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
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
QUESTION 1: Approximate the relationship between period and
amplitude for a linear oscillator:
- 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
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?
- 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
- 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
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
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Copyright: 1996 by the Regents of the University of Minnesota.
Department of Mathematics. All rights reserved.
Last modified: Jan 21, 1997
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