Mathematics Homework Help. 5 Linear and Nonlinear Systems of Differential Equations questions
Exercise 1
Consider the dierential equation
_ = (
p
2
2
cos )( cos )
on the circle, where is a real parameter.
(a) Compute all bifurcation points (; ):
(b) Draw all qualitatively dierent phase portraits on the circle.
(You do not need to draw the bifurcation diagram.)
(c) What types of bifurcations occur at the bifurcation points?
Exercise 2
(a) Classify the xed point at the origin and draw the phase portrait of the dierential
equation
x_ = 2x y;
y_ = y:
(b) Determine all a 2 R so that the dierential equation
x_ = (1 + 4a)x 5ay;
y_ = ax y
has a stable spiral at the origin.
(c) Determine all a 2 R so that the dierential equation
x_ = a2(a + 2)x;
y_ = 3x + (a 2)a(a + 1)y
has a Lyapunov stable xed point at the origin.
Exercise 3
A model for the population sizes of A, B 0 is given by
dA
dt
= r1A k1AB;
dB
dt
= r2B k2AB;
where r1; r2; k1; k2 > 0 are constants.
(a) By rescaling A;B and t, derive the following non-dimensionalized version of the
model
dx
d
= x(1 y);
dy
d
= y( x):
(b) Sketch the nullclines for the non-dimensionalized model, including the vector eld
along the nullclines.
(c) Based on the nullclines, sketch a plausible phase portrait.
Exercise 4
Consider the dierential equation
x_ = r x(2×2 + 3x 12):
It undergoes saddle-node bifurcations at (r; x) = ( 8; 1); (20; 2) and there are
no other bifurcations. The corresponding bifurcation diagram, with all xed points
plotted in blue, is
Let x(t) be the solution with x(0) = 4 for r = 0: Explain what happens to x(t)
as …
(a) … r is increased continuously to r = 30 and then lowered back to r = 0:
(b) … r is increased continuously to r = 30, then decreased to r = 10; and nally
increased back to r = 0:
Remark. You may answer the question in words and/or with diagrams. No computa-
tions necessary.
Exercise 5
Consider the dierential equation
x_ = 2x 3y y3;
y_ = 12x + 2y y5:
(a) Show that the origin is the unique xed point.
(b) Classify the xed point using the linearization.
What can you deduce about the stability of the xed point and the phase portrait
of the non-linear system near the xed point?
(c) Draw a qualitative phase portrait for the non-linear system. Indicate the stable
and unstable manifolds.
