Chap 41 Exercises

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Exercise 1 Active R chunk 1 defines a dynamical function named $f()$. The differential equation \[\partial_t x = f(x)\ .\] has two fixed points which you can easily find by graphing $f()$. Add R/mosaic statement to Active R chunk 1 that will locate all of the fixed points in the interval $-10 \leq x \leq 10$. For each fixed point find the stability (stable or unstable) and find the parameter in the exponential solution $e^{at}$ near the fixed point. (Hints: Zeros() and D().)

Active R chunk 1

Answer

f <- doodle_fun(~ x, seed=385)
slice_plot(f(x) ~ x, bounds(x=-10:10)) %>%
  gf_hline(yintercept = ~ 0, color="magenta")

Pts <- Zeros(f(x) ~ x, bounds(x=-10:10))
df <- D(f(x) ~ x)
Pts |> mutate(slope = df(x))

# A tibble: 2 × 3
      x     .output. slope
  <dbl>        <dbl> <dbl>
1 0.542 -0.000103     6.10
2 7.22   0.000000706 -1.22

Exercise 7 MC_xref("fig-cow-consumption") shows a model of a cow’s consumption of vegetation as a sigmoid function of the amount of vegetation. In addition to the mathematics of the function, it is helpful to be able to translate the shape of the function into terms that make sense in the context of the model.

At what level of available biomass is the cow most hungry at the end of the day? (Hint: Think about how you would measure how hungry a cow is at the end of the day.)

Activities

Exercise 9 The graphs below show two dynamical functions, $f(x)$ and $g(x)$, in the two differential equations $\partial_t x = f(x)$ and $\partial_t x = g(x)$.

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

For function $f(x)$,
1. locate all of the fixed points in the domain $-10 \leq x \leq 10$ and determine whether each one is stable or not
2. find the state value $x$ where the state is growing the fastest.
3. find the state value $x$ where the state is decreasing the fastest.
Answer the questions in (a) but for $g(x)$.

Exercise 10 The dynamical function in the graph is a sixth-order polynomial. It has several fixed points, labelled “a”, “b”, “c”, and so on.

The numbers i, ii, iii, etc. mark some initial conditions.

For each of the initial conditions marked in the graph, say whether the long-term behavior of the trajectory from that point will lead to one of the fixed points (and which one) or to $-\infty$ or $\infty$. (Hint: That the function is a sixth-order polynomial should tell you something about the functions behavior beyond the domain shown in the graph.)

Exercise 11 Here is a generic, linear differential equation:

\[\partial_t x = a x + b\ .\] 1. How many fixed points are in this system?

For one of the fixed points you identified in (1), what is the value for the location $x^\star$ in terms of the parameters $a$ and $b$?
For one of the fixed points, translate the equation to be in terms of $y$ in \[\partial_t y = ???\ .\]
Write down the symbolic solution to the equation in (3). (Hint: it will involve the parameter $a$ but not $b$.)
Translate the solution in (4) back into terms of $x$. (Hint: The solution will involve both parameters $a$ and $b$.)

Exercise 12 Use differentiation to verify that \[y(t) = 4 e^{2 t} + 7\] is a solution to the differential equation \[\partial_t y = 2 y - 14\ .\]

Exercise 13

For the dynamical system graphed above, sketch a plausible time series solution starting from initial condition $x_0=-2.5$.

Exercise 14

Suppose that $p(t)$ is the population density of squirrels over time. Draw the dynamical function $s(p)$ in the differential equation \[\partial_t p = s(p)\] that will produce a population that grows from a very small density of 4 squirrels per hectare to a carrying capacity of 50 squirrels per hectare. Mark the units on your graph.
Challenge: Arrange the units on the output of $s(p)$ so that it will take roughly 3 months to reach the carrying capacity.

Exercise 15 Coffee at an initial temperature of $T=$ 80 degrees C is in a closed thermos at a (very cold) construction site where the ambient temperature is -20 C. Suppose, with the thermos closed, it takes 4 hours for the coffee to fall to a tepid 30 degrees C. The thermos cools in accord with Newton’s Law of Cooling.

Draw a quantitatively reasonable graph of the dynamical function consistent with these facts in the differential equation \[\partial_t T = f(T)\ .\]

Exercise 16 A first-order dynamical system $\partial_t x = g(x)$ can have any number of fixed points, depending on the shape of $g(x)$.

A. Draw a graph of a $g(x)$ that has seven fixed points.

B. Explain why, when there are multiple fixed points, a stable fixed point will have a neighboring fixed point that is unstable, and vice versa.

Exercise 17 A debit card is a way to make withdrawals from a bank account. Funds in the account earn interest. In the economic environment of 2021, the interest rate is practically zero. But to make it possible to see the growth on a graph, we will stipulate it to be 4% per year. But if the account is in debt, then the debt grows at a much faster rate, perhaps 15% per year.

Suppose the amount in the bank account is $m$ (in dollars) and time is $t$ (in years). If $m$ is positive, you are earning interest. If negative, you are in debt and the debt will grow.

Which of the following graphs reflects the dynamics of the system?

On the graph paper below, sketch out the modifications of the bank-card dynamics described. Identify any fixed points by marking them on the graph and say if they are stable or unstable.

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In addition to the normal bank-card dynamics, your employer makes a steady retirement deposit into your account at the rate of $100 per year.
Your darling aunt looks at your account balance. If you have less than $300 in your account, she will add money daily at a rate of $100 per year. Otherwise, she does nothing.

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