You will have one week to complete this lab, professionally write up
the answers, and submit it for a grade.
There is also an exam next week, so do not wait until the last
minute to complete the lab! (Hint: only two of the questions below
require the use of the computer; you are encouraged to do some/all of the
other questions * before* coming to lab!)

The purpose of the lab is

- to gain experience with ODEs that model forced oscillators
- to understand differences and similarities between linear and nonlinear oscillators

** Background:**
A linear (harmonic)
oscillator may be modeled by the differential equation

Here is a parameter called the coefficient of damping. It reflects the amount of dissipation in the oscillator (think ``spring-mass system''). If , then there is no friction; if is large, then trajectories are quickly pulled into a sink (meaning that the motion stops). The parameter reflects a constant external force that affects the system. For example, the spring-mass system may be affected by gravity, or the mass may be a piece of metal in a constant magnetic field.

In this portion of the lab, you will discover the relationship between the period of oscillation and the amplitude of oscillation for linear oscillators.

Set the parameters to and .

The * period* of an oscillation is the time that it takes for the position
and velocity to return to their initial values. This corresponds to the time
it takes for a trajectory in phase space to make a closed path. For our
current values of the parameters, the * amplitude* of a trajectory is the
largest value of **x** that the trajectory passes through. This will always
occur when **v=0**.

Fix **v=0** and let . The amplitude for this trajectory will be
(why?), but what is the period? We can estimate the period
numerically by generating a trajectory, graphing versus **t**,
and estimating the time that it takes for the oscillator to return to
**v=0**.

- Plot the period of
each solution versus the amplitude of that solution for
initial conditions with
**v=0**and . - For the linear oscillator, you can explicitly check your numerical work. Explicitly compute the period of oscillation for and by computing the general solution, , of the linear oscillator. (Hint: recall that the period of is .) How does the period change as you change the initial condition?

- For small positive values of , the mass still oscillates (note
that the position alternates between positive and negative values). What is
the critical value of the damping at which the system changes from
being a focus to being a node? (Hint: use the ``pitchfork diagram.'')
- Use this result to sketch a curve (on Figure ) in
the -plane that corresponds to the parameter values
that separate node equilibria from focus equilibria. This curve is
called a
*bifurcation curve*in which eigenvalues of the linearization change from being real to being complex. - When the parameter is large, then the oscillator doesn't
oscillate much. We say that the system is
*overdamped*. One application for having a spring-mass system with large damping is a screened-door on a porch. Why is overdamping desirable in this scenario? Can you think of another application for an overdamped oscillator?

- Compute the location of equilibria in phase space in terms of the parameters and .
- How does the value of (forcing) affect the location of equilibria? Does this make sense physically?
- How does the value of the affect the location of equilibria? Justify your answer (in words) by discussing a physical oscillator.
- On Figure , sketch a curve
that corresponds to
the parameter values that separate systems that have
equilibria from those systems without equilibria. (This is called a
curve of
*saddle-node bifurcation*.) If no such curve exists, explain why.

For this portion of the lab, you will explore a

**Figure:** Bifurcation curves for a linear and
nonlinear oscillator.

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Last modified: Jan 21, 1997

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