Lesson test: Nonlinear systems and what solutions say

1.[2p]
What does linearising a nonlinear system at an equilibrium give you?

The local portrait, since the trace and determinant of the partial derivatives classify it

Correct

The exact solution near that point

The answer is: The local portrait, since the trace and determinant of the partial derivatives classify it

A conserved quantity for the whole plane

The answer is: The local portrait, since the trace and determinant of the partial derivatives classify it

The period of any closed orbit

The answer is: The local portrait, since the trace and determinant of the partial derivatives classify it

2.[3p]
Why is a centre in the linearisation an unreliable conclusion?

With zero trace the neglected quadratic terms decide, and they can turn it into a spiral either way

Correct

Because the equilibrium may not exist in the nonlinear system

The answer is: With zero trace the neglected quadratic terms decide, and they can turn it into a spiral either way

Because centres are always unstable

The answer is: With zero trace the neglected quadratic terms decide, and they can turn it into a spiral either way

Because the determinant is negative there

The answer is: With zero trace the neglected quadratic terms decide, and they can turn it into a spiral either way

3.[3p]
For $x' = x (3 - x - 2 y)$ , $y' = y (2 - x - y)$ , what is the determinant $Δ$ of the linearisation at the equilibrium $(0, 2)$ ?

CorrectNot quite: 2

4.[3p]
A pendulum of length 1.50 m hangs at rest. What angular speed at the bottom, in radians per second, is just enough to carry it over the top?

CorrectNot quite: 5.115

5.[3p]
By what factor does the exact pendulum period exceed the small-swing period at an amplitude of 30 degrees? Use $1 + θ_{0}^{2} / 16$ .

CorrectNot quite: 1.017

6.[3p]
A Lotka-Volterra system has $a = 0.4$ , $b = 0.01$ , $c = 0.3$ and $d = 0.015$ per year. What is the period, in years, of small oscillations about the coexistence equilibrium?

CorrectNot quite: 18.1

7.[3p]
Volterra's principle says that harvesting both predator and prey at the same rate has what effect on the averages?

It raises the average prey population and lowers the average predator population

Correct

It lowers both averages in proportion to the harvest

The answer is: It raises the average prey population and lowers the average predator population

It leaves both averages unchanged, since they are set by the initial conditions

The answer is: It raises the average prey population and lowers the average predator population

It raises both averages, since harvesting reduces crowding

The answer is: It raises the average prey population and lowers the average predator population

8.[2p]
Sensitive dependence on initial conditions means the Lorenz equations no longer have unique solutions.

True

The answer is: False

False

Correct

9.[3p]
Which statements about nonlinear systems in the plane are correct?

Select all that apply

A limit cycle is an isolated closed orbit that nearby trajectories approach

Correct

No linear system can have a limit cycle, since scaling a solution gives another solution

Correct

A bounded trajectory that avoids equilibria must approach a closed orbit

Correct

Chaotic motion is possible in two dimensions if the nonlinearity is strong enough

The answer is: A limit cycle is an isolated closed orbit that nearby trajectories approach, No linear system can have a limit cycle, since scaling a solution gives another solution, A bounded trajectory that avoids equilibria must approach a closed orbit

1.[2p] What does linearising a nonlinear system at an equilibrium give you?

2.[3p] Why is a centre in the linearisation an unreliable conclusion?

3.[3p] For x'=x(3-x-2y), y'=y(2-x-y), what is the determinant Δ of the linearisation at the equilibrium (0,2)?

4.[3p] A pendulum of length 1.50 m hangs at rest. What angular speed at the bottom, in radians per second, is just enough to carry it over the top?

5.[3p] By what factor does the exact pendulum period exceed the small-swing period at an amplitude of 30 degrees? Use 1+θ02/16.

6.[3p] A Lotka-Volterra system has a=0.4, b=0.01, c=0.3 and d=0.015 per year. What is the period, in years, of small oscillations about the coexistence equilibrium?

7.[3p] Volterra's principle says that harvesting both predator and prey at the same rate has what effect on the averages?

8.[2p] Sensitive dependence on initial conditions means the Lorenz equations no longer have unique solutions.

9.[3p] Which statements about nonlinear systems in the plane are correct?

1.[2p]
What does linearising a nonlinear system at an equilibrium give you?

2.[3p]
Why is a centre in the linearisation an unreliable conclusion?

3.[3p]
For $x' = x (3 - x - 2 y)$ , $y' = y (2 - x - y)$ , what is the determinant $Δ$ of the linearisation at the equilibrium $(0, 2)$ ?

4.[3p]
A pendulum of length 1.50 m hangs at rest. What angular speed at the bottom, in radians per second, is just enough to carry it over the top?

5.[3p]
By what factor does the exact pendulum period exceed the small-swing period at an amplitude of 30 degrees? Use $1 + θ_{0}^{2} / 16$ .

6.[3p]
A Lotka-Volterra system has $a = 0.4$ , $b = 0.01$ , $c = 0.3$ and $d = 0.015$ per year. What is the period, in years, of small oscillations about the coexistence equilibrium?

7.[3p]
Volterra's principle says that harvesting both predator and prey at the same rate has what effect on the averages?

8.[2p]
Sensitive dependence on initial conditions means the Lorenz equations no longer have unique solutions.

9.[3p]
Which statements about nonlinear systems in the plane are correct?