Method of Multiple Scales

Example 1

We are interested in the behavior of the solutions of the differential equation
y''+((ϵ(y'))^3)+y=0

ϵ=.001;
sol=NDSolve[{y''[t]+ϵ*(y'[t])^3+y[t]==0,y[0]==1,y'[0]==0},y[t],{t,0,1000}]
Plot[{1/Sqrt[1+3*ϵ*t]*Cos[t],Evaluate[y[t]/.sol],1/Sqrt[1+3*ϵ*t],-1/Sqrt[1+3*ϵ*t]},{t,0,1000},PlotRange->{-1,1}]

Example 2

We are interested in the behavior of the solutions of the differential equation
y''+ϵ*y'((y^2)-1)+(1+ϵ*k)*u=ϵ*cos(t)

ϵ=.1;
k=-.2;
sol=NDSolve[{y''[t]+ϵ*y'[t]*(y[t]^2-1)+(1+ϵ*k)*y[t]==ϵ*Cos[t],y[0]==1,y'[0]==0},y[t],{t,0,1000}]
Plot[{Evaluate[y[t]/.sol]},{t,0,1000}]
sol=NDSolve[{-2*r[t]*ϕ'[t]+r[t]*k==Cos[ϕ[t]]/2,2*r'[t]==-r[t]^3+r[t]-Sin[ϕ[t]]/2,r[0]==1,ϕ[0]==1},{r,ϕ},{t,1000}]
Plot[Evaluate[r[ϵ*t]*Cos[t+ϕ[ϵ*t]]/.sol],{t,0,1000}]

Example 3

We are interested in the behavior of the solutions of the differential equation

ϵ=.1;
k=-5/24;
sol=NDSolve[{y''[t]+(1+k*ϵ^2+ϵ*Cos[t])*y[t]==0,y[0]==1,y'[0]==1},y[t],{t,0,1000}]
Plot[{Evaluate[y[t]/.sol]},{t,0,1000}]

Simplify[Sin[ArcCos[2*x]]]

k=.49

Solve[r-r^3+Sqrt[1-4*k^2]==0,r]

Plot[r-r^3,{r,-2,2}]