**============== update - animation ==============** 

This animation has about 50 frames - meaning it's obtained from tracking 50 independent events during single run of numerical solver:

![enter image description here][1]

Code is simply given by:

    evs = Mod[t, 2 \[Pi]] == # & /@ (Range[0, 2 Pi - #, #] &@(2 Pi/50));

    data = Block[{\[Delta] = 0.15, \[Gamma] = 0.3}, 
       Reap[NDSolve[{x''[t] + \[Delta] x'[t] - x[t] + 
            x[t]^3 == \[Gamma] Cos[ t], x[0] == 0, x'[0] == 0, 
          WhenEvent[Evaluate@evs, 
           Sow[{x[t], x'[t]}, Round[100 Mod[t, 2 \[Pi]]]]]}, {}, {t, 0, 
          200000}, MaxSteps -> \[Infinity]]]];


    Export["test.gif",
     Table[ListPlot[data[[2]][[k]], PlotStyle -> PointSize[0], 
       PlotRange -> {1.8 {-1, 1}, 1.3 {-1, 1}}, AspectRatio -> 1, 
       Frame -> True, ImageSize -> 450], {k, 1, 50, 1}]
     ]

**============== older - testing grounds ==============** 

Put independent events in `List` when you need a few of them. Then to separate while Sow/Reap - use tags in Sow - for example for 4 different events. I'll show all of them on a single plot to compare. Just use Manipulate or Animate or ListAnimate for your goal.

    data = Block[{δ = 0.15, γ = 0.3}, 
       Reap[NDSolve[{x''[t] + δ x'[t] - x[t] + 
            x[t]^3 == γ Cos[ t], x[0] == 0, x'[0] == 0, 
          WhenEvent[{Mod[t, 2 π] == 0, Mod[t, 2 π] == π/2, 
            Mod[t, 2 π] == π, Mod[t, 2 π] == (3 π)/2}, 
           Sow[{x[t], x'[t]}, Round@Mod[t, 2 π]]]}, {}, {t, 0, 
          100000}, MaxSteps -> ∞]]];


    ListPlot[data[[2]], ImageSize -> Medium, 
     PlotRange -> {{-1.5, 1.5}, All}, PlotStyle -> PointSize[0.0025]]

![enter image description here][2]


  [1]: https://i.sstatic.net/sHG3f.gif
  [2]: https://i.sstatic.net/jGa7I.jpg