Math Insight

Solving linear discrete dynamical systems

Math 1241, Fall 2020
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Due date: Sept. 25, 2020, 11:59 p.m.
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Total points: 3
1. Consider the discrete dynamical system \begin{align*} y_{ n+1} &= 0.3 y_n \qquad \text{for $n = 0,1,2, \ldots$}\\ y_0 &= 15 \end{align*}

Calculate
$y_{1} =$
$\times y_0$
$=$

$y_{2} =$
$\times y_1$
$=$
$\times y_0$
$=$

$y_{3} =$
$\times y_2$
$=$
$\times y_0$
$=$

This means that to calculate $y_{3}$, you start with $y_0$ and multiply by
how many times?

Continuing the pattern $n$ times, to calculate $y_n$, you start with $y_0$ and multiply by
how many times?

Since repeated multiplication can be written as exponentiation, the formula for $y_n$ is:
$y_n =$
$\times y_0$
$=$

(When answering online, you can use ^ for exponentiation, so enter $a^b$ as a^b.)

Calculate $y_{17}=$

(Keep at least 6 significant digits in your response. Online, you can enter a number like $1.2352 \times 10^{-5}$ as either 1.2352*10^-5 or 1.2352E-5.)

2. Consider the discrete dynamical system \begin{align*} z_{ t+1} &= 2.1 z_t \qquad \text{for $t=0,1,2, \ldots$}\\ z_0 &= P \end{align*} where the parameter $P$ indicates the value of the initial condition. Calculate
$z_{1} =$

$z_{2} =$

$z_{3} =$

Continuing the pattern, calculate: $z_t =$

(When answering online, you can use ^ for exponentiation, so enter $a^b$ as a^b.)

Calculate $z_{33}=$

If we start with the tiny initial condition $P=7.4 \cdot 10^{-9}$, calculate $z_{47} =$

(Keep at least 6 significant digits in your response.)

3. Consider the dynamical system \begin{align*} x_{ t+1} &= c x_t \qquad \text{for $t = 0,1,2,\ldots$}\\ x_0 &= D. \end{align*}
1. Calculate:
• $x_1 =$
$\times x_0$
$=$
• $x_2 =$
$\times x_1$
$=$
• $x_3 =$
$\times x_2$
$=$
• . . . .
• $x_t =$
2. If $c=3$ and $D=2$, then
• $x_t =$
• $x_{10} =$
• $x_{16} =$

4. Before you solve this dynamical system, \begin{align*} u_{n+1} &= 0.3 u_n \qquad \text{for $n=0,1,2,\ldots$}\\ u_0 &= 20,000,000. \end{align*} do you think that the solution should grow or shrink?

Solve the system and use the solution to calculate $u_5$, $u_{15}$, and $u_{25}$. Include a least 6 significant digits in your response.

$u_n =$

$u_5 =$

$u_{15} =$

$u_{25} =$

5. The above dynamical systems were written in function iteration form, given the value of the state variable as a function of the old value. Sometimes, though, we might have a dynamical system in difference form, where we are given the change in the state variables. In this case, we want to convert to function iteration form before solving.

Consider the dynamical system where the state variable $y$ decreases by 60% each time step: \begin{align*} y_{ t+1} - y_t &= -0.6 y_t \qquad \text{for $t=0,1,2\ldots$}\\ y_0 &= 19. \end{align*} Each time step, the change in $y$, i.e., $y_{ t+1} - y_t$, is -60% of the old value of $y$.

1. To solve the dynamical system, we don't want to view it as subtracting 60% at each time step. Instead, we want to view it as multiplying by some number at each time step. In other words, we want write the dynamical system in function iteration form, where the new $y_{ t+1}$ is set equal to some number times the old $y_t$.

Converting to function iteration form is easy, we just need to add $y_t$ to both sides of the equation so we get $y_{ t+1}$ alone on the left side of the equation. The evolution rule for the dynamical system becomes

$y_{ t+1} = -0.6y_t +$
or, combining the terms on the right hand side,
$y_{ t+1} =$
.
(Online, enter $y_t$ as y_t.)

The function iteration form makes it clear that after each time step, the new value of $y$ is
% of the previous value of $y$. In other words, at each time step, we multiply by
.

2. The rest of the procedure for finding the solution should be familiar. Calculate:
• $y_1 =$
$\times y_0$
$=$
• $y_2 =$
$\times y_1$
$=$
• $y_3 =$
$\times y_2$
$=$
• . . . .
• $y_t =$

6. Consider the dynamical system \begin{align*} y_{ t+1} - y_t &= k y_t \qquad \text{for $t=0,1,2,\ldots$} \\ y_0 &= L. \end{align*}
1. This dynamical system is written in
form. To solve the system, we want to rewrite it in
form. To convert it, we need to add
to both sides of the equation. The result is:
$y_{ t+1} = ky_t +$
To write the result in a slightly more simple form, factor out the $y_t$ on the right hand side so that it is just some quantity times $y_t$. We end up with the following dynamical system in
form:
$y_{ t+1} =$

$y_0 =$
,
where we repeated the initial condition for completeness.

The result is a little uglier than the previous problems, but not bad.

2. Calculate:
• $y_1 = ($
$) \times y_0 =$
• $y_2 = ($
$) \times y_1 =$
• $y_3 = ($
$) \times y_2 =$
• . . . .
• $y_t =$
3. If $k=2$ and $L=4$, then
• $y_t =$
• $y_{10} =$
• $y_{15} =$
4. The dynamical system would look a lot prettier in function iteration form if we could write is as \begin{align*} y_{ t+1} &= K y_t \qquad \text{for $t =0,1,2,\ldots$}\\ y_0 &= L. \end{align*} If we wanted to transform our system into this prettier form, what must we make $K$? (It must depend on our original parameter $k$.)

$K =$

With this definition of $K$, what is the solution to the dynamical system? (Your answer shouldn't have $k$ in it, only $K$.)

$y_t =$

5. If $K=3$ and $L=4$, then
• $y_t =$
• $y_{10} =$
• $y_{15} =$

7. Consider the discrete dynamical system \begin{align*} m_{ t+1} - m_t&= 0.7 m_t \qquad \text{for $t=0,1,2,\ldots$}\\ m_0 &= c \end{align*}

Calculate
$m_{1} =$

$m_{2} =$

$m_{3} =$

Continuing the pattern, calculate: $m_t =$

If $c=3.2$, calculate $m_{70}=$

(Keep at least 6 significant digits in your response.)

8. In the first days of life, the cells in a human embryo divide into two cells approximately every day. Our goal is to write down a discrete dynamical system describing the growth of the embryo. After solving the model, we can use the model to predict the embryo growth. We can also evaluate the merits of the model.
1. Step 1: formulate the model

The first step is to determine the state variable of the model. Given the above description, what is the most natural quantity to describe what quantity about the embryo is changing from day to day? The
of the embryo

Choose a variable to represent this quantity:
. Choose a variable for time (either n or t):
.

With your choice of variables, $＿_{ ＿ }$ represents the
of the embryo that are present
days after conception.

After fertilization (in day 0), the embryo consists of a single cell. (OK, it should be called a zygote until if divides into multiple cells, but you get the idea.) Therefor the initial condition is $＿_0=$

Write the dynamical system both in difference form as

$＿_{ ＿+1} - ＿_{ ＿ } =$

$＿_0=$
and in function iteration form as
$＿_{ ＿+1} =$

$＿_0=$

The function iteration result is easy to understand because the number of cells in one day is
times the number from the previous day. To understand the difference form result, you need to deduce that for any given day, the change in the number of cells from the previous day to that day is equal to
times the number from the previous day.

2. Step 2: solve the model

Solve the dynamical system to obtain an expression for the number of cells as a function of the number of days since fertilization.

$_＿^{＿} =$

3. Step 3: predict embryo growth and evaluate the merits of the model

After one week, how many cells would there be in the embryo?
After two weeks?
After three weeks?
These values seem
, so we might
that our model is capturing the dynamics of the embryo growth during these first weeks.

What about the predictions of the model for the later stages of the pregnancy? If a pregnancy lasts 40 weeks, what does the model predict for the number of cells that the baby would have upon being born?

How does this number compare to the number of atoms in the observable universe, which is estimated to be about $10^{80}$? The number is about
times
than the number of atoms in the observable universe.

Do you believe the results of the model for the end of the pregnancy?
What does the model assume about the rate of cell division? It assumes that the rate of division
throughout the pregnancy. That assumption must be
.

What must be true of the rate of cell division? Can the cells continue to divide into two each day for the whole course of the pregnancy?