ESAM 438

Problem Set 5

For Discussion Section November 1

1. Damped Driven Pendulum

   Consider a pendulum with friction that is exposed to a constant torque G,
   

   ml2 d2 q dt2 +b d q dt = -mgl sinq+G. (1)

   1. Rewrite (1) as a system of first-order equations and determine all fixed points of this system.
   2. Perform a linear stability analysis of all the fixed points. Discuss what happens physically when the fixed points become unstable? Can you use this to make an educated guess as to what kind of bifurcation the instability leads to?

2. Weakly Nonlinear Analysis of Hopf Bifurcation

   Consider the two coupled equations
   

   dx dt = mx +y + a x3, (2) dy dt = -x+my+by3. (3)

   1. Sketch the nullclines of (2,3) and use them to determine qualitatively the flow as much as that is possible from that information.
   2. Perform a linear stability analysis of the fixed point (0,0) including the eigenvectors associated with the relevant eigenvalues.
   3. Use the eigenvectors as the first-order terms in a weakly nonlinear analysis of the Hopf bifurcation. Depending on a and b, can the bifurcation be supercritical and/or subcritical?
   4. Solve (2,3) numerically (write your own code or use pplane.m¹) and compare with the weakly nonlinear analysis. Specifically, measure (roughly) the amplitude of the oscillations and their frequency for a suitable range of the control parameter m.

3. Relaxation Oscillations

   Read Ch. 7.5 in Strogatz and do problem 7.5.4. Solve also the equations numerically and measure (roughly) the period of the oscillation. Compare the numerically determined period with that from the analytical approximation obtained for m >> 1.

Footnotes:

¹See the class web site for downloading that software.