Q1: Suppose that x, y evolve according to the ODE

d

dtx = α(y − x)

d

dty = f(y) − x

(1)

where f(y) is a specified smooth function and α > 0 is a real parameter.

(a) Find all equilibria for this ODE and determine the Jacobian at an equilibrium.

(b) Show that if f is odd then there is an equilibrium at x = y = 0. Find the stability of this

equilibrium in terms of f. Hence find an example such that (1) has a saddle at x = y = 0

and sketch a phase portrait in this case.

(c) Find an example of f(y) such that (1) has precisely two equilibria. Show that in such a case

it is never possible for both of these equilibria to be hyperbolic saddles.

(d) Show that it is possible to re-write (1) as a second order ODE of with a single dependent

variable. Verify this has the same number of equilibria as (1).

Q2: Which of the following statements are always true? For each statement, give a short justification if true, give a counterexample if not.

(a) If the real n×n matrix A with n ≤ 3 is such that all entries of exp(At) are periodic in t, then

all entries must have a common period.

(b) If the real n×n matrix A with n ≥ 4 is such that all entries of exp(At) are periodic in t, then

all entries must have a common period.

(c) If ρ is the largest real part of an eigenvalue of the real n×n matrix A then there is an M > 0

such that | exp(At)y0| ≤ Meρt|y0| for all y0 and t > 0.

(d) If all eigenvalues of a real n×n matrix A have negative real parts then all entries of exp(At)y0

limit to zero monotonically as t → ∞ Q3: Suppose that x ∈ R satisfies

d

dtx = f(x, a, b, c) := x

4 − ax2 + bx + c

where a, b and c are real constants.

(a) For b = 0, find all equilibria of this system in terms of c and a.

(b) For fixed a and b = 0, sketch a bifurcation diagram of x against c in the cases a > 0 and

a < 0.

(c) Show that if a < 0 then for any b and c there are at most two equilibria.

(d) For any a > 0, show that there are two choices of (x, b, c) such that f(x, a, b, c) = f

0

(x, a, b, c) =

f

00(x, a, b, c) = 0, where f

0 denotes derivative with respect to x. Find an expression (independent of a and x) for the curve where this occurs in the (b, c) plane.

Q4: Let T : [0, 1) → [1, 0) be the “tripling map” T(x) = 3x mod 1. You may find it helpful to

write points in ternary expansion x =

P∞

k=1

ak

3

k where ak ∈ {0, 1, 2}.

(a) Sketch T(x): show that there are two fixed points in [0, 1) that you should find. Show that

all periodic points of T are linearly unstable.

(b) Explicitly find all points for this map with minimal period 2. Show that every periodic orbit

of T is at a rational value of x.

(c) Now define h : [0, 1) → [−1, 1] by h(x) = cos(2πx) and let yn := h(xn). Show that yn+1 =

G(yn) where G : [−1, 1] → [−1, 1] is a map that you should explicitly find, sketch and show

that this is continuous. Show that h is not invertible but has at most two preimages.

(d) Show that for any periodic orbit y of G there is an x with y = h(x) where x is a periodic orbit

of T. Are the minimal periods of x and y always the same? Justify or give a counterexample.

(e) Verify that G has a horseshoe.