Assignment 6

Due Wednesday, October 21

1

Consider the system

𝑥' = 𝑥 + 2 𝑦, 𝑦' = 3 𝑥 + 2 𝑦 .

(a)

Decide which case this system falls into: two independent real eigenvectors, one, or none.

(b)

Find the solution with initial condition $𝑥 (0) = 𝑥 0$ , $𝑦 (0) = 𝑦 0$ . Then find the solution with $𝑥 (0) = 1$ , $𝑦 (0) = 4$ .

2

Consider the system

𝑥' = 𝑥 - 4 𝑦, 𝑦' = 2 𝑥 - 3 𝑦 .

(a)

Decide which case this system falls into: two independent real eigenvectors, one, or none.

(b)

Find the solution with initial condition $𝑥 (0) = 𝑥 0$ , $𝑦 (0) = 𝑦 0$ . Then find the solution with $𝑥 (0) = 1$ , $𝑦 (0) = 1$ .

3

Consider the system

𝑥' = - 𝑥 - 4 𝑦, 𝑦' = 𝑥 - 5 𝑦 .

(a)

Decide which case this system falls into: two independent real eigenvectors, one, or none.

(b)

Find the solution with initial condition $𝑥 (0) = 𝑥 0$ , $𝑦 (0) = 𝑦 0$ . Then find the solution with $𝑥 (0) = 3$ , $𝑦 (0) = 1$ .

4

The eigenvalues tell us how a flow scales its eigenvector directions. In this problem, we investigate what they tell us about area.

(a)

Return to the system in Problem 1. Consider the parallelogram whose sides are the eigenvectors

𝑣 1 = (2, 3), 𝑣 2 = (1, - 1) .

Under the flow, by what factor does each side scale after time $𝑡$ ? By what factor does the area change? Explain why the area grows even though one side contracts.

(b)

A linear map with matrix $𝑀$ multiplies areas by $| d e t 𝑀 |$ . For the system $𝑋' = 𝐴 𝑋$ , the flow matrix is $𝑒 𝑡 𝐴$ , so its determinant tells us how the flow changes area.

Suppose $𝐴$ has two independent real eigenvectors. Show that

d e t (𝑒 𝑡 𝐴) = 𝑒 𝑡 t r 𝐴 .

(Hint: use diagonalization and the identity $d e t (𝑋 𝑌) = d e t (𝑋) d e t (𝑌)$ to simplify your calculation.)

(c)

Give an example of a system $𝑋' = 𝐴 𝑋$ whose flow preserves area but has trajectories that travel arbitrarily far from the origin. You may choose $𝐴$ to be diagonal. Explain how your example has both properties.

The formula in part (b) also holds when $𝐴$ has only one independent real eigenvector or none. For every real $2 \times 2$ matrix, the trace determines the area factor: positive trace gives exponential area growth, negative trace gives exponential area decay, and zero trace gives area preservation.

5

Consider the family of systems

𝑥' = - 𝑥 + 𝑐 𝑦, 𝑦' = - 2 𝑦,

where $𝑐 > 0$ is a constant.

(a)

Find the eigenvalues and classify the flow. Explain why every solution approaches the origin as $𝑡 \to \infty$ , no matter how large $𝑐$ is.

(b)

Now follow the solution starting at

𝑥 (0) = 0, 𝑦 (0) = 1 .

You may use the eigenvectors $(1, 0)$ and $(𝑐, - 1)$ . Write the initial state as a combination of these vectors, then find $𝑥 (𝑡)$ and $𝑦 (𝑡)$ .

When does $𝑥 (𝑡)$ reach its largest value for $𝑡 \geq 0$ ? How does that value depend on $𝑐$ ? Use your answer to choose a system whose trajectory starts one unit from the origin, travels more than $100$ units away, and eventually returns to the origin.

Every system in this family is a sink, yet you have shown that a state can travel far from the origin before returning. Being a sink describes what happens in the long run; it does not require trajectories to move steadily closer to the origin.