Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 4

Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 4
University of Vermont, Spring 2016
Dispersed: Thursday, February 11, 2016.
Due: By start of lecture, 10:05am, Thursday, February 18, 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. Let G(x) = 4x(1 − x). Prove that for each positive integer k, there is an orbit of period
k.
"
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a b
2. For a general matrix M =
, what are the conditions on a, b, c and d which
c d
make the two-dimensional map M v (mod 1) continuous? Hint: prove the converse of
Step 1, page 93.
3. Construct a periodic table (up to period 6) for the cat map. To do so, you will need to
look at Steps 6, 7, and 9 on pages 95, 96 of the book.
4. Consider the two-dimensional map given by
f~(x, y) = (2x + y, a − y 2 )
(a) Solve for all fixed points of f~. For what range of the parameter a do (real) fixed
points exist?
(b) Fix a = 0 and for each of the fixed points from part (a), determine the stability (i.e.
is it a sink, source, saddle, something else?)
(c) Find the critical value of a above which the fixed points are of the same type.
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5. Use the code henon_iteration.m on the course website to recreate figure 2.17. In
addition, find and plot a periodic orbit higher than period-16 (panel b). To do so, you’ll
need to increase the number of iterates performed by the code.
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