(Over-damped) bead on a rotating hoop, analysis

Fixed points, graphically

Other fixed points

Clear[r,ϕ,y]
Manipulate[
y1=Cos[ϕ];
y2=1/r;
pts=Quiet[Solve[y==y1&&y==y2,{ϕ,y}]];
p1=Plot[{y1,y2},{ϕ,-π,π},PlotRange->{-1,2},AxesLabel->{ϕ,y},LabelStyle->17,ImageSize-> Large];
If[r≥1,
Show[p1,Graphics[{PointSize[Large],Red,Point[{ϕ,y}/.pts]}]],
p1],
Control[{{r,0.6,"Proportional to velocity, γ"},0.5,20,Appearance->"Labeled"}],TrackedSymbols:>{r},SaveDefinitions->True]

Bifurcation diagram of the bead on a rotating hoop

sol=Solve[Cos[ϕ]-1/γ==0,ϕ]
p1=Plot[Evaluate[Pi],{x,0,15}, PlotStyle->{Black, Dashed}];
p2=Plot[Evaluate[-Pi],{x,0,15}, PlotStyle->{Black, Dashed}];
p4=Plot[Evaluate[0],{x,0,1}, PlotStyle->{Black}];
p3=Plot[{Evaluate[ϕ/.sol/.{C[1]-> 0}],0},{γ,0,15},FrameLabel->{γ,"(SuperStar[ϕ])"},LabelStyle->17, PlotStyle->{Black,Black, {Black, Dashed}},Frame->True];
Show[p1,p2,p3,p4, PlotRange-> {-Pi,Pi},Frame->True,FrameLabel->{γ,"(SuperStar[ϕ])"},LabelStyle-> 16,ImageSize-> Large]

Creation of this notebook was supported by IT Academy program of Information Technology Foundation for Education (HITSA).