C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
EXAMPLE SHEET 5
Stars indicate difficulty of questions.
1.
Let v1, v2, …, vn be n vectors in Rn. Recall that we say that these are linearly independent if α1v1 +
α2v2 + … + α nvn = 0 implies α 1 = α 2 = … = α n = 0. Show that v1, v2, …, vn are linearly independent
if and only if the matrix P = [v1| v2| …| vn] is invertible (where [v 1| v 2| …| vn] is the matrix whose
columns are v1, v2, …, vn).
2.
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, …, (A - λI)vn= vn-1 , as in Q4, Ex. Sheet 4. Show that v1, …, vn are linearly independent.
3.
Let A be a symmetric n×n matrix, so that A † = A. Show that the eigenvectors corresponding to
different eigenvalues are orthogonal, ie if vi and v j are eigenvectors corresponding to λi and λ j
with λi ≠ λj, then vi†.vj = 0.
4.
_
Suppose that A is a 2×2 matrix with a complex conjugate pair of eigenvalues λ = α + iβ, λ = α - iβ,
with β ≠ 0. Let u + iv be the complex eigenvector corresponding to λ, so that
A(u+iv)
=
(α+iβ)(u+iv)
Let P = [u|v] be the matrix which conjugates A to JNF. By noting that e iϕ (u+iv) is also a (complex)
eigenvector of A, or otherwise, show that
P'
[cosϕ.u - sinϕ. v| sinϕ. u + cosϕ.v]
=
also conjugates A to JNF. Find a matrix R such that P' = PR. Relate this result to the extra
normalization condition required when finding a Hopf bifurcation numerically, as discussed in
the final C3 lecture.
5.
Consider the 2nd order linear equation
..
.
x + b x + ax
=
0
(1)
Suppose that λ is a double root of the polynomial y2 + by + ay. Show that a solution of (1) has
the form
x(t)
=
αeλt + βteλt
where α and β are determined by the initial conditions. Obtain a matrix P such that
 x(0)
˙ 
 x(0)
=
α 
P 
β
and hence compute α and β. Find a matrix A such that (1) is equivalent to
C2 Exercise Sheet 5
d
dt
x
 
y
2
x
A 
y
=
.
where y = x. Compute P -1AP and exp At. Deduce that (1) always corresponds to a non-trivial
Jordan block when it has a repeated eigenvalue.
6.
Consider the 2nd order linear equation
..
.
x + b x + ax
=
0
(2)
Suppose that the polynomial y 2 + by + α y has a complex conjugate pair of eigenvalues µ ± iω .
Show that a solution of (2) has the form
x(t)
(α cos ωt + α sin ωt)e µt
=
where α and β are determined by the initial conditions. Obtain a matrix P such that
 x(0)
˙ 
 x(0)
α 
P 
β
=
and hence compute α and β. Find a matrix A such that (2) is equivalent to
d
dt
x
 
y
x
A 
y
=
.
where y = x. Compute P -1AP and exp At.
7.
C2 May 1995, Q2i) and ii)
8.
C2 May 1994, Q6.
9.
C2 May 1995, Q6.
10. Let B be the non-trivial JNF
B
=
λ

0
1

λ
Show that curves of the form (ϕc(y),y) are invariant under the linear map xn+1 = Bxn, where
ϕc (y)
=
y log cy
λ log λ
11.* Let A = diag(λ,λ) and B be as in Q10. By mapping the line (αy,y) to the curve (ϕc (y),y) in Q10, or
other wise, show that A and B are topologically conjugate. Hint:
α
=
log c
λ log λ
.
.
12.** With A and B as in Q11, show that the flows given by the differential equations x = Ax and y = By
are topologically equivalent.
C2 Exercise Sheet 5
3
_
13.* Let B be the JNF corresponding to a complex conjugate pair of eigenvalues λ = α + iβ, λ = α - iβ,
with β ≠ 0:
B
 α β


 −β α 
=
Show that for an appropriate choice of ω and ρ , spirals of the form (rcosϕc(r), rsinϕc(r)) are
invariant under the linear map xn+1 = Bxn, where
ϕc (r)
c + ω
=
log r
log ρ
14.* Let B be as in Q13, and A = diag(ρ,ρ ) with ρ as in Q13. By mapping a straight line through the
origin to the curve in Q13, or other wise, show that the map
 x
h 
y
 x cosϕ(x , y ) − y sin ϕ(x , y )


 x sin ϕ(x , y ) + y cosϕ(x , y )
=
where
ϕ(x,y)
=
ω
log x 2 + y 2
log ρ
Show that this is the map given in lectures, but expressed in Cartesian co-ordinates.
.
.
15.** With A and B as in Q14, show that the flows given by the differential equations x = Ax and y = By
are topologically equivalent.