MAT 335 Homework #6
Due: Friday, April 11
Please hand in to the math office (SS 4072). Late penalalty: -15%
per day
Please hand in solutions to the problems that have a * . Note that some parts of multipart questions
may have a *, so look carefully for the questions to hand in.
You may work in groups of up to two.
Please check these questions regularly as sometimes errors are corrected and the choice of problems to
be handed in may be changed.
Last updated: April 1
Some useful formula
- (1)
Let A be a 2 by 2 diagonal
matrix with entries a and b (so [A]11 = a and [A]22 = b).
Sketch phase portraits of the two dimensional discrete dynamical systems F(z) in the following cases:
- a < -1 < b < 0 where F(z) = Az.
- a < -1 < 0 < b < 1 where F(z) = Az.
- 0 < a < b < 1 where F(z) = Az + (1,0). Hint: what is the
fixed point of F?
- (2*)
Consider the (nonlinear) two dimensional discrete dynamical system
defined by the function F(x,y) = r2*A(x,y), where r2
= (x2 + y2)
and A is the matrix that is rotation by pi/4 counterclockwise.
- (a) Write out the functions f1(x,y) and f2(x,y) where
F(x,y) = (f1(x,y), f2(x,y)) (you will need this for part (d)).
- (b) Sketch the phase portrait of F.
- (c) Are there any periodic points? What are the invariant sets of F
and their basins of attraction?
What is the basin of attraction of infinity? (i.e., which points have
orbits that tend to infinity?)
- (d) In what regions of the plane does F contract areas? Expand
areas? Is this what you would have expected by just looking at the
phase portrait of F? (To answer this question you need to calculate
the determinant of the Jacobian matrix of F.)
- (e) Find four sets D1... D4 that are invariant under F, i.e., such
that F(D1) in D1, F(D2) in D2, etc. What is the set
Ai = int_1^infinity Fn(Di) in each case i = 1,..4? (Here, int_1^infinity
Fn(D) means
the intersection of all the sets F1(D),F2(D), .....) What are the basins of
attraction of the sets A1, ..., A4?
- (3)
- Show that z + (z) = 2 * real part of z (here, (z) denotes the conjugate of z).
- Show that z - (z) = 2i * imaginary part of z.
- If z = r*eit and w = s*eiu where t,u are real numbers, show that
z/w = (r/s)*ei(t-u).
- Show that 1/w = (w)/|w|2.
- (4*)
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?
- (5)
Sketch the orbits of the points 1/2, 1/2*exp(i*pi),
2*exp(i*pi), and 2*exp(i*pi/4) under the quadratic function q0(z) = z2.
(Here, exp(x)
is the exponential function ex.)
- (6)
Show that for each n = 1,2,..., the points
z(n,m) = exp(itm) , m = 0,1,... 2n-1, where
tm = m*(2*pi)/(2n-1),
are periodic points of q0(z) of period n.
From this deduce that periodic
points of q0(z) "fill out" J0.
Note that all these periodic points
are repelling (i.e., unstable) periodic points
of q0(z). (If z = eit where t is an irrational
number, then the orbit of z is ergodic on J0, thus "most" points
on J0 are ergodic, but at the same time there are periodic points "everywhere". This is the same
for all Julia sets Jc.)
- (7)
- What is the Julia set of fc(z) = z+c for various (complex) c?
What is the Julia set of g(z) = zn, where n>2?
- Find some points in the escape set and some points in the prisoner
set for the function h(z) = z2 + z.
(z is a complex number)
- (8*)
Verify that there are two fixed points of qc(z) = z2 + c for
c = -(1/2) + i(1/2) and that one fixed point is attracting
(stable) and the other is
repelling (unstable).
- (9*)
Explain why W s(z2) is a subset of Pc, for c = -1/2 + 1/2*i. Here,
z2 is the stable fixed point of qc(z) for c = -1/2 + 1/2*i, and
W s(z2) is the stable set of z2.
- (10*)
Let c=0 and let W = W1 U W2 be the IFS for
(q0)-1(z) (as
described in Section 13.7).
- (a) Plot the points of the "chaos game" using the sequence {1,1,2,1,2}
starting with the points z0 = 9, z0 = 0.5i, and z0 = 1.
- (b) Plot the game points using the same initial points z0 given in (a)
but use the sequences {1,1,1,1,1} and {2,2,2,2,2}.
Where do the game points go? What are the fixed points of W1 and
W2?
- (c) Locate the addresses on J0 (via the IFS given above).
- (d) What image of J0 may result if a random sequence is not
used in this chaos game?
(11)
Explain why Qc^(-n) tends to Pc as n tends to infinity. In other words,
show that if z is in Pc, then z is in Qc^(-n) for all n, and
conversely, if z is in Qc^(-n) for all n, then z is
Pc. Here
Qc^(-n) is as defined in the text on page 796.
(12)
Show that P_c and E_c are invariant sets of q_c(z).
(13)
Consider the quadratic function q(z) = z2. 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 t1 < t2 be any two angles,
then there is a t, t1 < t < t2, such that the complex number
z = eit is a periodic point of q(z). (Hint: zn = qn(z) =
ei(2^n)t.
Find a t, t1 < t < t2, such that
2nt = t + 2k*pi for some integers n and k.)
(14*)
Let (c) denote the complex conjugate of c. What is the relation between J(c)
and Jc?
(15)
Show that the prisoner set of the Cantor tent function,
TC(x) is the Cantor set. (Here,
TC(x) = 3x if x is less than 1/2, and = 3-3x if x
is greater than 1/2. See pages 826-829.)
(16*)
Why is the Mandelbrot set M symmetrical with respect to
the (real) x-axis? That is, why is (M) = M? (Here, like
in the questions above, (z) denotes the complex conjugate of z.)
Hint: Use an appropriate definition of M.
(17)
Consider Figure 14.27 in the text. Explain the relationships between
the two objects in the figure (the final state diagram of the real
quadratic function x^2 +c
and the Mandelbrot set).
(18)
What do the Julia sets look like for those values of c where
the periodic orbit is parabolic (indifferent, neutral)? Where are these
c-values located on the Mandelbrot set?
(19)
Suppose Pc has zero area. Is c necessarily in the
Mandelbrot set M?