(*******************************************************)
(* Nonlinear oscillator with cubic nonlinear damping
y'' + y + ϵ (((y'))^3) = 0;
   y(0) = 1;  y'(0) = 0                               *)
(*******************************************************)

Remove["Global`*"]
(* Exact solution *)
ϵ=-0.1;
s=NDSolve[{y''[t] + y[t] + ϵ ((y'[t])^3)==0,y[0]==1,y'[0]==0},y[t],{t,0,10}]
ypert= ((Cos[t])/(Sqrt[1+((3 ϵ t)/4)]));

Plot[{Evaluate[y[t]/.s],ypert},{t,0,10},PlotStyle->Thick]