# How to plot vector field on a circle

I want to plot

1) vector field on a circle for the nonlinear equation and

2) theta dot and theta for different values of mu for the equation

\!$$\*OverscriptBox[\(θ$$, $$.$$]\) == μSin[θ] -
Sin[2 θ]


I'm not getting any idea, I tried with the code as below:

Clear [θ];
eqn=Overscript[θ, .]==μSin[θ]-Sin[2θ]
sol=θ/.DSolve[{eqn,θ[0]==0},θ,t][[1,1]];
ParametricPlot[Evaluate[Table[Limit[{sol[t],D[sol[t],t]},μ->i],{i,{-3,-2,-1,0,1,2,3}}]],{t,0,100},AspectRatio->1,PlotRange->{{-5,5},{0,50}},PlotPoints->1000]
tTicks=Range[-24,24 30,24];
tGrid=Range[-60,24 30,6];
[Frame->True,FrameTicks->{tTicks,Automatic},FrameTicksStyle->Directive[Red,Thick],GridLines->{tGrid,Automatic},GridLinesStyle->LightGray,FrameLabel->(Style[#,14,Bold]&/@{θ,Overscript[θ,"."]}),AspectRatio->1]


Sincerely,

This is really just for fun. For simplicity $\theta$ has been changed to $u$ and $\mu$ changed to $m$ and the initial condition changed to $u(0)=1$:

{p1, p2} = {u, u'} /.
ParametricNDSolve[{u'[t] == m Sin[u[t]] - Sin[2 u[t]],
u[0] == 1}, {u, u'}, {t, 0, 4 Pi}, {m}]
Manipulate[
Row[{
Show[ParametricPlot[{p1[m][t], p2[m][t]}, {t, 0, 4 Pi},
Epilog -> {Red, PointSize[0.02],
Point[{p1[m][time], p2[m][time]}]}, Frame -> True,
PlotRange -> Table[{-3, 3}, {2}]],
Plot[m Sin[v] - Sin[2 v], {v, 0, 2 Pi}, PlotStyle -> Dashed],
ImageSize -> 300, FrameLabel -> {"\[Theta]", "\[Theta]'"},
BaseStyle -> 20],
Plot[p1[m][t], {t, 0, 2 Pi},
Epilog -> {Red, PointSize[0.02], Point[{time, p1[m][time]}]},
Frame -> True, FrameLabel -> {"t", "\[Theta](t)"},
ImageSize -> 300, BaseStyle -> 20]

}], {m, -2, 2, Appearance -> "Labeled"}, {time, 0, 4 Pi, Animator}]


The (hopefully) instructive points:

• need for numerical approach NDSolve and related
• use of ParametricNDSolve to explore variation of parameter m ($\mu$)
• superimposition of phase space relation and solution