Discrete-Time Dynamical Systems

Question 1

Population growth and heart and lung dynamics are examples of what type of change?

Answer 1

Discrete change

Question 2

A discrete-time dynamical system models what type of change?

Answer 2

Discrete change

Question 3

What relationship does a discrete-time dynamical system describe?

Answer 3

The relationship between a quantity at the beginning and end of a time interval.

Question 4

In a discrete-time dynamical system, what does m(t) represent?

Answer 4

A measurement at the beginning of the time interval

Question 5

In a discrete-time dynamical system, what does (t + 1) represent?

Answer 5

The time one step into the future

Question 6

In a discrete-time dynamical system, what does m(t+1) represent?

Answer 6

A measurement at the end of an experiment

Question 7

What types of numbers make up the domain of a discrete-time dynamical system?

Answer 7

Whole numbers only

Question 8

How do you graph a discrete-time dynamical system?

Answer 8

By plotting points on the graph. This is because the domain only contains those points.

Question 9

How do you find the point of equilibrium of a DTDS?

Answer 9

Sub in L (limit) for m(t) and m(t + 1). Solve for L.

Question 10

What is the updating function of a DTDS?

Answer 10

The formula of the dynamic rule.

Question 11

How do you find the updating function?

Answer 11

Replace m(t) with x and m(t + 1) with f(x)

Question 12

What is cobwebbing?

Answer 12

A way to find values for m(t + 1) using graphing.

Question 13

In cobwebbing, which two functions are graphed?

Answer 13

The updating function and y = x.

Question 14

What do the x-coordinates where the updating function intersects with y = x signify?

Answer 14

Where the DTDS reaches equilibrium.

Question 15

Given the updating function x(t + 1) = f(xt), what is the equilibrium of the DTDS?

Answer 15

The value L for which L = f(L)

Question 16

If you start near x = L and the points approach x = L, is the equilibrium stable or unstable?

Answer 16

Stable

Question 17

If |f’(L)| < 1, is the equilibrium stable or unstable?

Answer 17

Stable

Question 18

If |f’(L)| > 1, is the equilibrium stable or unstable?

Answer 18

Unstable

Question 19

If |f’(L)| = 1, is the equilibrium stable or unstable?

Answer 19

Could be either. Need to use cobwebbing

Question 20

In medication concentration problems, is the equilibrium stable or unstable?

Answer 20

It is always stable

Question 21

In population growth problems, is the equilibrium always stable or unstable?

Answer 21

Could be stable or unstable