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C2 Fundamental Theory of Dynamical Systems
MSc Examination, May 2000
2 hours.
You are advised to attempt FOUR questions, though credit will given for all material handed in.
To pass, you will need to correctly answer the equivalent of about two questions, whilst for a
distinction you will need to answer the equivalent of four questions.
1.
Consider the non-autonomous differential equation
dxt
dt
=
xt∈Rn
F(xt ,t)
Show how this can be transformed into an autonomous equation. Suppose the time dependence is such that for some T > 0 we have
F(xt , t+T)
=
-F(xt ,t)
Describe the resulting phase space of the autonomous equation, and explain how the dynamics
of this equation can be reduced to a map g : R n → Rn and explain the relationship between the
periodic orbits of g and of the original differential equation. Deduce that all periodic orbits of
the differential equation must have a period which is a multiple of 2T.
Now suppose that F is of the form F(x,t) = A(t).x(t) where A(t) is a n×n matrix such that A(t+T) =
–A(t). Let
Ψ(t0,t)
=
t
∫ A(s)ds
t0
Show that the solution of the above equation with initial condition x(t0) at t = t0 is given by x(t) =
exp[Ψ(t0,t)].x(t0) and hence deduce that the time-2T map is given by the identity g(x) = x.
2.
Describe the possible Jordan Normal Forms (JNF) of a 2×2 matrix and briefly indicate how to
compute the exponential of the matrix in each case. Find the JNF of the matrix
 −α

 1
−4 

−α 
where α∈R is a parameter. Hence, or otherwise, compute the general solution of the differential
equation
ẋ
ẏ
=
=
-αx - 4y
x - αy
Sketch the phase portrait for α = 1 and α = -1.
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3.
Suppose that f : [0,1] → [0,1] is given by
f(x)
=
3x


3x − 1

3x − 2

0≤x < 1
3
1 ≤x<2
3
3
2 ≤ x ≤1
3
Explain how the dynamics of f can be understood in terms of sequences of symbols. Hence or
otherwise determine how many orbits of minimal period 4 does f have and show that f has a
dense orbit.
Now consider the map g : [0,1]×[0,1] → [0,1]×[0,1]
g(x,y)
=
(3x , 1 y )

5

(
3
x
−
1, 1 y + 2 )

5
5

(3x − 2, 1 y + 4 )
5
5

0≤x < 1
3
1 ≤x<2
3
3
2 ≤ x ≤1
3
Sketch the image of the square under the first few iterates of g, and hence or otherwise, describe
an attracting invariant set for g. How many periodic orbits of minimal period 4 does g have, and
what are the eigenvalues of each.
4.
Define a periodic point, a closed invariant set and a dense orbit for a homeomorphism f : Rn →
Rn. Show that if an invariant set X consists of a finite number of points, then they must all be periodic. Does such an invariant set necessarily have to contain a dense orbit?
Suppose that a closed invariant set X contains no proper closed invariant subsets, ie if Y ⊂ X and
Y is a closed invariant set then either Y = ∅ or Y = X. Show that if X contains an infinite number
of points, then it cannot contain any periodic orbits.
5.
Define the terms stability and asymptotic stability for a periodic orbit of a map f : Rn → R n. Derive an equation describing the linearized dynamics about such a periodic orbit and show that if
all the eigenvalues λ of the periodic orbit satisfy |λ| < 1, then the orbit is asymptotically stable
(you may restrict yourself to the case where all the eigenvalues are real and distinct).
In the case n = 1, show that if a periodic orbit has an eigenvalue λ > 1, then it cannot be stable.
Give an example of an unstable periodic orbit for which the linearized dynamics are stable.
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6.
Define the term topological conjugacy and show that the only map conjugate to the identity on
Rn is the identity map itself. Suppose that h : Rn → Rn is a topological conjugacy between f and g,
where f,g : Rn → Rn. Show that h(ω(y )) = ω(h(y)), where ω(y) is the ω-limit set of y.
Suppose that f,g : R → R are given by
f(x)
=
g(x)
=
1
2
x + x2
3x + x3
Show that f cannot be locally topologically conjugate to g in a neighbourhood of 0. State the
Hartman-Großman theorem for a fixed point of a map f : Rn → R n and explain its significance.
Deduce that f and g -1 are locally topologically conjugate in a neighbourhood of 0.
7.
Define hyperbolicity and local structural stability for a fixed point and a periodic orbit of a map f
: Rn → Rn.
Suppose that fλ is a one parameter family of maps f : R n → R n such that x 0 is a fixed point of f0
and D x f 0 has no eigenvalue equal to 1 then for all sufficiently small λ , the map fλ has a fixed
point close to x0. Give an example for which this holds but which is not structurally stable.
0
Suppose f : R2 → R2 is given by f (x) = A.x where A is the matrix
A
=
λ

0
1

λ
with λ > 0. Sketch a proof that the origin is locally structurally stable as long as λ ≠ 0,1. You may
assume that under suitable conditions (which should be clearly stated) there exist topological
conjugacies between the following pairs of linear maps
λ

0
1

λ
and
λ

0
0

λ
 ξ 0


0 ζ 
and
λ

0
0

λ
 α β


 −β α 
and
α 0 


 0 α
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