\!\(w\  = 1\[IndentingNewLine]
  F\  =  .1\[IndentingNewLine]
  solq = NDSolve[{\(x'\)[t] \[Equal] x[t]\^2 - x[t]*y[t], \(y'\)[t] \[Equal]
    F*Cos[w*t] - x[t] +
      y[t]\^2, \ x[0] \[Equal] 1, y[0] \[Equal] 1}, {x[t], y[t]}, {
            t, 0, 1000}, MaxSteps \[Rule] Infinity, MaxStepSize \[Rule]
          0.1]\[IndentingNewLine]
  \)

1

0.1

\!\(\*
  RowBox[{\(NDSolve::"ndsz"\), \(\(:\)\(\ \)\), "\<\"At \\!\\(t\\) == \\!\\(
      1.8277755354892973`\\),
      step size is effectively zero; singularity or stiff system suspected. \
\\!\\(\\*ButtonBox[\\\"More:\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \
ButtonFrame->None, ButtonData:>\\\"NDSolve::ndsz\\\"]\\)\"\>"}]\)

{{x[t]\[Rule]
      InterpolatingFunction[{{0.,1.82778}},<>][t],y[t]\[Rule]\
InterpolatingFunction[{{0.,1.82778}},<>][t]}}