11 Drill Questions: Dynamics

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The dynamical rule describes for each location in the state space how fast and in what direction the state will move from that location. For each of these questions, a very small region of state space is shown with one or two arrows indicating which way the flow moves from the circle at the root of the arrow.

Drill 11. 1

  1. Which of these best describes the change in the x-component of the state as it moves through this region of state space?
flow-crc-1-3k
decreasing quickly       decreasing slowly       not changing       increasing slowly       increasing quickly      

  1. Which of these best describes the change in the y-component of the state as it moves through this region of state space?
flow-crc-2-kus
decreasing quickly       decreasing slowly       not changing       increasing slowly       increasing quickly      

Drill 11. 2 1. Which of these best describes the change in the x-component of the state as it moves through this region of state space?

flow-gwp-1-8e
decreasing       not changing       increasing      

  1. Which of these best describes the change in the y-component of the state as it moves through this region of state space?
flow-gwp-2-kus
decreasing       not changing       increasing      

Drill 11. 3

  1. Which of these best describes the change in the x-component of the state as it moves through this region of state space?
flow-bsc-1-83
decreasing quickly       decreasing slowly       not changing       increasing slowly       increasing quickly      

  1. Which of these best describes the change in the y-component of the state as it moves through this region of state space?
flow-bsc-2-p2
decreasing quickly       decreasing slowly       not changing       increasing slowly       increasing quickly      

Drill 11. 4

  1. Which of these best describes the change in the x-component of the state as it moves through this region of state space?
flow-lrk-1-82s
decreasing       not changing       increasing      

  1. Which of these best describes the change in the y-component of the state as it moves through this region of state space?
flow-lrk-2-982
decreasing       not changing       increasing      

  1. Notice that the previous questions didn’t offer choices like “decreasing slowly” or “increasing quickly.” That’s for a good reason. What is it?
flow-lrk-3-u3s

There is only one flow arrow.

“slowly” and “quickly” are relative. Here, the rate of change has about the same magnitude in both the x and y directions.

The flow is toward the south-east rather than the south-west.

Drill 11. 5

Which of these best describes how fast the state will move through this section of space?

flow-ors-1-sy

The state moves slowly.

The state moves quickly.

No way to say.

Drill 11. 6

  1. Which starting point—green or blue—will bring the state to the other flow arrow?
flow-csl-1-lw
green       blue      

  1. Following the flow through this small region of space, which of the following best describes the shape of the trajectory?
flow-csl-2-lw
straight       curved clockwise       curved counterclockwise      

Drill 11. 7

  1. Following the flow through this small region of space, which of the following best describes the shape of the trajectory?
flow-nds-1-jw
straight       curved clockwise       curved counterclockwise      

  1. Which of the following best describes the speed of movement through the state-space region?
flow-nds-2-xls
steady speed       speeding up       slowing down       not enough information       blue      

In each of the following questions, a flow field is shown. In general, the rate of change of one component of the state can vary (or not) depending on itself or the other components of the state. (Here we have just two components.)

Drill 11. 8  

  1.    flow-tss-1-k3
    True     or       False      
    The y-component of the flow depends strongly on the value of x.

In fact, the flow arrows don’t vary at all with x.

  1.    flow-tss-2-ws
    True     or       False      
    The x-component of the flow depends strongly on the value of y.

The x-component of flow is sometimes up and sometimes down. Which it is depends on the value of y.

  1.    flow-tss-3-8d

    True     or       False      

    The x-component of the flow depends strongly on the value of x.

  2.    flow-tss-4-is

    True     or       False      

    The y-component of the flow depends strongly on the value of y.

Drill 11. 9  

  1.    flow-gsl-1-k3
    True     or       False      
    The y-component of the flow depends strongly on the value of x.

Focus, for instance, on the column of flow arrows stacked up at y = 0. For different values of x in that column, the flow arrows have different y components.

  1.    flow-gsl-2-ws
    True     or       False      
    The x-component of the flow depends strongly on the value of y.

Focus, for instance, on the row of flow arrows where x = 0. Moving left to right along the y axis, the x-component of the flow goes from zero to strongly negative.

  1.    flow-gsl-3-8d

    True     or       False      

    The x-component of the flow depends strongly on the value of x.

  2.    flow-gsl-4-is

    True     or       False      

    The y-component of the flow depends strongly on the value of y.

Drill 11. 10  

  1.    flow-ftb-1-k3

    True     or       False      

    The y-component of the flow depends on the value of x.

  2.    flow-ftb-2-ws

    True     or       False      

    The x-component of the flow depends strongly on the value of y.

  3.    flow-ftb-3-8d

    True     or       False      

    The x-component of the flow depends strongly on the value of x.

  4.    flow-ftb-4-is

    True     or       False      

    The y-component of the flow depends on the value of y.

Drill 11. 11  

  1.    traj-ftd-1-lx

    True     or       False      

    For the flow shown above, red is a trajectory.

  2.    traj-ftd-2-l7y

    True     or       False      

    For the flow shown above, green is a trajectory.

  3.    traj-ftd-3-kw

    True     or       False      

    For the flow shown above, red is a trajectory.

  4.    traj-ftd-3-luy

    True     or       False      

    For the flow shown above, gray is a trajectory.

  1. Time series 1 matches which component of which trajectory?

    1. gray-x
    2. gray-y
    3. red-x
    4. red-y
    5. green-x
    6. green-y
    7. blue-x
    8. blue-y
ts-ftd-5-wkd
a       b       c       d       e       f       g       h       None of the others.      

  1. Time series 2 matches which component of which trajectory?

    1. gray-x
    2. gray-y
    3. red-x
    4. red-y
    5. green-x
    6. green-y
    7. blue-x
    8. blue-y
ts-ftd-6-wkd
a       b       c       d       e       f       g       h       None of the others.      

Drill 11. 12  

  1.    traj-rls-1-lx

    True     or       False      

    For the flow shown above, red is a trajectory.

  2.    traj-rls-2-l7y

    True     or       False      

    For the flow shown above, green is a trajectory.

  3.    traj-rls-3-kw

    True     or       False      

    For the flow shown above, blue is a trajectory.

  4.    traj-rls-3-luy

    True     or       False      

    For the flow shown above, gray is a trajectory.

  1. Time series 1 matches which component of which trajectory?

    1. gray-x
    2. gray-y
    3. red-x
    4. red-y
    5. green-x
    6. green-y
    7. blue-x
    8. blue-y
ts-rls-5-wkd
a       b       c       d       e       f       g       h       None of the others.      

  1. Time series 2 matches which component of which trajectory?

    1. gray-x
    2. gray-y
    3. red-x
    4. red-y
    5. green-x
    6. green-y
    7. blue-x
    8. blue-y
ts-rls-6-wkd
a       b       c       d       e       f       g       h       None of the others.