poincare[xold_, vold_, γ_, d_] := 
      NDSolve[{x[0] == xold, 
          v[0] == vold, x'[t] == 
            v[t], v'[
              t] ==  -(x[t] + x[t]^3)  - γ v[t] + 
              d Cos[ω t]}, {x[t], v[t]}, {t, 0, 
          2  Pi/ω }] 
xo[xold_, vold_, γ_, 
        d_] := ((x[t]  /. poincare[xold, vold, γ, d])  /.  
            t  ->  
              2 Pi/ω)[[1]]
vo[xold_, vold_, γ_, 
        d_] := ((v[t]  /. poincare[xold, vold, γ, d]) /.  
            t  ->  
              2  Pi/ω)[[1]]
new[old_] := {xo[old[[1]], old[[2]], γ, d], 
        vo[old[[1]], old[[2]], γ, d]}

old = {0, 0};  γ =  .05;  
    d =  .3; ω = 1/ .7
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True,  
      PlotRange -> {{-1, 1}, {-1, 
            1}}]

old = {0, 
        0};  γ =  .05;  
    d =   .6; ω = 1/ .7

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  Frame -> True, 
      PlotRange -> {{0, 2}, {-2, 2}}]

old = {0, 0};  γ =  .05;  
    d =  1.5; ω = 1/ .7

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  Frame -> True, 
      PlotRange -> {{0, 2}, {-2, 2}}]

old = {0, 0};  γ =  .05;  
    d =  .3; ω =  .7
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

old = { .1, 0};  γ =  .05;  
    d =  .5; ω =  .7
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

old = {0, 0};  γ =  .05;  
    d = 2; ω = 1/ .7
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

old = {0, 0};  γ =  .05;  
    d = 4; ω = 1/ .7
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  d = 4; ω = 1/ .7

    init = {0, 0}

    old = Nest[new, init, 50]

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  d = 5; ω = 1/ .7

    init = {0, 0}

    old = Nest[new, init, 1]

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 10; ω = 1/ .7

    init = {0, 0}

    old = Nest[new, init, 1]

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  
      Frame -> True]

γ =  .05;  
    d = 20; ω = 1/ .7

    init = {0, 0}

    old = Nest[new, init, 100]

    NestList[new, old, 50];

    ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  d = 50; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 1]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  d = 68; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 68.1; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 68.2; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 68.3; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 68.4; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
NestList[new, old, 50];
ListPlot[%,  Axes -> False,  Frame -> True]

γ =  .05;  
    d = 68.6; ω = 1/ .7
init = {0, 0}
old = Nest[new, init, 200]
w = NestList[new, old, 30000];
lp = 
      ListPlot[w,  Axes -> False,  Frame -> True, 
        PlotRange -> {{-1, 6}, {-20, 20}}, 
        PlotStyle -> {AbsolutePointSize[ .001]}]