvectors of A with eigenvalue λ, as well as the zero vector.) Show that Eλ is a subspace of Rn.

(1)The coefficients of ~v1 and ~v2 should be dependent on t, which is why we’re writing them as c1(t) and c2(t).