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Math 211 Activities Workbook: Adapted from: Understanding Linear Algebra by David Austin

David Austin, Erica Bernstein

Section 2.3 Dynamical Systems

Activity 2.3.1.

Suppose we run a company that has two warehouses, which we will call \(P\) and \(Q\text{,}\) and a fleet of 1000 delivery trucks. Every morning, a delivery truck goes out from one of the warehouses and returns in the evening to one of the warehouses. It is observed that
  • 70% of the trucks that leave \(P\) return to \(P\text{.}\) The other 30% return to \(Q\text{.}\)
  • 50% of the trucks that leave \(Q\) return to \(Q\) and 50% return to \(P\text{.}\)
The distribution of trucks is represented by the vector \(\xvec=\twovec{x_1}{x_2}\) when there are \(x_1\) trucks at location \(P\) and \(x_2\) trucks at \(Q\text{.}\) If \(\xvec\) describes the distribution of trucks in the morning, then the matrix transformation \(T(\xvec)\) will describe the distribution in the evening.
  1. Suppose that all 1000 trucks begin the day at location \(P\) and none at \(Q\text{.}\) How many trucks are at each location that evening? Using our vector representation, what is \(T\left(\ctwovec{1000}{0}\right)\text{?}\)
    So that we can find the matrix \(A\) associated to \(T\text{,}\) what does this tell us about \(T\left(\twovec{1}{0}\right)\text{?}\) Hint: Use the second bullet in the proposition above.
  2. In the same way, suppose that all 1000 trucks begin the day at location \(Q\) and none at \(P\text{.}\) How many trucks are at each location that evening? What is the result \(T\left(\ctwovec{0}{1000}\right)\) and what is \(T\left(\twovec01\right)\text{?}\)
  3. Find the matrix \(A\) such that \(T(\xvec) = A\xvec\text{.}\)
  4. Suppose that there are 100 trucks at \(P\) and 900 at \(Q\) in the morning. How many are there at the two locations in the evening?
  5. Suppose that there are 550 trucks at \(P\) and 450 at \(Q\) in the evening. How many trucks were there at the two locations that morning?
  6. Suppose that all of the trucks are at location \(Q\) on Monday morning.
    1. How many trucks are at each location Monday evening?
    2. How many trucks are at each location Tuesday evening?
    3. How many trucks are at each location Wednesday evening?
  7. Suppose that \(S\) is the matrix transformation that transforms the distribution of trucks \(\xvec\) one morning into the distribution of trucks in the morning one week (seven days) later. What is the matrix that defines the transformation \(S\text{?}\)
Solution.
  1. If 1000 trucks begin at \(P\text{,}\) that evening we find that 70% of them are at \(P\) with the remaining 30% at \(Q\text{.}\) Therefore, \(T\left(\ctwovec{1000}{0}\right) = \twovec{700}{300}\text{.}\) Since \(T\left(\ctwovec{1000}{0}\right) = 1000A\twovec{1}{0}\text{,}\) we see that \(T\left(\twovec{1}{0}\right) = \twovec{0.7}{0.3}\text{.}\)
  2. In the same way, we see that \(T\left(\ctwovec{0}{1000}\right) = \twovec{500}{500}\) so that \(T\left(\twovec{0}{1}\right) = \twovec{0.5}{0.5}\text{.}\)
  3. The columns of \(A\) are \(T\left(\twovec{1}{0}\right)\) and \(T\left(\twovec{0}{1}\right)\) so that \(A=\left[\begin{array}{rr} 0.7 \amp 0.5 \\ 0.3 \amp 0.5 \\ \end{array}\right] \text{.}\)
  4. Evaluate \(T\left(\twovec{100}{900}\right) = A\twovec{100}{900} = \twovec{520}{480}\text{.}\)
  5. We solve \(T(\xvec) = A\xvec = \twovec{550}{450}\) to find \(\xvec=\twovec{250}{750}\text{.}\)
  6. We denote the distribution of trucks Monday morning by \(\xvec_0 = \ctwovec{0}{1000}\text{.}\)
    1. Monday evening, we have \(\xvec_1=A\xvec_0 = \twovec{500}{500}\text{.}\)
    2. Tuesday evening, we have \(\xvec_2=A\xvec_1 = \twovec{600}{400}\text{.}\)
    3. Wednesday evening, we have \(\xvec_3=A\xvec_2 = \twovec{620}{380}\text{.}\)
  7. The matrix is \(A^7 = \left[\begin{array}{rr} 0.625 \amp 0.625 \\ 0.375 \amp 0.375 \\ \end{array}\right] \text{.}\)
As we will see later, this type of situation occurs frequently. We have a vector \(\xvec\) that describes the state of some system; in this case, \(\xvec\) describes the distribution of trucks between the two locations at a particular time. Then there is a matrix transformation \(T(\xvec) = A\xvec\) that describes the state at some later time. We call \(\xvec\) the state vector and \(T\) the transition function, as it describes the transition of the state vector from one time to the next.
Beginning with an initial state \(\xvec_0\text{,}\) we would like to know how the state evolves over time. For instance,
\begin{equation*} \begin{aligned} \xvec_1 \amp = T(\xvec_0)=A\xvec_0 \\ \xvec_2 \amp = T(\xvec_1) = (T\circ T)(\xvec_0)= A^2\xvec_0 \\ \xvec_3 \amp = T(\xvec_2) = A^3\xvec_0 \end{aligned} \end{equation*}
and so on.
We call this situation where the state of a system evolves from one time to the next according to the rule \(\xvec_{k+1}=A\xvec_k\) a discrete dynamical system.
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