Jim Belk Cornell University

Takehome Midterm

Due Date: Monday, October 26

Rules: This is a midterm exam, not a homework assignment. You must solve the problems entirely on your own, and you should not discuss the problems with any other students in the class, or with anyone on the internet. Most of these problems require the use of Mathematica, and you should feel free to use Mathematica's online help system, as well as other Mathematica resources available on the web.

Note: For all questions, please make sure to fully explain the method you used to find the answer, and include any Mathematica code that was helpful. Feel free to use multiple notebooks to organize your work, but make sure to send me all of your notebooks.

    1. Use NDSolve and ParametricPlot to plot the first 50 seconds of the orbit of the point \((1,1)\) under the following dynamical system:

      \(\displaystyle\frac{dx}{dt} \;=\; -0.1x -y\),\(\displaystyle \frac{dy}{dt} \;=\; -0.1y+x\).

    2. Create a Manipulate that plots the first 50 seconds of an orbit of this dynamical system in the range \(-2\leq x \leq 2\) and \(-2\leq y \leq 2\), using a Locator to specify the initial state.
    3. Describe the dynamics of this system. Are there any attractors? Classify each attractor as a fixed point, a periodic cycle, or something else.
    4. Repeat parts (b) and (c) for the system

      \(\displaystyle\frac{dx}{dt} \;=\; x + y -x^3\),\(\displaystyle \frac{dy}{dt} \;=\; 0.5\bigl(1-x^2\bigr) -y\).

    5. Repeat parts (b) and (c) for the system

      \(\displaystyle\frac{dx}{dt} \;=\; 0.1\bigl(1-x^2-y^2\bigr)x-y\),\(\displaystyle \frac{dy}{dt} \;=\; 0.1\bigl(1-x^2-y^2\bigr)y + x\).

  2. Let \(f(x) = (1+c)x^3 - cx\), where \(c\) is a constant between 0 and 3.
    1. Find any value of \(c\) for which the function \(f(x)\) has an attracting 4-cycle.
    2. Find any value of \(c\) for which the function \(f(x)\) has an attracting 3-cycle.
  3. Let \(T\) be the tent map: \[ T(x) \;=\; \begin{cases}2x & \text{if }0\leq x \leq 1/2,\\[3pt] 2 - 2x & \text{if }1/2 < x \leq 1.\end{cases} \] Note: The following three questions do not depend on one another, so you can work on them in any order. For example, it's possible to solve part (c) without solving parts (a) or (b).
    1. Find a point \(p\) such that \(0.4 < p < 0.4001\) and \(T^{14}(p)=0\). Express your answer as a fraction in lowest terms.
    2. The point \(\dfrac{20}{3433}\) is periodic under \(T\). What is its period?
    3. Find a point \(q\) in the orbit of \(\dfrac{20}{3433}\) such that \(0.7 < q < 0.701 \). Express your answer as a fraction in lowest terms.
  4. The notebook MidtermOrbit.nb contains a list of the first 1000 points in the orbit of 0.47 under a certain function \(f(x)\). Find the formula for \(f(x)\).
  5. Let \(f(x)\) be the following function on the interval \([0,2]\). \[ f(x) \;=\; \begin{cases}x+1 & \text{if } 0\leq x \leq 1, \\[3pt] 4-2x & \text{if } 1 < x \leq 2.\end{cases} \] For the following questions, express your answers as fractions in lowest terms.
    1. Find a point \(p\in[0,2]\) whose orbit under \(f\) has Lyapunov number \(2\).
    2. Find a point \(q\in[0,2]\) whose orbit under \(f\) has Lyapunov number \(\sqrt{2}\).
    3. Find a point \(r\in[0,2]\) whose orbit under \(f\) has Lyapunov number \(\sqrt[5]{8}\).