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.
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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.
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