Lab #10

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.

QUESTION 1: Approximate the relationship between period and amplitude for a linear oscillator:
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?

QUESTION 2:
• 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?

QUESTION 3:
• 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 nonlinear oscillator that models a spring-mass system in which the spring is not perfectly linear. The differential equations are

QUESTION 4--6: Repeat Questions 1--3 for this nonlinear oscillator. For Question 2, recall that the linearization of a vector field at equilibria (almost always) classifies that equilibrium. Make the bifurcation curves in a different color from the linear bifurcation curves.

QUESTION 7: Write a paragraph describing differences between linear and nonlinear oscillators. Be careful not to over-generalize based on the single nonlinear oscillator that you've seen.

Figure:   Bifurcation curves for a linear and nonlinear oscillator.