11 Exercises: Dynamics

Exercise 11. 1 The figure shows six different instantaneous configurations for a pendulum as it swings. The curved arrow shows the velocity of the bob. The length of the curved arrow is the velocity times one second.

Which frame shows the pendulum where it is near to the topmost point on its swing.

For drawing purposes, Fig E11. 2 gives you an empty image of the state space for the pendulum.

gf_blank(0 ~ 0) |>
  gf_lims(y = c(-1,1)) |>
  gf_labs(y = "velocity", x = (expression(paste(theta, " (deg)")))) |>
  gf_refine(
    scale_x_continuous(
      limits = c(-100,100), 
      breaks = c(-90,-60, -45, -30, -15, 0, 15, 30, 60, 45, 60, 90))) |> 
  gf_theme(theme_minimal())

Your task: For each of the six pendulum pictures in Figure 11.1, write the corresponding letter a-f in the right position in the state space. Imagine that the fastest velocity in Figure 11.1 is 1. Your answers can be approximate, but they should be correct order relative to one another.

Return to Figure 11.2, adding in a vector showing the rate of change of state for each picture with respect to time. Hint: You can tell the rate of change of $\theta$ by referring to the arrows in Figure 11.1. But for the rate of change with respect to $\theta$, you’ll have to draw on your intuition about how a pendulum works. Your answers will necessarily be approximate, but make sure to get the sign right for each quantity.

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Exercise 11. 2 For each of the pendulum pictures in Fig E11. 3, mark where you think the pendulum will be in one second. Remember that the length of the curved arrow corresponds to the velocity of the pendulum bob.

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Exercise 11. 3 Fig E11. 4 show the state space for the pendulum system. Five points in the state space have been marked. For each, draw a corresponding pendulum picture of the sort shown in Exercise 11.1 (but those pictures are not necessarily answers for this question).

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Exercise 11. 4 Show arrows for SIR and ask what’s going on at that point in the epidemic.

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Exercise 11. 5

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In discussing the formation of new physicians, we combined two facts: the total number of medical students is 100,000 and medical school takes four years. This led to our estimate that there will be 25,000 graduates each year. Revisit this model taking into account that not all entering medical students graduate. Assume that attrition in the first year is 10%, then 5% in each of the two successive years.

If there are 100,000 students all together, how many graduate each year?

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Exercise 11. 6 EXERCISE: Show some random dynamical rules and have students sketch a trajectory from each of several initial conditions.

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Exercise 11. 7 Referring to Figure 11.4, why is there no place where infectives are growing and susceptibles aren’t decreasing?

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Exercise 11. 8 Recall that in Chapter 1, the SIR model had two parameters: the social distance between people and the fraction of infectives who isolate. The dynamical rule shown in Figure 11.4 is drawn for a social distance of 20 and an isolation fraction of 10%.

One of the flow fields in Fig E11. 5 is for a different social distance than in Figure 11.4. The other is for a different isolation fraction. Identify which flow is for each changed parameter and say whether the parameter was raised or lowered. (Note: Both might be lowered, or both might be raised, or one lowered and one raised. You have to figure it out!)

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Exercise 11. 9 Refering to Figure 11.4, three nearby points in state space, marked , have mainly a westward flow. But only the middle dot is purely westward, the neighboring positions have flows that are a little bit inclined to the north or south. Explain what it is about the mechanics of the epidemic that leads to the north or south component of the flows.

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Exercise 11. 10 Other dynamics examples: Arms race, battles.

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Exercise 11. 11 What to do about acceleration, e.g. pendulum,

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Often, the motivation behind model building is to make predictions. For instance, domestic policy planners have to make decisions about how many new roads to build over the next 30 years, or how to staff education or expand health care facilities over the next decade. To make such decisions sensibly and effectively, the decision-maker must have good predictions what will be the demand for such services and facilities will be.

Prediction models for such purposes have a characteristic form that draws on two distinct types of information:

What is the current state of the situation?
What are the expected patterns of change?

These two types of information draw on different sources. Quantifying the current state of the situation can be based on existing records, e.g., how many nurses and other licensed medical professionals are currently active. In contrast, describing the pattern of change often involves figuring out what is the mechanism of change. For instance, the change in the number of available medical professionals comes primarily from two sources: the number of students enrolled in appropriate training programs and the retirement or burn-out rates of existing professionals.

Consider one aspect of health-care provision: the number of active physicians. Government statistics provide indicators of the present state: as of 2025, there are very close to one-million active physicians and about 100,000 medical students in the US.

The change in state is another matter; we have to know about the mechanisms. For instance, medical school is typically four years long. (Med school is followed by additional years of training, but those people are usually already providing medical care.) So, a reasonable estimate is that there will be roughly 25,000 new physicians per year entering the system.

What about retirement? A ballpark estimate will get us started. If a typical physician works for 35 years, roughly 1/35 of the physician population will retire each year: about 3% per year. Following along these lines, a quantitative model for the yearly change in the physician population is:

\[\text{Yearly change in }P \equiv \underbrace{25,000}_\text{new graduates} - \underbrace{0.03 P}_\text{retirements} \tag{1}\]

Where $P$ is the number of physicians.

Now for the prediction itself. This is a matter of book-keeping. For each year, the number of physicians at the end of the year will be the number at the start plus the yearly change in $P$. Arithmetic suffices to construct the prediction which we do year by year. Apply the mechanism of the change in state to the starting 2025 state to find the ending 2025 state. Each row of @tab-physician-prediction shows one such calculation. We can then move on to the row for the next year, noting that the ending 2025 state will be the starting 2026 state.

Year	Starting state	Change in State	Ending state
2025	1,000,000	25,000 - 0.03 $\times$ 1,000,000	995,000
2026	995,000	25,000 - 0.03 $\times$ 995,000	990,150
2027	990,150	25,000 - 0.03 $\times$ 990,150	985,446
2028	985,446	25,000 - 0.03 $\times$ 985,445	980,883

Accumulating the change in state over several years.

Really? 980,883 physicians to start 2029? The simple model of Equation 1 doesn’t itself justify the many digits being given in @tab-physician-prediction for the physician population each year. The reason for the digits is something else. [[We’re just mechanically applying the dynamical rule. It’s a matter of book-keeping, like keeping track of the pennies when calculating credit-card debt.]]

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Exercise 11. 12 TURN THIS INTO THE CIRCLE MODEL of circadian rhythms. I tried the VDP system, but really I should just show the simplified approximate model.

Van der Pol

$\mu = 5.25, P = 2.1$ gives a 25 hour internal day.

vdp <- makeODE(dx ~ mu*(x - x^3/3 - y)/P, 
               dy ~ (x / mu)/P, 
               mu = 5.25, P = 2.1)
streamlines(vdp, domain(x = -2:2, y = -.5:.5), npts = 20, nsteps=10)

vdp_soln <- integrateODE(vdp, domain(t = 0:26), x=0.25, y=0.25)

Solution containing functions x(t), y(t).

traj_plot(x(t) ~ y(t), vdp_soln, nt = 27)

# Need to write functions to set initial values of x and y from r, theta, and Pval
Pval = 25
orbit <- makeODE(dtheta ~ 2*pi/P, dr ~ r*(1-r),
                 dx ~ r*(1-r)*cos(theta) - r*sin(theta)*2*pi/P,
                 dy ~ r*(1-r)*sin(theta) + r*cos(theta)*2*pi/P,
                 P = Pval)
orbit_soln <- integrateODE(orbit, domain(t = 0:26), 
                           theta = 0, r = 1, x=0, y=1*2*pi/Pval)

Warning in makeODE(...): Parameters <x> & <y> are not used in the differential
equations.

Solution containing functions theta(t), r(t), x(t), y(t).

traj_plot(x(t) ~ y(t), orbit_soln, nt = 27)

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Exercise 11. 13 BUILD A SLIDER APP that let’s the reader choose the position and velocity and returns the rate of change of the position and velocity. Also plot out the pendulum with a little arrow indicating the direction and speed.

THE DYNAMICAL RULE must have a pair of rates of change (one for each component of the state) for any given value of the state. THE APP shows this.

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Exercise 11. 14 [[Population with predation]]

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Exercise 11. 15 [[Compound interest, rule of 72]]

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Exercise 11. 16 Dynamical rules for systems with a single state quantity. The skeleton will look like $a + b P + c P^2$. Enumerate the possibilities. For simplicity, let’s consider first $a=0$ and look at positive, negative, and zero values for $b$ and $c$.

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Exercise 11. 17 A better way to see the smooth flow. The display of the straight arrows for the dynamical rule is mathematically correct and shows the instantaneous motion over a wide domain of state space. But it is visually somewhat confusing. On the other hand, the display of the trajectory, also mathematically correct, shows the motion over a very limited domain.

We can design a display that, for many people, gives a less distracting view of the flow induced by the dynamical rule. We pick many more-or-less evenly spaced points in the state space at random. Then draw a trajectory starting at each of those point. Track the trajectory for a short interval at time.

Figure E11. 6: The flow field for the system of `r xf('fig-pendulum-gif')`.

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Exercise 11. 18 Continuous-time Solow growth model. See www/CEdmond_SolowModel.pdf

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AI generated:

Exercise 11. 19 AI Generated:

A freelance consultant charges a flat fee of $75 per hour for her services. Which type of growth describes her total earnings as she works more hours?

Think about whether the ‘bonus’ for working one extra hour changes or stays the same regardless of how many hours have passed.

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