C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
EXAMPLE SHEET 2
Stars indicate difficulty of questions.
1.** Consider the differential equation
dx
dt
=
x - y + αx(x2 + y2)
dy
dt
=
x + y + αy(x2 + y2)
Let Σ = { (x,y) : x > 0, y = 0 }, verify that this is a valid Poincaré section and compute the return
map (hint: polar co-ordinates, see Q12, Sheet 1).
2.
Consider the map of R 2 given by F(x,y) = (f(x),g(y)), where f and g are two maps on R. Suppose
that x is a periodic orbit of f of minimal period p and y is a periodic orbit of g of minimal period
q. Show that (x,y) is a periodic orbit of F and calculate its minimal period.
3.* Suppose f : [0,1] → [0,1] is given by
f(x)
=
2x

2(1 − x )
0≤x ≤
1
2
1
2
≤ x ≤1
By considering the binary expansion of x investigate the dynamics of f. In particular show that f
has periodic orbit of any period, and that it has a dense orbit.
4.* Find all the period 3 and period 4 orbits of the map in Q3. How many orbits are there of least
period 6?
5.
6.
1
Let g be the logistic map g(x) = 4x(1 - x) and define ϕ(x) = sin2 (2πx), for x∈[0,1]. Show that ϕ is
invertible and ϕ°f = g°ϕ , where f is as in Q3. Hence deduce that g has a dense orbit. How many
orbits of least period 3 does it have?
Investigate the dynamics of the map f in Q3, g in Q5 and of h(x) = 2x mod 1 (as described in the
lectures) on a computer. Do the results agree with our theoretical analysis. If not, why not?
7.* Let f be the rigid rotation of the unit circle S 1 given by f( θ ) = θ + ω (mod 1). Show that if ω is
irrational, then every orbit of f is dense in S 1. What if ω is rational?
C2 Exercise Sheet 2
8.
2
Consider the differential equation (recall Q1)
dx
dt
=
x - y - x(x2 + y2)
dy
dt
=
x + y - y(x2 + y2)
Let f be the time τ map of this flow. Find a nontrivial invariant set of f (i.e. an invariant set
which is not just a single point). For what values of τ does this invariant set have a dense orbit?
9.
Let f be the map of the two dimensional torus S1×S1 given by
θ 
f 
ϕ 
=
θ + ω

ϕ + µ
(mod 1)

(mod 1)
Find conditions on ω, µ for every orbit of f to be dense (see Q7).
10.* C2 May 1995, Q1.
11.* C2 May 1995, Q4.
12. C2 May 1994, Q1i).
13. C2 May 1994, Q1ii).
14*. Let Γ be a periodic orbit of (minimal) period q of a (continuous) map f : Rn → Rn. Show that the
two definitions of a stability given in lectures are equivalent. Thus show that x∈Γ is stable as a
fixed point of fq if and only if given ε > 0, there exists a δ > 0 such that fn(Nδ(Γ)) ⊂ Nε (Γ) for all n >
0. Deduce that if Γ is stable then for any ε > 0, there exists a δ > 0 such that if y∈Nδ(x0) for some
x0∈Γ then fn(y)∈Bε(xp) where n = kq + p, for 0 ≤ p < q.
15*. Repeat Q14 for asymptotic stability instead of stability.
16. Let Γ be a periodic orbit of (minimal) period q of a (continuous) map f : R n → Rn. Show that the
definition of stability/asymptotic stability in terms of a fixed point of fq is independent of the
choice of x∈Γ (hint: use Q14 and Q15).
17.** Show that the two definitions of a stable periodic orbit of a differential equation given in
lectures are equivalent. Thus, let Γ be a periodic orbit of a differential equation on Rn, let Σ be a
Poincaré section such that Γ ∩ Σ is a single point x*. Let g be the corresponding first return map;
thus x* is a fixed point of g. Show that x* is stable as a fixed point of g if and only if given ε > 0,
there exists a δ > 0 such that f t(Nδ(Γ)) ⊂ N ε (Γ) for all t ≥ 0. Deduce that the definition of stability
in terms of a fixed point of g is independent of the choice of Poincaré section Σ.