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微积分代写-MATH 2391

时间：2021-04-17

University of Leeds MATH 2391

Nonlinear Differential Equations

Extended Coursework for April/May 2020

To be submitted by 2 pm BST on 14 May. You must submit the end-of-semester coursework using

Gradescope.

Q1 (a) For the one-dimensional dynamical system

dx

dt

= (1 + x)(x2 − µ) (1)

find the fixed points for any given fixed value of µ, with−∞ < µ <∞. Determine

the stability of each fixed point, giving ranges of µ. Find the bifurcation points

and sketch the bifurcation diagram, identifying the type of bifurcation for each

bifurcation point.

(b) Choose 3 values of µ to illustrate the features of this bifurcation diagram. For

each of these, sketch the graph of f(x, µ) and plot the vector field (sign of x˙)

on the x−axis. Do this also for the bifurcation values of µ. You should sketch

these diagrams in a sequence of increasing µ, describing the stability of the fixed

points at each value of µ.

Q2 For the following system, find all the fixed point in the given region:

x˙ = sinx cos y, y˙ = cosx sin y, −pi ≤ x ≤ pi, −pi ≤ y ≤ pi.

As well as the obvious 2pi periodicity of the individual cosine and sine functions, prove

that the dynamical system and its Jacobian have the symmetry (x, y) 7→ (x±pi, y±pi),

as a consequence of the products which occur in the formulae.

Show that, as a result of this symmetry, only 4 forms of Jacobian matrix occur at the

13 fixed points in this region and use this to classify these.

Show that all lines x = mpi, y = mpi, y ± x = mpi are invariant (in particular, that

the axes, as well as the edges and diagonals of this square region, are invariant under

the flow).

Hence, sketch the phase portrait of the nonlinear system in this region.

1

Q3 (a) Consider the dynamical system

dx

dt

= X(x, y),

dy

dt

= Y (x, y),

with X and Y quadratic functions. Determine the most general forms of X and

Y , which leave the x and y axes invariant, and have fixed points

(0, 0), (1, 0), (0, 1) and (1, 1).

Choosing the surviving parameters in some simple (but nontrivial) way, classify

the 4 fixed points and sketch the phase portrait.

Which lines are invariants of this dynamical system?

(b) Consider the system

dx

dt

= µ− x2, dy

dt

= −y, where µ is a real parameter.

Consider the number of fixed points and their nature as µ varies on the real line.

Sketch the phase portrait for 3 values of µ, chosen to illustrate any interesting

change of behaviour. Comment on your results.

Q4 (a) For the linear system

dx

dt

= y,

dy

dt

= −1

4

x,

determine the type of fixed point at the origin and hence construct a first integral

H(x, y). Sketch the level curves of H(x, y) and use the system to place arrows

in the correct direction.

(b) Show that (0, 0) is the only fixed point of the system

dx

dt

= y − x3y2, dy

dt

= −1

4

x− y3. (2)

Show thatH(x, y) (found in Part (a)) is a weak Liapunov function for this system.

Sketch again the level curves of H(x, y) and a typical orbit of the dynamical

system (2), commenting on any consequence of the weakness of the Liapunov

function.

Comment on the stability of the point (0, 0).

(c) Show that the system

x˙ = y, y˙ = −1

4

x+

1

8

y(4− x2 − 4y2)

has just one fixed point, which is at the origin. What is the nature of this fixed

point?

For the function H(x, y) = x2 + 4y2, consider the derivative dHdt and use this to

show that there exists a stable limit cycle. What is the explicit formula for this?

Sketch the phase portrait of this dynamical system.

2

学霸联盟

Nonlinear Differential Equations

Extended Coursework for April/May 2020

To be submitted by 2 pm BST on 14 May. You must submit the end-of-semester coursework using

Gradescope.

Q1 (a) For the one-dimensional dynamical system

dx

dt

= (1 + x)(x2 − µ) (1)

find the fixed points for any given fixed value of µ, with−∞ < µ <∞. Determine

the stability of each fixed point, giving ranges of µ. Find the bifurcation points

and sketch the bifurcation diagram, identifying the type of bifurcation for each

bifurcation point.

(b) Choose 3 values of µ to illustrate the features of this bifurcation diagram. For

each of these, sketch the graph of f(x, µ) and plot the vector field (sign of x˙)

on the x−axis. Do this also for the bifurcation values of µ. You should sketch

these diagrams in a sequence of increasing µ, describing the stability of the fixed

points at each value of µ.

Q2 For the following system, find all the fixed point in the given region:

x˙ = sinx cos y, y˙ = cosx sin y, −pi ≤ x ≤ pi, −pi ≤ y ≤ pi.

As well as the obvious 2pi periodicity of the individual cosine and sine functions, prove

that the dynamical system and its Jacobian have the symmetry (x, y) 7→ (x±pi, y±pi),

as a consequence of the products which occur in the formulae.

Show that, as a result of this symmetry, only 4 forms of Jacobian matrix occur at the

13 fixed points in this region and use this to classify these.

Show that all lines x = mpi, y = mpi, y ± x = mpi are invariant (in particular, that

the axes, as well as the edges and diagonals of this square region, are invariant under

the flow).

Hence, sketch the phase portrait of the nonlinear system in this region.

1

Q3 (a) Consider the dynamical system

dx

dt

= X(x, y),

dy

dt

= Y (x, y),

with X and Y quadratic functions. Determine the most general forms of X and

Y , which leave the x and y axes invariant, and have fixed points

(0, 0), (1, 0), (0, 1) and (1, 1).

Choosing the surviving parameters in some simple (but nontrivial) way, classify

the 4 fixed points and sketch the phase portrait.

Which lines are invariants of this dynamical system?

(b) Consider the system

dx

dt

= µ− x2, dy

dt

= −y, where µ is a real parameter.

Consider the number of fixed points and their nature as µ varies on the real line.

Sketch the phase portrait for 3 values of µ, chosen to illustrate any interesting

change of behaviour. Comment on your results.

Q4 (a) For the linear system

dx

dt

= y,

dy

dt

= −1

4

x,

determine the type of fixed point at the origin and hence construct a first integral

H(x, y). Sketch the level curves of H(x, y) and use the system to place arrows

in the correct direction.

(b) Show that (0, 0) is the only fixed point of the system

dx

dt

= y − x3y2, dy

dt

= −1

4

x− y3. (2)

Show thatH(x, y) (found in Part (a)) is a weak Liapunov function for this system.

Sketch again the level curves of H(x, y) and a typical orbit of the dynamical

system (2), commenting on any consequence of the weakness of the Liapunov

function.

Comment on the stability of the point (0, 0).

(c) Show that the system

x˙ = y, y˙ = −1

4

x+

1

8

y(4− x2 − 4y2)

has just one fixed point, which is at the origin. What is the nature of this fixed

point?

For the function H(x, y) = x2 + 4y2, consider the derivative dHdt and use this to

show that there exists a stable limit cycle. What is the explicit formula for this?

Sketch the phase portrait of this dynamical system.

2

学霸联盟