C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
EXAMPLE SHEET 4
Stars indicate difficulty of questions.
1.
Let Γ be a periodic orbit of a differential equation on Rn of period T. Let Σ be a Poincaré section
and g the corresponding return map. Let x*∈Σ ∩ Γ be the corresponding fixed points of g. Show
that if the linearization MT around the orbit has an eigenvalue λ = 1 of multiplicity k, then D x*g
has an eigenvalue λ = 1 of multiplicity k-1. Similarly, show that if MT has an eigenvalue λ ≠ 1 of
multiplicity k, then the same holds for Dx*g. Make sure that you take due consideration of the
possibility of non-trivial Jordan Normal Forms.
2.
Suppose that A = P -1BP. Show that
a) det (A) = det (B)
b) the eigenvalues of A are the same as those of B.
3.
Show that trace (BA) = trace (AB) for any two matrices A and B.
4.
Let A be a n×n matrix with an eigenvalue λ of multiplicity n. Suppose that A has only one linearly
independent eigenvector v1, and v2, …, vn are generalized eigenvectors obtained by solving
(A - λI)v2
=
v1
M
(A - λI)vi
=
vi-1
M
(A - λI)vn
=
vn-1
with vi ≠ 0 for i = 2, …, n. Let P = [v 1, v2, …, vn] be the matrix whose columns are v 1, v2, …, vn.
Show that P -1AP = B, where B is the n×n Jordan block
B
5.
=
λ
0

M

0

0
1 0 K
λ 1 L
M O M
0 L λ
0 L 0
0
0

M

1

λ
Find an example of two matrices A and B such that
e Ate Bt
≠
e (A+B)t
6.
Show that e Ate -At = Id for any matrix A.
7.
Show that e At = Id if A = 0.
C2 Exercise Sheet 4
8.
Consider the linear differential equation
d
dt
x
 
y
λ

0
=
1

λ
x
 
y
Show that x(t) satisfies
d -λt
(e x(t))
dt
=
y(0)
and hence that
x(t)
=
e λt(x(0) + y(0)t)
Use this to compute expBt where
B
9.
=
λ

0
1

λ
Recall that in Q12, Ex. Sheet 1, you solved the linear differential equation
d x
dt  y 
x
B 
y
=
using polar co-ordinates, where
B
=
a

b
−b 

a
Use this to compute expBt.
10. Compute expBt by direct summation of the series, where
B
=
λ

0
1

λ
Hint: consider the expression for Bn derived in lectures.
11. Compute expBt by direct summation of the series, where
B
=
a

b
−b 

a
Hint: consider
∞
∑ n! λ
1
n
(cos nϕ + i sin nϕ)tn
n=0
where a = λcos ϕ and b = λsin ϕ.
12. C2 May 1994 Q2i) and ii)
2