fx = r*x + x^3 - x^5;
Manipulate[
FP = x/.Solve[fx==0,x]/.r->param;
fixedpoints=Table[{FP[[i]],0},{i,Length[FP]}];
Show[{
Plot[fx/.r->param,{x,-1.5,1.5},PlotRange->{-0.2,0.2}],
ListPlot[fixedpoints]
}],
{{param,-1,"r"},-1,1}]

FP=Solve[fx==0,x]
StabilityFP = FullSimplify[D[fx,x]/.FP]
Plot[x/.FP,{r,-2,2}]

Manipulate[
s=NDSolve[{x'[t]==r*x[t] + x[t]^3 - x[t]^5,x[0]==initial},x,{t,0,30}];
Plot[Evaluate[x[t]/.s],{t,0,30},PlotRange->{-2,2}],
{r,-1,1},{initial,-2,2}]