Math 260 Lab #11 Scroll circuit: beyond limit cycles

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Monday 4pm. Since the plots take up a lot of memory, delete all of them except the very last one before you email me your file (leave the plot commands, so I can see that you did the exercise--just delete the graphs themselves).

A scroll circuit is a double looped circuit with two capacitors, an inductor, and a nonlinear
resistor.  After scaling, the circuit is described by the following system of equations:

dx/dt  = −c*f(y−x)     (x is the voltage drop across the first capacitor)
dy/dt  = −f(y−x) − z    (y is the voltage drop across the second capacitor)
dz/dt  =  y              (z is the current)

where f(v) =−av +0.5(a +b)( |v +1| − |v −1| ) describes the current passing through the nonlinear resistor as a function of the voltage drop across the resistor (after scaling).  We want to explore the dynamics of this system, to see the kinds of interesting pictures that are possible with nonlinear systems and just to have a bit of fun.  

This lab is adapted from an article by Borelli and Coleman in Revolutions in Differential Equations.

Exercises

The function f(v) can be entered into Mathematica as

a=.07;b=.1; (* "realistic" values for the parameters a and b *)
f[v_]:=-a v+0.5(a+b)(Abs[v+1]-Abs[v-1]);(* sawtooth function *)

Exercise 1.   Plot the function f(v) on the interval [-3,3].

Exercise 2.   Let c=0.01.  Use the following NDSolve command to plot the trajectory in the xz-plane (ParametricPlot) and in xyz-space (ParametricPlot3D).  The BoxRatios option of ParametricPlot3D sets the aspect ratio so the graph doesn't look squashed, while the PlotPoints option ensures that enough points are plotted to make the graph look smooth (this is more necessary when chaos appears).

c=0.01;

sol=Flatten[NDSolve[{x'[t]==-c f[y[t]-x[t]],y'[t]==-f[y[t]-x[t]]-z[t],z'[t]==y[t],x[0]==1,y[0]==1,z[0]==1},{x[t],y[t],z[t]},{t,0,700},MaxSteps->25000,MaxStepSize->0.5]];

ParametricPlot[{x[t],z[t]}/.sol,{t,0,700},AspectRatio->1,AxesLabel->{x,z},PlotPoints->10000]

ParametricPlot3D[{x[t],y[t],z[t]}/.sol,{t,0,700},BoxRatios->{2,2,1},AxesLabel->{x,y,z},PlotPoints->10000]

Exercise 3.   Repeat Exercise 2 using c=0.1.

Exercise 4.   Repeat Exercise 2 using c=1.

Exercise 5.   Repeat Exercise 2 using c=5.

Exercise 6.   Repeat Exercise 2 using c=10.

Exercise 7.   Repeat Exercise 2 using c=13.  You should see some random swings starting to happen.

Exercise 8.   Repeat Exercise 2 using c=33.  This is a picture of "organized chaos" where the voltages and current of the circuit wander apparently at random between two distinct oval regions.