Homework Problems: MAT335 - Chaos,
Fractals and Dynamics, Winter 2000
- (1) (A) Using Charkovsky's Theorem (see page 639), prove that if a
continuous function f from [0,1] to [0,1] has only finitely many
distinct periodic points, then the period of any one of them must be
a power of 2 (i.e., 2^m for some m > 0).
(B) State three periods other than odd ones that have the
following property : If a function like the one in part (A) has a periodic
orbit of that prime period, then the function has infinitely many
distinct periodic orbits.
- (2) You are observing a dynamical system as a parameter a
increases. Initially, you
observe a steady state (i.e., system doesn't change in time) until at
parameter value a = a_1 = 7 the system begins a periodic
oscillation. As a continues to increase you observe that
the periodic oscillations suddenly double their
period at parameter value a = a_2 = 10.
Assuming Universality, at what parameter value do you expect to observe
the onset of chaos in this system?
- (3) Consider the function f(z) = Az where z = (x,y) is
a two dimensional vector and A is a 2 by 2 matrix.
(A) Suppose A has entries
[A]_11 = a, [A]_12 = 0, [A]_21 = 1, and
[A]_22 = a where 0 < a < 1
(here [A]_ij denotes the element in the ith row and
jth column of the matrix).
Show that f takes lines through the origin to lines through the origin, i.e.,
if l is such a line, then so is f(l). Which lines through
the origin (if any) are invariant sets for f? What are their basins of
attraction?
(B) Suppose A has entries [A]_11 = a,
[A]_12 = 0, [A]_21 = 0, and [A]_22 = 1/a,
where 0< a < 1. Show that the orbits of f move along hyperbolae xy = k.
- (4) (A) Show that z + conj(z) = 2Re(z), and that z - conj(z) = 2iIm(z).
Here, conj(z) denotes the complex conjugate of the complex number z,
Re(z) and Im(z) denote the real and imaginary parts of z respectively,
and i^2 = -1.
(B) Recall the polar form of a complex number; z = r e^(it) where r
is the norm of z and t is the argument of z (angle). Show that if
z = r e^(it), w = s e^(ip), then z/w = r/s e^i(t-p).
(C) Show that 1/w = conj(w)/|w|^2 (here |w| denotes the norm of w).
- (5) (A) Consider the quadratic function q(z) = z^2. Sketch on the complex
plane the orbits of 1/2, (1/2)e^(i*pi), (1/2)e^(i*pi/4), and 2e^(i*pi/4).
(B) Sketch the orbits of 0, 3, i, -i, (2+i), (2-i) under iteration by the
function g(z) = (1/2)z + 1. What do you conclude about the dynamics of
this function?
- (6) Consider the quadratic function q(z) = z^2. Recall that the
circle of radius 1 is invariant under q(z). Show that periodic points of
q(z) are dense on the circle, i.e., let t_1 < t_2 be any two angles,
then there is a t, t_1 < t < t_2, such that the complex number
z = e^(it) is a periodic point of q(z). (Hint: z_n = q^n(z) = e^(i2^nt).
Find a t, t_1 < t < t_2, such that
2^nt = t + 2k*pi for some integers n and k.)
- (7) Verify that there are two fixed points of q_c(z) = z^2 + c for
c = -(1/2) + i(1/2) and that one fixed point is attracting
(stable) and the other is
repelling (unstable).