noisydat = 
      NDSolve[{x''[t] + 0.15 x'[t] - x[t] + x[t]^3 == 
            0.3 Cos[t], x[0] == -1, x'[0] == 1}, 
        x, {t, 0, 90}]

Plot[x[t] /. noisydat, {t, 0, 90}]

Plot[(x[t] /. noisydat)*(x[t + 0.0] /. noisydat),  {t, 0, 
        70}]

cnoisy[
        y_] := (∫_0^80)(x[t] /. noisydat)*(x[t + y] /. 
              noisydat) \[DifferentialD]x

Evaluate[cnoisy[1]]

Plot[cnoisy[y] /. noisydat, {y, 0, 80}]