Deck 9: Nonlinear Differential Equations and Stability

Question 1

Consider the linear system = x.
Identify all the eigenvalues.

A) $\lambda$ = -8i
B) $\lambda$ = 8
C) $\lambda$ = -8
D) $\lambda$ = 8i
E) $\lambda$ = 0

Answer 1

$\lambda$ = -8i
$\lambda$ = 8i

Question 2

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Answer 2

center

Question 3

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Answer 3

a stable

Question 4

Consider the linear system = x.
Identify all the eigenvalues.

A) $\lambda$ = -15i
B) $\lambda$ = 3i
C) $\lambda$ = 15i
D) $\lambda$ = -6i
E) $\lambda$ = 6i
F) $\lambda$ = -3i

Question 5

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 6

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 7

Consider the linear system = x.
Identify all the eigenvalues.

A) $\lambda$ = 7i
B) $\lambda$ = -11i
C) $\lambda$ = -7i
D) $\lambda$ = -2i
E) $\lambda$ = 11i

Question 8

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 9

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 10

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = -7
B) $\lambda$ = 7
C) $\lambda$ = 0
D) $\lambda$ = 5
E) $\lambda$ = -5

Question 11

Consider the linear system = x.
Identify which two of the following are fundamental solution vectors for this system.

A)
B)
C)
D)
E)
F)

Question 12

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 13

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 14

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = 5
B) $\lambda$ = -5
C) $\lambda$ = 0
D) $\lambda$ = 2
E) $\lambda$ = -2

Question 15

Consider the linear system = x.
Identify which two of the following are fundamental solution vectors for this system.

A)
B)
C)
D)
E)
F)

Question 16

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 17

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 18

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = -6
B) $\lambda$ = -10
C) $\lambda$ = 0
D) $\lambda$ = 10
E) $\lambda$ = 6

Question 19

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 20

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 21

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = -8
B) $\lambda$ = -4
C) $\lambda$ = 0
D) $\lambda$ = -16
E) $\lambda$ = 4
F) $\lambda$ = 16
G) $\lambda$ = 8

Question 22

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 23

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 24

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = -4
B) $\lambda$ = 4
C) $\lambda$ = 0
D) $\lambda$ = -9
E) $\lambda$ = 9
F) $\lambda$ = -10
G) $\lambda$ = 10

Question 25

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 26

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 27

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = 10
B) $\lambda$ = -10
C) $\lambda$ = 0
D) $\lambda$ = -9
E) $\lambda$ = 9
F) $\lambda$ = -2
G) $\lambda$ = 2

Question 28

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 29

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 30

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = 0
B) $\lambda$ = 4
C) $\lambda$ = -4
D) $\lambda$ = 3
E) $\lambda$ = -3

Question 31

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 32

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 33

Consider the linear system = x.
Identify all of the eigenvalues.

A) $\lambda$ = 0
B) $\lambda$ = -3
C) $\lambda$ = 3
D) $\lambda$ = -4
E) $\lambda$ = 4

Question 34

Consider the linear system = x.
The origin is a ________.

A) spiral sink
B) center
C) spiral source
D) nodal sink
E) saddle point
F) degenerate node
G) nodal source

Question 35

Consider the linear system = x.
The origin is ________ critical point.

A) a stable
B) an unstable
C) an asymptotically stable

Question 36

For which of the following systems is the origin a saddle point?

A)
B)
C)
D)

Question 37

The trajectories of some nonzero solutions of this system converge to the origin as t $\rightarrow$ $\infty$ while many other solutions do not.

A)
B)
C)
D)
E)

Question 38

Every nonzero solution of this system spirals away from the origin.

A)
B)
C)
D)
E)

Question 39

For which of the following systems do all solution trajectories converge to the origin as t $\rightarrow$ $\infty$ ?

A)
B)
C)
D)
E)

Question 40

For which of the following systems is the origin a degenerate node?

A) $\mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 5 \\ 0 & -7\end{array}\right) \mathbf{x}$
B) $x^{\prime}=\left(\begin{array}{ll}-2022 & 20 \\ 0 & -5\end{array}\right) x$
C) $\mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 0 \\ 0 & 2022\end{array}\right) \mathbf{x}$
D) $x^{\prime}=\left(\begin{array}{ll}5 & -4 \\ 4 & 0\end{array}\right) \mathbf{x}$
E) $\mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 4 \\ -1 & 0\end{array}\right) \mathbf{x}$

Question 41

For which of the following systems is every solution periodic?

A) $\mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & -25 \\ 0 & -4\end{array}\right) \mathbf{x}$
B) $x^{\prime}=\left(\begin{array}{ll}-2022 & -60 \\ 0 & -4\end{array}\right) x$
C) $\mathbf{x}^{\prime}=\left(\begin{array}{ll}2022 & 0 \\ 0 & 2022\end{array}\right) \mathbf{x}$
D) $x^{\prime}=\left(\begin{array}{ll}4 & -3 \\ 3 & 0\end{array}\right) x$
E) $\mathbf{x}^{\prime}=\left(\begin{array}{cc}0 & 1 \\ -3 & 0\end{array}\right) \mathbf{x}$

Question 42

Which of the following is a critical point of this nonlinear system?

Select all that apply.

A) (7, -4)
B) (-7, 4)
C)
D) (0, 0)
E)
F)

Question 43

Which of the following is a critical point of this nonlinear system?

Select all that apply.

A) $\left(\frac{5}{4}, \frac{1}{2}\right)$
B) $\left(-\frac{4}{5},-2\right)$
C) $\left[-\frac{5}{4}, \frac{1}{2}\right)$
D) $\left(\frac{5}{4},-\frac{1}{2}\right)$
E) $\left(\frac{4}{5}, 2\right)$
F) $\left(\frac{4}{5},-2\right)$
G) $\left(-\frac{5}{4},-\frac{1}{2}\right)$
H) $\left(-\frac{4}{5}, 2\right)$
I) $(0,0)$

Question 44

Which of the following is a critical point of this nonlinear system?

Select all that apply.

A) (1, -1)
B) (-7, -7)
C) (1, 7)
D) (-1, 1)
E) (-1, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)

Question 45

Find an equation of the form H(x, y) = C, where C is an arbitrary real constant, satisfied by the trajectories of the following nonlinear system:
= 6y
= -10 x

A) $y=\frac{3}{5} x$
B) $6 y^{2}--10 x^{2}=C$
C) $6 y^{2}+-10 x^{2}=C$
D) $y=\frac{5}{3} x$

Question 46

Find an equation of the form H(x, y) = C, where C is an arbitrary real constant, satisfied by the trajectories of the following nonlinear system:

A)
B)
C)
D)

Question 47

Consider the following nonlinear system:
= +
Let Express x and y using polar coordinates and determine which of these statements is true.

A) The system is locally linear near the origin.
B) The system is not locally linear near the origin because
C) The system is not locally linear near the origin because
D) The origin is not an isolated critical point.

Question 48

Consider the following nonlinear system:

Express x and y using polar coordinates and determine which of these statements is true.

A) The system is locally linear near the origin.
B) The system is not locally linear near the origin because
C) The system is not locally linear near the origin because .
D) The origin is not an isolated critical point.
E) Both B and C.

Question 49

Consider the following nonlinear system:

Which of the following is a critical point of this nonlinear system? Select all that apply.

A) (2, -2)
B) (-7, -7)
C) (2, 7)
D) (-2, 2)
E) (-2, 7)
F) (-7, 7)
G) (7, 7)
H) (0, 0)

Question 50

Consider the following nonlinear system:

Question 51

Consider the following nonlinear system:

Which of the following statements are true? Select all that apply.

A) (-3, 3) is an asymptotically stable node.
B) (3, -3) is an unstable spiral node.
C) (-3, 3) is an unstable node.
D) (6, 6) is an unstable node.
E) (6, -6) is an unstable node.
F) (6, 6) is an asymptotically stable node.
G) (0, 0) is a saddle point.
H) (-6, -6) is an asymptotically stable node.

Question 52

Consider the following nonlinear system:

Which of the following is a complete list of the critical points of this nonlinear system?

A)
B)
C)
D)
E)

Question 53

Consider the following nonlinear system:

Question 54

Consider the following nonlinear system:

Which of the following statements are true? Select all that apply.

A) $\left(\frac{\pi}{6}, \frac{5 \pi}{18}\right)$ is an unstable saddle point.
B) $\left(-\frac{\pi}{6}, \frac{\pi}{18}\right)$ is a stable center.
C) $\left(-\frac{\pi}{6}, \frac{\pi}{36}\right)$ is a stable center.
D) $\left(\frac{\pi}{3},-\frac{5 \pi}{18}\right)$ is an asymptotically stable spiral point.
E) $\left(\frac{3 \pi}{6},-\frac{\pi}{18}\right)$ is an improper node.
F) $\left(\frac{\pi}{3}, \frac{\pi}{36}\right)$ is a stable center.

Question 55

Consider the following nonlinear system:

Which of the following statements is true?

A) The origin is an unstable spiral point.
B) The origin is an asymptotically stable node.
C) The origin is an asymptotically stable spiral point.
D) The origin is a stable sink.
E) The origin is a stable improper node.

Question 56

Consider this competing species model:

Which of these are critical points for this system? Select all that apply.

A) $\left(0, \frac{4}{7}\right)$
B) $\left(\frac{4}{7}, 0\right)$
C) $\left(0, \frac{8}{7}\right)$
D) $\left(0, \frac{7}{8}\right)$
E) $\left[-\frac{4}{7}, 0\right)$
F) $\left(0,-\frac{8}{7}\right)$
G) $(0,0)$
H) $\left(2,-\frac{10}{7}\right)$

Question 57

Consider this competing species model:

Which of the following statements are true? Select all that apply.

A) The entire first quadrant is the basin of attraction for the critical point $\left(-\frac{1}{3}, \frac{5}{3}\right)$ .
B) The origin is an unstable node.
C) Both $\left(-\frac{6}{7}, 0\right)$ and $\left(-\frac{2}{3}, 0\right)$ are saddle points.
D) Both $\left(\frac{6}{7}, 0\right)$ and $\left(0, \frac{2}{3}\right)$ are saddle points
E) All solution trajectories approach the origin as $t \rightarrow \infty$ .
F) The critical point $\left(-\frac{1}{3}, \frac{5}{3}\right)$ corresponds to coexistence in this model.

Question 58

Suppose ? is a real parameter. Consider this competing species model:

Which of these are true? Select all that apply.

A) The origin is a critical point of this system for all values of $\alpha$ .
B) The $x$ -nullcline is the curve $\frac{6}{3} x^{2}+\frac{4}{3} x$ .
C) The $y$ -nullcline is the horizontal line $y=\frac{2}{7} \alpha$ .
D) The system has no critical points for values of $\alpha<-\frac{7}{9}$

Question 59

Suppose ? is a positive real parameter. Consider this competing species model:

What are the nullclines for this system?

A) x = 0
B) y = 0
C)
D) 4x + 3y = 1
E)
F)
G)
H)

Question 60

Suppose ? is a positive real parameter. Consider this competing species model:

Which of these are critical points for this system? Select all that apply.

A) $\left(0, \frac{2}{\alpha}\right)$
B) $(0,0)$
C) $\left(\frac{2}{\alpha}, 0\right)$
D) $\left(0, \frac{\alpha}{2}\right)$
E) $\left(\frac{2}{3}, 0\right)$
F) $\left(\frac{3}{2}, 0\right)$
G) $\left(\alpha-2, \frac{-3 \alpha-8}{2}\right)$
H) $\left(-\alpha+2, \frac{-3 \alpha-8}{2}\right)$
I) $\left(2-\alpha, \frac{3 \alpha+8}{2}\right)$

Question 61

Suppose α is a positive real parameter. Consider this competing species model:
‪

Question 62

Suppose ? is a positive real parameter. Consider this competing species model:

Which of the following statements is true?

A) The origin is an unstable node.
B) The origin is an unstable saddle point.
C) The origin is an asymptotically stable node.
D) The origin is a stable center.

Question 63

Consider the following Lotka-Volterra system of equations:

Determine all critical points for this system.

Question 64

Consider the following Lotka-Volterra system of equations:

Question 65

Consider the following Lotka-Volterra system of equations:

Which of these statements are true? Select all that apply.

A) The solution trajectories all spiral away from the point $\left(-\frac{4}{5},-\frac{3}{2}\right)$ as $t \rightarrow \infty$
B) The solution trajectories are closed curves encircling the point $\left(\frac{4}{5}, \frac{3}{2}\right)$ .
C) The point $\left(\frac{3}{2}, \frac{4}{5}\right)$ is an unstable node.
D) The predator and prey populations exhibit a cyclic variation.
E) The origin is a saddle point.
F) The period of the solution trajectories is $\frac{2 \pi}{\sqrt{6.0}}$

Question 66

Consider the function V(x, y) = 7 + $\alpha$ xy + 5 , where $\alpha$ is a real number.Which of these statements is true?

A) V(x, y) is negative definite, for every nonzero real number $\alpha$ .
B) V(x, y) is positive definite, for every nonzero real number $\alpha$ .
C) V(x, y) is negative definite, for every real number $\alpha$ for which > 140.
D) V(x, y) is positive definite, for every real number $\alpha$ for which < 140.

Question 67

Consider the following nonlinear system:

Question 68

Consider the following nonlinear system:

What can you conclude about this nonlinear system?

A) The origin is asymptotically stable.
B) The origin is an unstable node.
C) The basin of attraction for the origin is the entire xy-plane.
D) The origin is a center and all solution trajectories encircle it.

Question 69

Consider the following nonlinear system:

Convert this system into an equivalent system in polar form.

Question 70

Consider the following nonlinear system:

Which of these statements is true? Select all that apply.

A) There are no isolated closed trajectories for this system.
B) For r > 0, the corresponding solution trajectories spiral outward away from r = 1 in a counterclockwise fashion.
C) For 0 < r < 1, the corresponding solution trajectories spiral toward the origin in a counterclockwise fashion.
D) The unit circle is a semistable limit cycle.
E) The unit circle is an unstable limit cycle.

Question 71

Consider the following nonlinear system expressed in polar form:
= r( r - 2)( r - 8)
= 1
Which of these statements is true? Select all that apply.

A) r = 8 is a semistable limit cycle.
B) r = 2 is an asymptotically stable limit cycle.
C) For 2 < r < 8, the corresponding solution trajectories spiral outward away from r = 2 in a clockwise fashion.
D) For 0 < r < 2, the corresponding solution trajectories spiral toward the origin in a clockwise fashion.
E) For r > 8, the corresponding solution trajectories spiral outward away from r = 8 in a counterclockwise fashion.

Question 72

Consider the following nonlinear system expressed in polar form: = r ( r - 7)
= -1
Which of these statements is true? Select all that apply.

A) All solution trajectories approach either r = 5 or r = 7 in a counterclockwise fashion.
B) r = 5 is a semistable limit cycle.
C) r = 7 is an unstable limit cycle.
D) For 0 < r < 5, the corresponding solution trajectories spiral toward r = 5 in a clockwise fashion.
E) For r > 7, the corresponding solution trajectories spiral outward away from r = 7 in a counterclockwise fashion.

Question 73

Consider the van der Pol equation - 2.8(1 - ) + u = 0.Write this equation as a nonlinear system in x and y, where x = u and y = .

Question 74

Consider the van der Pol equation - 4.2(1 - ) + u = 0.
The origin is an unstable node.

Question 75

Consider the van der Pol equation - 3.6(1 - ) + u = 0.If a closed trajectory exists, it must encircle the origin.

Question 76

Consider the van der Pol equation - 2.4(1 - ) + u = 0.If a closed trajectory exists, then it must be contained within the vertical strip |x| < 1.