van der Pol Equation

Author

Eric W. Weisstein
February 27, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/vanderPolEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/vanderPolEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Analytic Solution

DSolve[y''[x]-μ(1-((y[x])^2))y'[x]+y[x]==0,y[x],x]

Plot

soln[μ_,t_]:=NDSolve[{y''[x]-μ(1-((y[x])^2))y'[x]+y[x]==0,y[0]==.01,y'[0]==0},y[x],{x,0,t}][[1,1,2,0]];

plot[t_,m_]:=Block[{$DisplayFunction=Identity},
{ParametricPlot[Evaluate[{x,soln[m,t][x]}],{x,0,t},PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,y[x]},PlotLabel->TraditionalForm[μ==m]],
ParametricPlot[Evaluate[{soln[m,t][x],D[soln[m,t][x],x]}],{x,0,t},AspectRatio->Automatic,PlotStyle->Red,PlotPoints->100,AxesLabel->TraditionalForm/@{y[x],y'[x]},
PlotRange->All]
}]

Show[GraphicsArray[
plot[100,#]&/@{.2,1,5}
],GraphicsSpacing->{-.15,0},ImageSize->400]