Three room feedback problem

This project is similar to the three room feedback problem that was discussed in lecture. The computations require some manipulations with determinants, trace, and eigenvalues of a $3\times3$ matrix. Possibly useful facts are posted here.

Triple room

Consider a three room apartment that has a heater/cooler in the first room, and two thermostats in the second and third rooms of the apartment. There are three temperatures $T_1$, $T_2$, $T_3$. Let $k_1$, … , $k_4$ be the heat conductance rates across walls as indicated in the drawing, and let $\alpha\ge0$ and $\beta\ge0$ be the feedback rates at which both thermostats report to the heater/cooler in the first room. Thus the equations are \begin{align*} T_1' & = k_2(T_2-T_1) - \beta T_2 - \alpha T_3 \\ T_2' &= k_1(T_1-T_2) + k_4 (T_3-T_2) \\ T_3' &= k_3(T_2-T_3) \end{align*}

Questions

  1. Write this system in matrix form $\frac{dX}{dt} = AX$, where \[ X = \begin{bmatrix} T_1 \\ T_2 \\ T_3 \end{bmatrix} \] and find the $3\times 3$ matrix $A$.
  2. Find the real parts of the eigenvalues of $A$ and indicate how they depend on the parameter $\beta$.
  3. For which values of $\beta$ does $A$ have only one real eigenvalue?
  4. For which values of $\beta$ does $A$ have eigenvalues with positive real part?
  5. For those values of $\beta$ for which $A$ has one real eigenvalue $\lambda$, and two complex eigenvalues $\mu\pm i\omega$ decide if the oscillating component of the solutions decays faster or slower than the component with the real exponential.

Parameter choices

—  $k_1$  $k_2$  $k_3$  $k_4$  $\alpha$
A 1 1 1 1 1
B 1 1 1 1 3
C 1 1 1 1 0
D 1 1 1 1 20
E 1 1 1 1 40
F 1 1 1 0 1
G 1 1 1 1 10
H 1 0 1 0 0
I 1 0 1 0 5
J 1 0 1 0 12
      
—  $k_1$  $k_2$  $k_3$  $k_4$  $\alpha$
K 1 1 1 1 20
L 1 1 1 1 40
M 1 1 1 0 1
N 2 2 1 1 0
O 2 2 1 1 0
P 3 3 1 1 0
Q 6 3 1 1 1
R 3 3 1 1 1
S 3 3 2 1 4
T 3 3 2 1 8