Mathematics 345
Homework Assignment 7
Due Tuesday 5 April 2016
1. Consider the cubic map xn+1 = f (xn ) = axn − x3n for real-valued parameter a. This is the map that
we studied in Question 4 of the previous assignment. The attached file hw7-1.ode is a file that helps
calculate the Liapunov exponents of the logistic map over a range of parameter values. Modify this
file to make it calculate the Liapunov exponents for the cubic map (do not touch the “ ’ ” symbol
in the file since the one on your keyboard may not work. Just replace the right-hand-side of the
equations.) Use the file to calculate Liapunov exponents for the cubic map for a ∈ [−1, 3]. Run
“Initialconds” and ”(R)ange”, say ok. Use the “(E)dit curve” to change the colour and to connect the
points to a continuous line (line type=1). Do not forget to change amin, amax, xlo, xhi, ylo, yhi
accordingly. Hand-in a copy of your calculation.
2. Get the file hw7-2a.ode from the course web page. This file is for the three-dimensional Rössler
system of ordinary differential equations
ẋ = −y − z,
ẏ = x + ay,
ż = bx − cz + xz,
(1)
where a, b, c are parameters. Start XPPAUT with the file.
(a) The file hw7-2b.ode sets up the main window to show a three-dimensional plot, and specifies
default parameter values a = 0.36, b = 0.4, c = 4.5 and initial conditions x(0) = 0, y(0) = −4.4,
z(0) = −0.054. Click Initialconds, Go to start computations using the default initial conditions. You should see (what is apparently) a complicated-looking attractor, possibly chaotic.
Print it and hand it in.
To investigate the possibility of chaos more closely, check for evidence of sensitive dependence
on initial conditions: select Xi vs t and arrange to plot y as a function of t. Freeze the curve
to a red colour (color=1). Select Initialconds, New and enter initial conditions (x0 , y0 , z0 ) =
(0, −4.4, −0.055). Freeze the new curve with some different colour. Then print out and hand
in the plot of the two solutions. Describe what the plot shows. Is there evidence of sensitive
dependence on initial conditions?
Use XPP to compute the largest, or maximal, Liapunov exponent using the most recently
computed solution. Select nUmerics, stocHast, Liapunov and in the command line 0 (so it
does not range over parameters). Repeat this for at least two more solutions with different initial
conditions. The initial conditions should be chosen on or near the supposed attractor to get the
most accurate possible Liapunov exponent calculation (open the Data Viewer window to see
numerical values of the computed solutions). Describe the results and explain their significance.
(b) Use XPP to compute a Poincaré map. Continuing from above, [Esc]-exit to the main menu,
then delete frozen curves in the buffer by selecting Graphic stuff, (F)reeze, (R)emove all.
Edit the curve (Graphic stuff, (E)dit curve, 0) and set Line type to 0 (see Question 2). Select nUmerics and change Total to 1000, then select Poincare map, Section. In the Poincare
map window, check that Variable: X, Section: 0, Direction: 1, Stop on sect: N and click
Ok. This chooses the section S as the x = 0 plane (i.e. the yz-plane), and plots only the points
where the orbit crosses S in the positive direction and calculations do not stop after returning
to the section. [Esc]-exit to the main menu, select Viewaxes, 2D and arrange to plot z vs. y
for this map, with y on the horizontal axis, −7 ≤ y ≤ 0, and z on the vertical axis, −1 ≤ z ≤ 9.
Erase the screen, and select Initialconds, Go: you should see iterates of the Poincaré map
on the yz-plane. Notice the iterates almost lie along a curve (just below the horizontal axis).
This indicates that the two-dimensional Poincaré map is approximately one-dimensional. Print
the plot and hand it in. (Note: Do not freeze this picture before saving to Postscript. Freezing
gives a less clear picture.)
Now, change z on the vertical axis to the range −0.1 ≤ z ≤ 0. For a closer look, select
Window/zoom, Fit to get XPP to fit the window to the data, and notice the scales on the axes.
Print and hand in this plot as well.
Make a sketch by hand of part of a trajectory on the Rössler attractor in three dimensions,
approximately to scale, also showing the section S, the initial value x0 and the first three
iterates x1 , x2 , x3 of the Poincaré map, showing the relationship between the trajectory and
iterates. Alternatively, you could figure out how to use XPP to plot the trajectory corresponding
to the first three iterates and indicate by hand the section S and the initial value and iterates
on the printed plot.
(c) Make a Ruelle plot: continuing in XPP from above, select Viewaxes, 2D and put y on both
axes, −7 ≤ y ≤ 0. The y-coordinate of each iterate is plotted against the y-coordinate, so of
course all the points lie on a diagonal line. Erase the plot, now select nUmerics, rUelle plot,
choose x-axis shift 1 and leave the remaining axes unshifted (0). This allows you to see yn+1
vs. yn . [Esc]-exit to the main menu and select Restore. The map looks as if yn+1 is nearly a
quadratic function of yn . Print and hand in the Ruelle plot. (Again, do not freeze before saving
to Postscript.)
(d) The curve seen in the Ruelle plot is approximated by the quadratic map
yn+1 = −6 + 0.65(yn + 3.5)2 .
i. Use XPP with the file hw7-2b.ode to iterate this quadratic map. (Set Line type to 0 as
described above for the logistic map in Question 2 and try the default initial condition.) Is
there a chaotic attractor? Explain.
ii. The graph of the quadratic map “looks like” an upside-down graph of the logistic map.
To make this more precise, show that a change of variables of the form y = αx + β, for
some appropriately chosen α and β (find the values), takes the quadratic map into the
logistic map xn+1 = rxn (1 − xn ). What value of r does the quadratic approximation of the
Poincaré map correspond to? Does the logistic map have a chaotic attractor for this value
of r? Explain.
(e) Since the logistic map xn+1 = rxn (1 − xn ) can be obtained through a linear transformation
of the approximate Poincaré map obtained from the Rössler chaotic attractor, summarize how
it models the attractor of the Rössler system (1). In particular, point out explicitly all the
approximations made in the analysis. Is the attractor seen in part (a) for the Rössler system
chaotic? Explain.
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