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.

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.

1. Plot the period of each solution versus the amplitude of that solution for initial conditions with v=0 and .
2. 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.

 
Figure:   Bifurcation curves for a linear and nonlinear oscillator.