Chap 41 Review
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Exercise 1 A physical analogy can give intuition about dynamics in a one-dimensional state space. We will imagine that the state space is represented by a road, the instantaneous state being the location of a car along that road.
On a real road, the motion of the car is set by the driver using the accelerator and brakes and constrained by other traffic and road conditions. In our state-space road, the velocity of the car is strictly dictated by an authoritarian government law: no exceptions. When the dictated speed is zero, the car will stand still and cannot start moving again. Ultimately, depending on the shape of the law, the car will head off to infinity or come to an eternal standstill. Two different cars can never collide on this road: if the cars are close together, they must be traveling at similar velocities. Cars cannot pass one another in different directions. Passing requires that the two cars have opposite velocities: one positive and one negative. If the mandated velocity at a point along the road is positive, every car that comes to that point must have the same positive velocity as it passes that point in the road.
The diagram in Figure @ref(fig:road-rules) shows one possible format for describing the regulations for a road. The road is shown as a grey line. The red numbers are mile-markers alongside the road. The function graph shows the mandated speed at any point: you can read off the numerical value of the mandated speed on the y-axis.
To better visualize the flow of traffic on the road, in Figure @ref(fig:road-flow) we draw vectors on the road. The direction of each vector will be either east or west; the length of the vector gives the required speed for a car passing that point.
Cars arriving from the west are obliged to stop their journey at about \(x=4\) where the arrows switch direction. Similarly, cars arriving from the east will be driving west, but their journey has a different destination, mile 13, where the velocity regulations change direction. The easterners will be obliged to stop at the point where the left-facing arrows meet the right-facing ones. A car starting near mile 10 will be obliged to drive east but can go only to mile 13; a car starting at mile 7 will drive west until reaching mile 4.
This is a lot of detail in the shape of the dynamical function—I mean, the road regulations. So far as the consequences of the road regulations go, the situation is simpler: cars standing still at mile 4 or at mile 13, depending on where they started from.
Zooming in on the dynamics near mile 4 or 13 helps to show what happens as the cars approach those points. (Figure @ref(fig:road-flow-zoom)) At either point, the action is similar. Cars coming from the west have to slow down as the arrows get smaller. Similarly for cars from the east. Where east meets west the cars are forced to a standstill; in order for the vectors to have different directions on the west end of the road than on the east end, the length of the vectors has to go to zero.
Such a point in the state space, where the flow arrows converge, is called a stable fixed point. The “fixed” merely means that the cars come to a standstill. “Stable” means that all cars starting near the fixed point are heading toward that point. Many details of the dynamics near the fixed point have no role in the eventual outcome. For instance, the dynamical function near mile-marker 4 is concave down, while that at mile-marker 13 is concave up. In both cases the behavior of the system is the same: convergence at the stable fixed point.
Answer these questions with respect to ?fig-road-rules.
- At mile marker 4, which way is the car going?
West East At a standstill
question id: dolphin-hang-dish-1
- How long will it take for the car to go from mile marker 4 to mile marker 0? (Choose the closest answer.)
5 minutes 15 minutes 25 minutes 45 minutes Can’t be done.
question id: dolphin-hang-dish-2
- A car starts at mile marker 8. Where will it end up?
All the way west
About mile 9
About mile 10
About mile 13
All the way east
question id: dolphin-hang-dish-3
- A car starts at mile marker 10. Where will it end up?
All the way west
About mile 9
About mile 10
About mile 13
All the way east
question id: dolphin-hang-dish-4
- You and your friend plan to bike along the road. (Bikes are not covered by the speed mandate.) You will stop at a point where you can sell snacks to motorists through their car windows. What’s the most profitable point for you to set up your business.
All the way west
About mile 9
About mile 10
About mile 13
All the way east
question id: dolphin-hang-dish-5
A government commission is exploring ways to improve road use by changing the speed mandate. They will add a constant to the existing speed mandate.
- One group of citizens wants to facilitate east-to-west travel. Which of these would be a suitable value of the constant to add to the speed mandate?
-75 mph -25 mph 15 mph 35 mph 60 mph
question id: dolphin-hang-dish-6
- Another group of citizens wants to facilitate west-to-east travel. Which of these would be a suitable value of the constant to add to the speed mandate?
-75 mph -25 mph 15 mph 35 mph 60 mph
question id: dolphin-hang-dish-7
- An angry chef has a road-side hamburger stand at mile 5. His business has been non-existent since the current speed mandate was implemented. To give his stand the best chance at success, which policy should he advocate?
-75 mph -25 mph 15 mph 35 mph 60 mph
question id: dolphin-hang-dish-8
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