Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 5

Math 266: Chaos, Fractals, & Dynamical Systems—Assignment 5
University of Vermont, Spring 2016
Dispersed: Thursday, February 18, 2016.
Due: By start of lecture, 10:05am, Thursday, February 25, 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 f : R → R be a continuous, differentiable map. Assume 0 is a sink and (-1,1) is the
largest interval that lies entirely inside the basin of 0. The points 1 and -1 are not in the
basin of 0.
(a): What are the possible trajectories of the points -1 and 1?
(b): What are the possible Lyapunov numbers for each of -1, 0, 1?
2. Show that if the Lyapunov number of the orbit of x0 under the map f is L, then the
Lyapunov number of the orbit of x0 under the map f k is Lk , whether or not x0 is
periodic. Note that if the orbit of x0 under f is {x0 , x1 , x2 , ...} then the orbit under f k
is {x0 , xk , x2k , x3k , ...}. Hint: Use the chain rule.
3. Begin by sketching g(x) = 2.5x(1 − x) and considering cobweb plots of typical orbits.
(a) What are the possible bounded asymptotic behaviors for all x ∈ R?
(b) Find the Lyapunov exponent shared by most bounded orbits of g(x).
(c) Do all bounded orbits have the same Lyapunov exponents?
4. Begin by sketching g(x) = 2 − x2 and considering cobweb plots of typical orbits.
(a) Find a conjugacy C between G(x) = 4x(1 − x) and g.
(b) Use the conjugacy to find the fixed points and period-2 orbits of g (if they exist)
and determine their stability.
(c) Show that g has chaotic orbits.
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5. For this problem, I suggest you modify the henon_bifurcation_example.m file on the
course website rather than starting from scratch.
(a) Write a program to draw the bifurcation diagram for the logistic map
ga (x) = ax(1 − x) for values of the parameter a ∈ [2, 4] in increments of 0.001. Iterate
the map 1000 times starting with a random initial condition x0 ∈ (0, 1), you can use the
matlab command rand. Plot the bifurcation diagram.
(b) Use iterates 101 to 1000 to approximate the Lyapunov exponent for each a. Graph
the Lyapunov exponent as a function of the parameter a. In this graph, plot a horizontal
line at the origin (the constant function at 0) as a guide to your eye.
(c) What do you learn from comparing the two pictures?
(d) Why does the Lyapunov exponent appear to bounce off of zero several times before
becoming positive?
(e) Will a new random initial condition x0 produce the same figure? Why or why not?
Turn in your code, a single picture with 2 figures (the bifurcation diagram above and
the Lyapunov exponent below, both as a function of a), and answers to these questions.
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