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C2 Fundamental Theory of Dynamical Systems
MSc Examination, May 1994
2 hours.
Full marks may be obtained by correct answers to FOUR questions.
1.
i)
Answer either i) or ii)
Show that periodic orbits of the forced differential equation
dx t
dt
=
F(xt) + µ cos ωt
xt∈Rn
must have a period which is a multiple of 2 πω-1. Explain how the dynamics of this equation can
be reduced to a map g : Rn → R n and describe the relationship between the periods of the periodic orbits of g and the periodic orbits of the original differential equation.
In the case n = 1 show that g cannot have periodic orbits of period greater than 1 (hint: think of
what an orbit of period 2 would look like in the phase space of the original differential equation).
ii) Define a Poincaré section and the corresponding Poincaré map. Explain the importance of this
construction to the study of differential equations. Consider the differential equation
ẋ
ẏ
θ̇
=
=
=
x2
2y
1
where x,y∈R and θ is an angle variable (so that θ = 0 and θ = 2π correspond to the same point).
Derive an explicit expression for the Poincaré map corresponding to the section θ = c for an
arbitrary constant c. Deduce that the Poincaré map is not globally defined and briefly comment on the reason for this.
2.
i)
Answer either i) or ii)
Suppose A is the matrix
A
=
λ

0
0

0
λ
0
0

1
λ 
What is An (justify your answer)? Hence or otherwise find the general orbit (xi,y i,z i) in terms of
the initial condition (x0,y 0,z 0) of the map on R 3 given by
xi+1
y i+1
z i+1
=
=
=
2xi + zi
2yi + zi
2zi
Is the origin a stable fixed point for this map?
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ii) Find the Jordan Normal Form of the matrix
 −1
 −1
 2α
−2α 

−1 
where α∈R is a parameter.
Hence, or otherwise, compute the general solution in terms of the initial condition (x(0),y(0)) of
the differential equation
ẋ
ẏ
=
=
-x - 2αy
2α-1x - y
Sketch the phase portrait for α = 1,2 and
3.
1
2
.
Define the terms stability and asymptotic stability for an equilibrium point of a differential
equation on Rn. Give an example of an equilibrium point which is stable but not asymptotically
stable.
Show that if an equilibrium point has distinct real negative eigenvalues then it must be asymptotically stable.
Consider the differential equation on R 2 defined in polar co-ordinates by
r˙
θ̇
=
=
r - 2r3 + r5
sin2 12 θ
Find the equilibrium points and determine their stability.
4.
Suppose that f : [0,1] → [0,1] is given by
f(x)
=
1− 2x


2x −1
0 ≤ x < 12
1
2
≤ x ≤1
Describe the action of f on the binary expansion of x. Hence or otherwise show that f has a
dense orbit. Find all the periodic orbits of least period 3 and compute their eigenvalues.
Now consider the map g : [0,1]×[0,1] → [0,1]×[0,1]
g(x,y)
=
( f (x), 1 y )

2

1
2
(
f
(x),

3 y + 3)
0 ≤ x < 12
1
2
≤ x ≤1
Sketch the image of the square under the first few iterates of g, and hence or otherwise, describe
an attracting invariant set for g.
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5.
Define the term topological conjugacy. Which properties of a map f : Rn → R n are preserved by
such a conjugacy and which properties are not? Suppose that f,g : R → R are given by
f(x)
=
g(x)
=
1
2
x + x2
- 12 x + x2
Show that f cannot be locally topologically conjugate to g in a neighbourhood of 0. State the
Hartman-Grobman theorem for a fixed point of a map f : Rn → R n and explain its significance.
Hence or otherwise deduce that f2 and g2 are locally topologically conjugate in a neighbourhood of 0.
6.
Define hyperbolicity and structural stability for an equilibrium point and a periodic orbit of a
differential equation
dx t
dt
=
xt∈Rn
F(xt)
Give sufficient conditions for each type of orbit to be structurally stable. Hence, or otherwise,
deduce that for β ≠ 0 and a sufficiently small ε, the forced Duffing equation
u˙˙ + δ u̇ - βu + u3
=
ε cosωt
u∈R
has a small amplitude periodic orbit of period 2πω-1.
End of paper