Quiz 2

Question 1

What is a system?

Answer 1

A differential equation

Question 2

What is the state of the system?

Answer 2

u

Question 3

What is a dynamic?

Answer 3

f(u)

Question 4

If the initial value of u(t)=u(0), what is the long-term behavior of the system?

Answer 4

What happens to lim u(t) as the t approaches infinity?

Question 5

If the initial value of u(t)= u(0) what is the short term behavior of the system?

Answer 5

Dependent on the application

Question 6

What are the axes for a dynamics plot?

Answer 6

x=u, y= f(u)

Question 7

What is an equilibrium point?

Answer 7

a constant solution for the function f(u)=0.

Question 8

If u* is an equilibrium point, what is f(u*) equal to?

Answer 8

f(u*)=0

Question 9

What does it mean for an eq pt to be isolated?

Answer 9

There is a small neighborhood containing u* s.t. no other U* exist within this neighborhood.

Question 10

What do eq pts show?

Answer 10

The direction of the movement of as time increases.

Question 11

What is a phase line?

Answer 11

Plots the growth rate vs the population. p on left p’ on right

Question 12

What is a phase line used for?

Answer 12

To determine if the points in the neighborhood containing u* move toward or away from the eq pt

Question 13

What is an attractor?

Answer 13

A stable eq pt. The points in the neighborhood (on both sides of u) are attracted to u

Question 14

What is a repeller?

Answer 14

An unstable eq pt. The points in the neighborhood (on both sides of u) are repelled or pushed away from u

Question 15

What is a semi-stable eq pt?

Answer 15

The points in the neighborhood containing u* are attracted to u* on one side and repelled on the other

Question 16

When is an eq pt globally stable?

Answer 16

When at no matter what value, all points go back to the eq. pt.

Question 17

Why do we want to work with a dimensionless model?

Answer 17

They’re easier to work with because they eliminate variables so that we can model the function.

Question 18

What happens when f’(u*)<0? (Stability Theorem)

Answer 18

u* is a stable eq. pt if f is autonomous and u* is isolated

Question 19

What happens when f’(u*)>0? (Stability Theorem)

Answer 19

u* is an unstable eq. pt if f is autonomous and u* is isolated

Question 20

What happens when f’(u*)=0? (Stability Theorem)

Answer 20

No information. Compute the next derivative.

Question 21

What is a bifurcation?

Answer 21

a sudden change in qualitative behavior of a system with a small change in behavior

Question 22

What is a Creation-Annihilation Bifurcation?

Answer 22

one or more eq pts. is either created/destroyed with a small change in parameter.

Question 23

What is a bifurcation diagram?

Answer 23

It plots u* versus h. Then, we draw a phase line through it and plug the values into f’(u*) to determine if the eq. pts. are stable, unstable, or semi-stable.

Question 24

What do you need to look for in a bifurcation diagram?

Answer 24

Changes in sign/stability.