3. Oscillating Systems

Legal Notice, Disclaimer, and Book Ordering Information:

Null

Setup

\(Clear["Global`*`"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

3.3 Problems

3.3.2  Nonlinear Oscillations

Problem 3:  van der Pole Oscillator and Limiting Cycles

<< "D:\Mma\Inits\ch3init.m"

Clear["Global`*`"];

Part (a)  Numerically solve the equations for a=2, ω0=1, with the initial conditions x[0]=0.06, y[0]=0, and x[0]=0, y[0]=4.  Plot the trajectories.

\(eq1 = {\(y'\)[t] == a \((1 - x[t]^2)\) y[t]\  - \ ω0^2\ x[t]

        				, \(x'\)[t] == y[t]}\)

values = {a -> 2, ω0 -> 1}

init1 = {x[0] == 0.06,  y[0] == 0}

init2 = {x[0] == 0,  y[0] == 4}

\(ndsol1 = 
      NDSolve[Join[eq1, init1] //. values
		, {x[t], y[t]}

          		, {t, 0, 25}
		, MaxSteps -> 700] // Flatten\)

\(x1[t_] = x[t] /. ndsol1; 
y1[t_] = y[t] /. ndsol1; \)

\(ndsol2 = 
      NDSolve[Join[eq1, init2] //. values
		, {x[t], y[t]}

          		, {t, 0, 25}
		, MaxSteps -> 700] // Flatten\)

\(x2[t_] = x[t] /. ndsol2; 
y2[t_] = y[t] /. ndsol2; \)

\(\(Show[
      GraphicsArray[{Plot[x1[t], {t, 0, 25}]

          			\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , Plot[y1[t], {t, 0, 25}]}]]; \)\)

\(\(Show[
      GraphicsArray[{Plot[x2[t], {t, 0, 25}]

          			\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , Plot[y2[t], {t, 0, 25}]}]]; \)\)

Part (b)  Plot the phase trajectories.

doublePlot[x1[t], y1[t], {t, 0, 25}];

doublePlot[x2[t], y2[t], {t, 0, 25}];

Part (c)  Animate the x behavior.

oscMovie[x1, 25, 40, {-2, 2}];