Homework #3: MAT335 - Winter, 2001
- (1) What are the parities of the following sequences;
.11101 and .01100 ?
- (2) Find a periodic point of T that lies in
the interval [9/32, 10/32]. (Hint: the binary
expansion of 9/32 is .01001). What is the smallest period such a
point can have? The longest period?
- (3) Use the Shadowing Lemma to justify the
following two statements;
-
The observed instability of periodic points of f(x)=4x(1-x)
(i.e., the computer experiments) implies the instability
of the exact (or true)
periodic points of f(x).
- If f(x) has an observed ergodic orbit, then f(x) has
an exact ergodic orbit.
- (4) Draw the bifurcation diagram and the final state diagram
of g_a(x) = x^3 + a. Do not try to solve the equation g_a(x) = x
to find the fixed points; instead, rely on your sketches of the graphs
of g_a(x) for various a to sketch the bifurcation diagram.
You will, however, need to determine the stability of the periodic
points of g_a(x), and to do this you will have to consider the
derivative g'_a(x) at the periodic points (which you do not know
exactly, but you can still figure out the stability).
- (5) Use the applet "Logistic Movie" to determine
the period-doubling points b_1 (period 6), b_2 (period 12) and
b_3 (period 24) in the period 3 window near a = 3.83 (see Figure 11.41).
Use these values to estimate the Feigenbaum constant 'delta'
(see pages 611 and 636) for
this period-doubling scenario. (You will have to estimate the b_i
to 4 decimal places.)
- (6) A dynamical system depends on a parameter
a. Initially, you observe a steady state (i.e., a period 1
orbit). As a increases you observe a period 2 oscillation
appearing at a = a_1 = 7.
Then at a = a_2 = 10 you observe
that the period 2 orbits splits into a period 4 orbit.
As a continues to increase
a series of period-doublings occurs. Assuming Universality,
at what a value would you expect to observe the onset
of chaos?
- (7) Answer the following questions about the final state
diagram of the logistic equation (see for example, Figure 11.5 in the
text).
- a) Explain the upper and lower boundaries
in the band region a greater than 3.57.
- b) Explain the two bands at a = 3.65. First look at orbits of f^2
with the applet "Graphical Iteration" and find two 'invariant boxes'.
Show that all orbits (i.e., all initial points) eventually stay in one
of the two boxes. However, when you look at one of these orbits (i.e., one
of these initial points) under iteration by f (use the applet), the
orbit fills out both bands (note that the 'gap' between the
two bands contains the (unstable) fixed point of f).
- c) Why is there only one band for a greater than 3.68?
(Think of those 'invariant boxes' of f^2.)
- d) Explain why when the curve f^3_a(v_a) meets the curve v_a
(that is, for what a value; see Figure 11.39) a period 3 window
appears in the final state diagram (see Figure 11.5). (Note that the
third curve down in Figure 11.39 (top) is the curve f^3(v_a), not
f^2(v_a) like I wrote on the handout.)
Could you predict the locations of other
period windows based on the curves f^k(v_a)?
- (8) ** New version **
Suppose f(x) is a function that has a fixed point
p and a period two orbit {q1, q2} where p is
between the two points q1 and q2.
Show that if p is an unstable fixed point and if
{q1, q2} is an unstable period 2 orbit, then f(x) must have
either a stable or unstable period 2 orbit
inside the interval [q1, q2].
Also show that if p is a stable fixed point
and if {q1, q2} is a stable period 2 orbit, then
f(x) must have either an unstable or a stable period 2 orbit inside
the interval [q1, q2].
(Hint: Sketch the graphs of f(x) and f^2(x).
Note also that
(f^2)'(p) = (f '(p))^2 and (f^2)'(q_1) = f '(q1)*f '(q2) = (f^2)'(q2),
by the Chain Rule - you need this to translate the stability of the
periodic points into the geometry of the graphs of f and f^2.)
- (9) Write down the relative ordering of the following
numbers using Charkovsky's ordering of the integers: 56, 31, 128, 160.
That is, which one of these numbers is the 'smallest', the next
'smallest', ... , the 'largest' according to Charkovsky's ordering?
- (10) Prove that if f(x) has an orbit of prime period 2^m,
then f(x) has an orbit of period 2^k for all k = 0,...,m.
Prove this without using Charkovsky's Theorem (just sketch
the graph of f^k(x)) (i.e., use graphical analysis).
- (11) Use Charkovsky's Theorem to prove that if f(x) is
a continuous function from [0,1] to [0,1] that has only finitely
many distinct periodic points, then the period of any one of
them must be 2^m for some m.
- (12) Consider the two dimensional discrete dynamical system
defined by the function F(x,y) = r^2*A(x,y), where r^2 = (x^2 + y^2)
and A is the matrix that is rotation by pi/4 counterclockwise.
- (a) Sketch the phase portrait of F.
- (b) 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?)
- (c) 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.)
- (d) Find two sets D1 and D2 that are invariant under F, i.e., such
that F(D1) in D1, and F(D2) in D2. What is the set
Ai = int_1^infinity F^n(Di) in each case? (Here, int_a^b F^n(D) means
the intersection of the sets F^a(D),...,F^b(D).) What are the basins of
attraction of the sets A1 and A2?
- (13) Sketch the orbits of the points 1/2, 1/2*exp(i*pi),
2*exp(i*pi) under the quadratic function q_0(z) = z^2. (Here, exp(x)
is the exponential function e^x.)
- (14)
- What is the Julia set of f_c(z) = z+c for various (complex) c?
What is the Julia set of g(z) = z^n, where n>2?
- Find some points in the escape set and some points in the prisoner
set for the function h(z) = z^2 + z.
(z is a complex number)
- (15) Explain why W^s(z_2) is a subset of P_c, for c = -1/2 + 1/2*i. Here,
z_2 is the stable fixed point of q_c(z) for c = -1/2 + 1/2*i, and
W^s(z_2) is the stable set of z_2.
- (16) Show that the prisoner set of the Cantor tent function,
T(x) = 3x if x less than 1/2, 3-3x if x greater than 1/2, is the
Cantor set. (See pages 826-829.)
- (17) Explain why Q_c^(-n) --> P_c as n --> infinity. In other words,
show that if z is in P_c, then z is in Q_c^(-n) for all n, and
conversely, if z is in P_c then z is in Q_c^(-n) for all n. Here
Q_c^(-n) is as defined in the text on page 796.
- (18) Show that P_c and E_c are invariant sets of q_c(z).
- (19) Show that for each n = 1,2,..., the points
z_(n,m) = exp(it_m) , m = 0,1,... (2^n)-1, where t_m = m*(2*pi)/((2^n)-1),
are periodic points of q_0(z) of period n ( note the correction
2n --> 2^n in the formulae!). From this deduce that periodic
points of q_0(z) "fill out" J_0. Note that all these periodic points
are repelling (i.e., unstable) periodic points
of q_0(z).
- (20) Let c=0 and let W = W_1 U W_2 be the IFS for q_0(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 z_0 = 9, z_0 = 0.5i, and z_0 = 1.
- (b) Plot the game points using the same initial points z_0 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 W_1 and
W_2?
- (c) Locate the addresses on J_0 (via the IFS given above).
- (d) What image of J_0 may result if a random sequence is not
used in this chaos game?
- Bonus homework problem Can you show
geometrically, i.e., by graphical analysis,
that if there is a (prime) period 3 orbit of a function f(x), then there
must be an orbit of (prime) period 2?
End of homework #3 (but see below).
The following problems are suggested problems for
you to work on.
- Let (c) be the complex conjugate of c. Is J_(c) = J_c?
- Why is the Mandelbrot set M symmetrical with respect to
the (real) x-axis? That is, why is (M) = M? (Here, like
in the question above, (z) denotes the complex conjugate of z.)
Hint: Use an appropriate definition of M.
- 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).
- Explain the shapes of the main bodies of the "Mandelbrot" sets
of the functions z^3 +c and z^4 + c (cf. page 8 of the handout).
Also explain the symmetries of the sets.
- Suppose q_c has a stable period 3 orbit. Make a sketch of what
the Julia set J_c looks like in general. Explain the shape of J_c (or P_c)
in terms of the stable set ( = basin of attraction) of the perioodic orbit.
(See April 9 notes and Figure 14.18).
- What to 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?
- Derive the formula for the threshold radius r(c) of q_c.
- Derive the formulae describing the main cardioid region of the
Mandelbrot set and the period 2 bulbs (including an explanation of
why we are considering stable periodic orbits of q_c).
- In what sense does the Mandelbrot set contain copies of Julia sets?
- Explain how one could use the chaos game to draw Julia sets. Is this
really a chaos game?
- Explain why the Julia set is a fixed point of an "IFS". Begin by
defining the "IFS".
- Sketch the encirclements Q_o^(-n) for the Julia set q_o(z) = z^2.
- Suppose P_c has zero area. Is c necessarily in the
Mandelbrot set M or is c necessarily not in
M, or could c be either in M or not
in M?
- Calculate the Jacobian matrix of the transformation W(w) = sqrt(w-c)
(i.e., one of the inverse transformations of q_c(z) = z^2 + c = w) and
determine the region in the complex plane where W contracts area.
Correction: It is easier to calculate the determinant of
the Jacobian matrix of q_c(z) and determine the regions where q_c
expands area. Then the regions where W(w) contracts areas would
be those (since W(w) is an inverse of q_c). To do this calculation
write q_c(z) as a two-dimensional function F(x,y) where z = x + iy.