Limit Cycle Example

The system we are interested in is the following

(dx/dt)=v
(dy/dt)=(1-3(x^2)-2(v^2))v-x

Null-Cline Analysis

Lets plot the vector field:

plot1=VectorPlot[{v,(1-3*x^2-2*v^2)*v-x},{x,-1,1},{v,-1,1}]

Lets plot the null-clines

plot2=ContourPlot[v==0,{x,-1.5,1.5},{v,-1.5,1.5},ContourStyle->{Thick,Red,Dashed}]

plot3=ContourPlot[(1-3*x^2-2*v^2)*v-x==0,{x,-1.5,1.5},{v,-1.5,1.5},ContourStyle->{Thick,Green,Dashed}]

Now lets combine the plots:

plotVec=Show[plot1,plot2,plot3]

Numerical Analysis

Now, lets numerically solve the equation using NDSolve. We can manually change initial conditions to generate the phase portrait. ParametricPlot is used to plot the trajectory.

s=NDSolve[{x'[t]==v[t],v'[t]==(1-3*x[t]^2-2*v[t]^2)*v[t]-x[t],x[0]==-1,v[0]==1},{x,v},{t,100}];
plottemp=ParametricPlot[Evaluate[{x[t],v[t]}/.s],{t,0,100},PlotStyle->{Thick,Purple}]
plot3=Show[plotVec,plot3,plottemp]

Lets plot how x evolves as a function of time.

plottemp=Plot[Evaluate[x[t]/.s],{t,0,40},PlotStyle->{Thick,Purple}]