780.20: 1094 Session 7
Handouts: Excerpts from Chaps. 9 and 11 on the nonlinear
oscillator and excerpts from Chap. 14 on chaos, eigen_basis_class.cpp
and diffeq_oscillations.cpp printouts.
Now that we've got routines to solve differential equations, we're going
to explore some interesting one: nonlinear oscillators. Today we'll
play with a program that solves for the time dependence of such an
Your goals for today (and ...):
- Do the plot from Session 6 of relative error at t=1 vs.
mesh size h.
- Think about how to enhance the eigen_basis code with more C++ classes.
- Run a code that solves the differential equation for a (driven)
nonlinear oscillator and explore how the time dependence changes as
various input parameters change.
- Add friction (damping) to the code.
- Use phase space plots as a tool to analyze the behavior of
Please work in pairs (more or less).
The instructors will bounce around 1094 and answer questions.
Leftover Task from Session 6
Spend about 45 minutes (or less) on this.
- Integrating a First-Order Differential Equation.
Try to finish through
part 7. If you find that the errors for Euler and Runge-Kutta lie
on top of each other, most likely you have not evaluated the exact
answer at precisely the same time as the last points. If you get
stuck, ask an instructor.
More on C++ Classes: eigen_basis_class
The code eigen_basis_class.cpp is a simple modification of
eigen_basis.cpp to use the Hamiltonian class we introduced
for eigen_tridiagonal_class.cpp. Here we'll take a few minutes
to think about how to introduce additional classes.
- Take a look at the eigen_basis_class.cpp printout
and note how the Hamiltonian class is re-used without modification. (If
you haven't done so yet, read the discussion of this class
in the Session 7 notes.) The only tricky change is
that matrix indices go from 1 to dimension rather than from
0 to dimension-1. What parts of the Hamiltonian class implementation
do you not yet understand?
- The potential is another good candidate for a class. We'd
like to just evaluate the potential at r without having to use
constructions like the switch statement in the Hij_integrand
function with all the messy void parameters. (Think about how
awkward and prone to error it is to add another potential.)
What would you like the declaration statement for the Potential
class to look like? What method(s) would you like the class to
- Give at least one example
of an additional classes that would be useful to define?
Driven Nonlinear Oscillations
The handout with excerpts from Chapters 9 and 11 describe the driven
nonlinear oscillator that is coded in diffeq_oscillations.cpp.
Note that the force depends on k and an exponent p,
the external force has a magnitude f_ext, a frequency w_ext, and a phase
phi_ext. The initial conditions in position and velocity are designated
x0 and v0. You also have control over the time interval (increase
t_end to see longer times), the step size h, and how often points are
printed to the file (plot_skip).
Use make_diffeq_oscillations to create diffeq_oscillations.
This code outputs to the file diffeq_oscillations.dat five columns of
data: t, x(t), v(t), kinetic energy, and potential energy. There are
four gnuplot plot files provided (diffeq_oscillations1.plt, etc.), each
of which generates a different type of plot. Run diffeq_oscillations
with the default values (enter "0" when it says "What do you want to
change?") to calculate a data set.
Start gnuplot and "load diffeq_oscillations1.plt" and then "load
diffeq_oscillations2.plt". (Once you've given these commands
once, you can use just use the arrows to go back and forth.)
Briefly, what do each of these plots show?
- Wouldn't it be convenient to generate all four plots at once in
separate files? Load "diffeq_oscillations_all.plt"!
- It's always a question whether or not you have coded a problem
correctly, so you should always seek ways to check your results.
One possibility is if we have a known solution. This works for p=2
(simple harmonic oscillator). What about other p? Another check
is to identify a quantity that shouldn't change with time. Create a
plot of such a quantity (you'll want to increase t_end)
and observe the effect of changing the step size
h to a larger value [e.g., try 10 and 100 times larger].
How do you decide on a reasonable h to use?
(The "plot_skip" parameter indicates how
often a point is written to the output file. So plot_skip=10 means
that every 10 points is output.)
- Verify that different amplitudes (e.g., different initial
conditions determined by x0 and v0) lead to different periods for
an anharmonic oscillator (p<2 or p>2). [Hint: You might find the
"append" option useful.] Can you identify a
qualitative rule? E.g., does larger amplitude mean shorter or
longer period always? Can you explain the rule?
- Go back to the original parameters (quit the program and start
it again), which has p=2.
Now add a driving force
f_ext=10 with w_ext=1 and look at the time dependence and phase-space
plots. Then increase w_ext to 3.14 and then to w_ext=6.28. What
are you observing? Now repeat with p=3 (starting with f=0).
Can you find resonant behavior?
Real-world systems have friction, which means the motion will be damped.
On page 150 of the text (see the back page of the Chap. 9/11 handout)
is a list of three simple models for friction. We'll implement
viscous damping: Ff = -b*v, where v(t) is the velocity.
- Introduce the damping parameter "b" into the code:
Try this part out before proceeding.
- add it to the force_parameters structure (with a comment!);
- add it to the list of local force parameters in the main
- give it an initial value;
- add a menu item (e.g., ) and a case statement to get
a new value.
- Modify the "rhs" routine to include damping (you're on your own
here!). What did you add?
- Test your routine starting with p=2 and a small damping and look at
both the time dependence and the phase-space plots.
Then try some other p values.
- Identify the three regimes described on page 150: underdamped,
critically damped, and overdamped.
Looking for Chaos (Part I)
Now we want to put it all together: a damped, driven, nonlinear
oscillator. The excerpt from chapter 14 describes a different system,
namely a realistic pendulum, but it has the same basic features.
- On page 191, there is a list of characteristic structures
that can be found in phase space, with sample pictures on page 190.
Can you find combinations of parameters that produce pictures like
these? (Try to imitate the x(t) vs. t pictures first.)
780.20: 1094 Session 7.
Last modified: 09:21 am, February 02, 2009.