Jim Belk University of Glasgow

Homework 5

Due Date: Friday, October 9

  1. Let \(\mathbf{f}(x,y) \;=\; \bigl(0.4\hspace{0.1em}x^2+y,\) \(0.5(x+1)y\bigr)\).
    1. Find the three fixed points of \(\mathbf{f}\).
    2. Use the eigenvalues of the Jacobian to classify each fixed point as a source, a sink, or a saddle.
    3. Make a plot of the first \(50\) points in the orbit of \((2.45,\hspace{0.1em}0.015)\) under \(\mathbf{f}\). Briefly describe how this orbit is affected by each of the three fixed points.
  2. Let \(\mathbf{f}(x,y) \;=\; \bigl(0.9x,\) \(-0.29x^2+1.1y\bigr)\), and note that \(\mathbf{f}\) has a fixed point at \((0,0)\).
    1. Set up a Manipulate with a Locator to investigate the dynamics of \(\mathbf{f}\) in the range \(-1.5\leq x \leq 1.5\) and \(-1\leq y \leq 1\).
    2. Use the Jacobian \(D\mathbf{f}(0,0)\) to show that the fixed point at \((0,0)\) is a saddle.
    3. Find the stable and unstable eigenvectors of \(D\mathbf{f}(0,0)\).
    4. Based on your Manipulate, what is the unstable manifold for \((0,0)\)?
    5. Use your Manipulate to guess the equation for the stable manifold for \((0,0)\).
    6. Show algebraically that the curve you guessed in part (e) is invariant under the function \(\mathbf{f}\).
  3. Let \(\mathbf{f}(x,y) \;=\; \bigl(x+0.05y,\) \(0.99y-0.05\sin x\bigr)\).
    1. Use Manipulate with a Locator to investigate the dynamics of this function in the range \(-2\leq x \leq 8\) and \(-2 \leq y \leq 2\). How many fixed points does the function have in this range? Do they seem to be sources, sinks, or saddles?
    2. Find the precise locations of each of the fixed points you observed in part (a).
    3. Use the Jacobian to classify each fixed point as a source, a sink, or a saddle.
    4. Compute an eigenvector \(\mathbf{v}\) of \(D\mathbf{f}(\mathbf{p})\) in the unstable direction, where \(\mathbf{p}\) is the saddle point.
    5. Use the following procedure to draw the unstable manifold for \(\mathbf{p}\):
      • Make a table of at least 100 different initial points on the line segment from \(\mathbf{p} - 0.01\mathbf{v}\) to \(\mathbf{p} + 0.01\mathbf{v}\).
      • For each of these initial points, compute at least 1000 points in the corresponding orbit under \(\mathbf{f}\).
      • Use ListPlot to draw all of these orbits together on a single graph. Use tiny points and a larger image size to increase the quality of the plot.