Deck 10: Plane Autonomous Systems

Question 1

The geometric configuration of the solutions of in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Answer 1

E

Question 2

Assume a bead of mass slides along the curve . Also assume that there is a damping force acting in the direction opposite to the velocity and proportional to the velocity, with proportionality constant . The differential equation that describes the horizontal position of the bead is

A)
B)
C)
D)
E)

Answer 2

C

Question 3

The values of that make the system stable are

A)
B)
C) It is locally stable for all values of .
D) It is unstable for all values of .
E)

Answer 3

A

Question 4

The constant solution of are

A)
B) and
C)
D)
E) and

Question 5

Which of the following systems are linear? Select all that apply.

A)
B)
C)
D)
E)

Question 6

The critical point of the system is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 7

The critical point of the system is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 8

The solution of the system is

A) ,
B) ,
C) ,
D) ,
E) ,

Question 9

The Jacobian matrix of the system at the critical point

A)
B)
C)
D)
E)

Question 10

The solution of the system is

A)
B)
C)
D)
E)

Question 11

The initial value problem can be rewritten as the system

A)
B)
C)
D)
E)

Question 12

The geometric configuration of the solutions of in the phase plane is

A) stable spiral point
B) unstable spiral point
C) stable node
D) unstable node
E) saddle point

Question 13

The critical points of the system are

A)
B)
C)
D)
E)

Question 14

Consider the differential equation . The point is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 15

The critical point of the system is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 16

Which of the following systems are autonomous? Select all that apply.

A)
B)
C)
D)
E)

Question 17

The critical points of the system are

A)
B)
C)
D)
E) none of the above

Question 18

The solution of the system is

A)
B)
C)
D)
E)

Question 19

The geometric configuration of the solutions of in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 20

Consider the differential equation . The point is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 21

The geometric configuration of the solutions of in the phase plane is

A) degenerate stable node
B) degenerate unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 22

The differential equation can be rewritten as the system

A)
B)
C)
D)
E)

Question 23

The Jacobian matrix of the system at the critical point

A)
B)
C)
D)
E)

Question 24

The critical point of the system is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 25

Consider the differential equation . The point is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 26

The critical point of the system is Select all that apply.

A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question 27

Consider the differential equation . The point is

A) a stable critical point
B) an unstable critical point
C) not a critical point

Question 28

The solution of the system is

A)
B)
C)
D)
E)

Question 29

The values of that make the system stable are

A)
B)
C) It is stable for all values of .
D) It is unstable for all values of .
E)

Question 30

The geometric configuration of the solutions of in the phase plane is

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 31

The critical points of the system are

A)
B)
C)
D)
E)

Question 32

Which of the following systems are autonomous? Select all that apply.

A)
B)
C)
D)
E)

Question 33

Which of the following systems are linear? Select all that apply.

A)
B)
C)
D)
E)

Question 34

The Jacobian matrix of the system at the critical point

A)
B)
C)
D)
E)

Question 35

The critical point of the system is a

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) center point

Question 36

The solution of the system is

A)
B)
C)
D)
E)

Question 37

The critical point of the system is a

A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question 38

The only constant solution of is

A)
B)
C)
D)
E)

Question 39

Assume that and represent the populations of two competing species at time . The Lotka-Volterra competition model is

A)
B)
C)
D)
E)

Question 40

The critical points of the system are

A)
B)
C)
D)
E) none of the above