How to Export this animation as a gif file for powerpoint presentation

You have to set values which are dynamic in Manipulate.

μ = .75; ω = .75; A = .075; xv0 = {1, 1};

Table pictures for different tt:

sol = Quiet@NDSolve[{x'[t] == v[t], v'[t] == μ (1 - x[t]^2) v[t] - x[t] + A*Cos[ω*t], 
                     x[0] == xv0[[1]], v[0] == xv0[[2]]
                    }, {x[t], v[t]}, {t, 0, 20}];

dat = Table[
            ParametricPlot[Evaluate[{x[t], v[t]} /. sol, {t, 0, tt}, 
                           PlotRange -> 4, AxesLabel -> {x[t], v[t]}]
            , {tt, .1, 20, .2}];

Create gif.

SetDirectory@NotebookDirectory[]
Export["gif.gif", dat]

You can capture the frames as you manually manipulate like this:

 frames = {}
 Animate[Manipulate[(AppendTo[frames, ParametricPlot[ ... ] ])[[-1]] .. ]  ]

disable dynamic updating when done, then

 Export["test.gif",frames]

warning this will quickly generate a large number of frames..

You can adjust the step size by changing the increment in the Table and you can adjust the animation rate using "DisplayDurations", e.g. for the Table above.

Export["dat.gif",dat, "DisplayDurations"->Table[t,{Length[dat]}]] 

where t is display duration in seconds (frame rate: 1/t). Other options for exporting are in the GIF documentation