A machine contains the grid of wires shown in the...

A machine contains the grid of wires shown in the accompanying sketch. At the seven indicated points, the temperature is kept fixed at the given values (in $^{\circ} \mathrm{C}$ ). Consider the temperatures $T_{1}(t), T_{2}(t),$ and $T_{3}(t)$ at the other three mesh points. Because of heat flow along the wires, the temperatures $T_{i}(t)$ changes according to the formula \[ T_{i}(t+1)=T_{i}(t)-\frac{1}{10} \sum\left(T_{i}(t)-T_{\mathrm{adj}}(t)\right) \], where the sum is taken over the four adjacent points in the grid and time is measured in minutes. For example, \[ \begin{aligned} T_{2}(t+1)=& T_{2}(t)-\frac{1}{10}\left(T_{2}(t)-T_{1}(t)\right)-\frac{1}{10}\left(T_{2}(t)-200\right) \\ &-\frac{1}{10}\left(T_{2}(t)-0\right)-\frac{1}{10}\left(T_{2}(t)-T_{3}(t)\right) \end{aligned} \] Note that each of the four terms we subtract represents the cooling caused by heat flowing along one of the wires. Let $\vec{x}(t)=\left[\begin{array}{l}T_{1}(t) \\ T_{2}(t) \\ T_{3}(t)\end{array}\right]$. a. Find a $3 \times 3$ matrix $A$ and a vector $\vec{b}$ in $\mathbb{R}^{3}$ such that \[ \vec{x}(t+1)=A \vec{x}(t)+\vec{b} \]. b. Introduce the state vector \[ \vec{y}(t)=\left[\begin{array}{c} T_{1}(t) \\ T_{2}(t) \\ T_{3}(t) \\ 1 \end{array}\right] \], with a "dummy" 1 as the last component. Find a $4 \times 4$ matrix $B$ such that \[ \vec{y}(t+1)=B \vec{y}(t) \] (This technique for converting an affine system into a linear system is introduced in Exercise $35 ;$ see also Exercise $32 .$ ) c. Suppose the initial temperatures are $T_{1}(0)=$ $T_{2}(0)=T_{3}(0)=0 .$ Using technology, find the temperatures at the three points at $t=10$ and $t=30$ What long-term behavior do you expect? d. Using technology, find numerical approximations for the eigenvalues of the matrix $B$. Find an eigenvector for the largest eigenvalue. Use the results to confirm your conjecture in part (c).

A machine contains the grid of wires shown in the accompanying sketch. At the seven indicated points, the temperature is kept fixed at the given values (in $^{\circ} \mathrm{C}$ ). Consider the temperatures $T_{1}(t), T_{2}(t),$ and $T_{3}(t)$ at the other three mesh points. Because of heat flow along the wires, the temperatures $T_{i}(t)$ changes according to the formula
\[
T_{i}(t+1)=T_{i}(t)-\frac{1}{10} \sum\left(T_{i}(t)-T_{\mathrm{adj}}(t)\right)
\], where the sum is taken over the four adjacent points in the grid and time is measured in minutes. For example,
\[
\begin{aligned}
T_{2}(t+1)=& T_{2}(t)-\frac{1}{10}\left(T_{2}(t)-T_{1}(t)\right)-\frac{1}{10}\left(T_{2}(t)-200\right) \\
&-\frac{1}{10}\left(T_{2}(t)-0\right)-\frac{1}{10}\left(T_{2}(t)-T_{3}(t)\right)
\end{aligned}
\]
Note that each of the four terms we subtract represents the cooling caused by heat flowing along one of the wires. Let $\vec{x}(t)=\left[\begin{array}{l}T_{1}(t) \\ T_{2}(t) \\ T_{3}(t)\end{array}\right]$.
a. Find a $3 \times 3$ matrix $A$ and a vector $\vec{b}$ in $\mathbb{R}^{3}$ such that
\[
\vec{x}(t+1)=A \vec{x}(t)+\vec{b}
\].
b. Introduce the state vector
\[
\vec{y}(t)=\left[\begin{array}{c}
T_{1}(t) \\
T_{2}(t) \\
T_{3}(t) \\
1
\end{array}\right]
\],
with a "dummy" 1 as the last component. Find a $4 \times 4$ matrix $B$ such that
\[
\vec{y}(t+1)=B \vec{y}(t)
\]
(This technique for converting an affine system into a linear system is introduced in Exercise $35 ;$ see also Exercise $32 .$ )
c. Suppose the initial temperatures are $T_{1}(0)=$ $T_{2}(0)=T_{3}(0)=0 .$ Using technology, find the temperatures at the three points at $t=10$ and $t=30$ What long-term behavior do you expect?
d. Using technology, find numerical approximations for the eigenvalues of the matrix $B$. Find an eigenvector for the largest eigenvalue. Use the results to confirm your conjecture in part (c).

Key Concepts

Affine Dynamical Systems

An affine dynamical system is one that evolves according to a linear transformation plus a constant vector, typically written as x(t+1) = A*x(t) + b. This formulation captures systems with both homogeneous dynamics (via the matrix A) and constant forcing (via the vector b), allowing one to model processes like heat transfer where fixed boundary conditions add a constant influence on the evolving state.

Matrix Representation of Difference Equations

Representing difference equations in matrix form is a powerful method because it organizes multiple interrelated equations into a compact vector and matrix framework. This approach facilitates analysis using linear algebra techniques, enabling efficient computation, clear visualization of the interactions between variables, and the application of tools like eigenvalue analysis to study system behavior.

Augmentation Technique for Converting Affine to Linear Systems

The augmentation technique involves adding an extra component (often a constant '1') to the state vector, which transforms an affine system into a homogeneous linear system. By forming an augmented state (such as y(t) = [T1(t), T2(t), T3(t), 1]?) and constructing a corresponding augmented matrix, one can address the constant term within a unified linear algebra framework, making it easier to apply analytical and computational methods.

Eigenvalue and Eigenvector Analysis

Eigenvalue and eigenvector analysis is a critical tool in understanding the dynamics of linear systems. The eigenvalues of the system matrix reveal the growth rates or decay rates of modes in the system, with the largest eigenvalue often dictating the long-term behavior. By studying the corresponding eigenvectors, one can identify invariant directions and predict how the system settles into equilibrium or diverges.

Stability and Long-term Behavior of Discrete Systems

The stability analysis of discrete dynamical systems involves examining whether the system converges to a fixed point, oscillates, or diverges as time progresses. By assessing the eigenvalues of the system matrix (particularly whether their magnitudes are less than, equal to, or greater than one), one can determine if the system will eventually settle into a steady state, which is critical in problems modeling processes such as heat diffusion with fixed boundary temperatures.

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