Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 10

Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 10
University of Vermont, Spring 2016
Dispersed: Thursday, April 7, 2016.
Due: By start of lecture, 10:05am, Thursday, April 14, 2016.
Some useful reminders:
Instructor: Chris Danforth
Office: 218 Farrell Hall, Trinity Campus
Twitter: @nonperiodicflow, #math266
E-mail: [email protected]
Office hours: Check Twitter
Course website: http://www.uvm.edu/~cdanfort/main/266.html
Instructions: Unless otherwise noted, use your brain and pencil to solve problems before
checking with Matlab. Graduate students (required) and those planning to go to graduate
school in a mathematical science (encouraged) should turn in their solutions in LATEX; you will
need to learn this language eventually. Check the course website for sample m-files.
Grading: All questions are worth 3 points unless marked otherwise (3 = perfect or nearly so, 2
= close but something missing, 1 = not close but a reasonable attempt, 0 = way off).
Excellent solutions will be returned with the graded HW.
Disclosure: Please show all your working clearly and list the names of other students with
whom you collaborated.
1. Challenge 5 Step 1: Assume that B(x0 ) differs from x0 by less (or equal to) d in each
coordinate. In Figure 5.20 we draw a rectangle centered at x0 with dimensions 3d in the
horizontal direction and 2d in the vertical direction (see below). Assume that the
rectangle lies near the bottom left of the unit square, nowhere near the line y = 1/2, so
that it is not chopped in two by the map. Then its image is the rectangle shown; the
center of the rectangle is of course B(x0 ), show that the image of the rectangle is
guaranteed to “map across” the original rectangle. Explain why there is a fixed point of
B in the overlapping region, within 2d of x0 .
B(So)
3d
S0
B(xo)
x0
2d
1
Hint: Show that the western and southern boundaries of B(S0 ) must overlap with
those of S0 , and that the northern (and eastern) boundaries of B(S0 ) map above
(inside) those of S0 . Note: For the Skinny Baker Map’s two fixed points, two of the
four edges of S and B(S) overlap. As such, the picture in the text is a bit misleading.
2. Challenge 5 Step 2: Now suppose our computer makes mistakes in evaluating B of size
at most 10−6 , and it tells us that B(x0 ) and x0 are equal to within 10−6 . Prove that B
has a fixed point within 10−5 of x0 .
3. Challenge 5 Step 3: Prove Theorem 5.19. Let B denote the Skinny Baker Map and let
d > 0. Assume that there is a set of points {x0 , x1 , · · · , xk−1 , xk = x0 } such that each
coordinate of B(xi ) and xi+1 differ by less than d for i = 0, 1, · · · , k − 1. Then there is
a periodic orbit {z0 , z1 , · · · , zk−1 , zk = z0 } such that |xi − zi | < 2d for
i = 0, 1, · · · , k − 1. Note: The sequence of points {xi } is not an orbit of B. Hint:
Draw a 3d × 2d rectangle Si centered at each xi as in Figure 5.21. Show that B(Si ) lies
across Si+1 by drawing a variant of Figure 5.20, and use Corollary 5.13.
4. Challenge 5 Step 4: Let f be a continuous map, and assume that there is a set of
rectangles S0 , S1 , · · · , Sk such that f (Si ) lies across Si+1 for i = 0, 1, · · · , k − 1, each
with the same orientation. Prove that there is a point x0 in S0 such that f i (x0 ) lies in
the rectangle Si for all i ∈ {0, 1, · · · , k − 1}. By the way, does k have to be finite?
5. Challenge 5 Step 7: Assume that a plot of length one million of the cat map is made on
a computer screen, and that the computer is capable of calculating an iteration of the
cat map accurately within 10−6 . Do you believe that the dots plotted represent a true
orbit of the map (to within the pixels of the screen)?
2