Math 260 Lab #12 Limit cycles

Save your work regularly--you never know when Mathematica may crash.  
Email your completed file to tleise@amherst.edu by Monday 4pm.

Plotting examples:

sol=NDSolve[{D[x[t],t]==y[t],D[y[t],t]==-x[t],x[0]==1,y[0]==0},{x[t],y[t]},{t,0,100}];
Plot[x[t]/.sol,{t,0,100},AxesLabel->{t,x}]
ParametricPlot[{x[t],y[t]}/.sol,{t,0,100},AxesLabel->{x,y}]

Today we'll explore the van der Pol equation  (Overscript[x, ¨])+μ((x^2)-1)(OverDot[x])+x=0 with μ>0.

Exercise 1.   Rewrite the DE as a system in the xy-plane (in the usual way is best for this lab, using y=(OverDot[x]), rather than the alternative approach used in lecture for analysis purposes).

Exercise 2.   Use NDSolve to obtain a solution with μ=1 and initial conditions x[0]==1, y[0]==0.  Plot this solution in the xy-plane (using ParametricPlot, with option PlotPoints->100 if the graph looks rough) and plot the waveform x[t] (using Plot).  Be sure to use a long enough time interval to clearly see the limit cycle and the steady oscillations in the waveform.

Exercise 3.   Repeat Exercise 2 with mu=2 and mu=10.  Observe the change in the limit cycle shape.

Exercise 4.   How is the period of the limit cycle changing as mu increases (estimate the period from the waveform plot)?

Exercise 5.    Particularly interesting for large values of mu is the appearance of two time scales: x(t) changes very slowly over a long interval of time with the waveform nearly horizontal, then suddenly x(t) changes extremely rapidly with the waveform nearly vertical.  What part of the limit cycle in the phase plane corresponds to the slow time scale?  (We will explore this phenomenon in more detail in section 7.5.)

Exercise 6.  Now repeat Exercise 2 with a small value of mu, like mu=1/10.  What happens to the limit cycle shape?  How rapidly is the trajectory approaching the limit cycle?

Exercise 7.  Try even smaller values of mu.  What is the radius of the nearly circular limit cycle for mu close to zero?