Needs["VisualDSolve`"]

\(\(pic07 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {y, \(-10\), 10}, {v, \(-5\), 5}, VectorField -> True, 
        FieldLogScale -> 10, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], 
	AxesLabel -> {y, v}]; \)\)

\(TraditionalForm
    \`\(Display["pic07.gif", pic07, "GIF", ImageSize -> {500, 309}]; 
    \)\)

\(\(pic08 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {t, 0, 50}, {y, \(-10\), 10}, {v, \(-5\), 5}, 
        VectorField -> True, FieldLogScale -> 10, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], 
        InitialValues -> {{\(-π\)/2, 3.5}}, 
	AxesLabel -> {y, v}, 
        PlotStyle -> Thickness[0.006], ShowInitialValues -> True, 
	
        Ticks -> {{\(-3\) π, \(-2\) π, \(-π\), 0, π, 
              2  π, 3  π}, Automatic}, 
	PlotPoints -> 100]; \)\)

\(TraditionalForm
    \`\(Display["pic08.gif", pic08, "GIF", ImageSize -> {500, 309}]; 
    \)\)

\(\(pic09 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {t, 0, 50}, {y, \(-10\), 10}, {v, \(-5\), 5}, 
        FlowField -> True, FieldLogScale -> 6, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], 
        InitialValues -> {{\(-π\)/2, 3.5}}, 
	AxesLabel -> {y, v}, 
        PlotStyle -> Thickness[0.006], ShowInitialValues -> True, 
	
        Ticks -> {{\(-3\) π, \(-2\) π, \(-π\), 0, π, 
              2  π, 3  π}, Automatic}, 
	PlotPoints -> 100]; \)\)

\(TraditionalForm
    \`\(Display["pic09.gif", pic09, "GIF", ImageSize -> {500, 309}]; 
    \)\)

\(\(pic10 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {y, \(-10\), 10}, {v, \(-5\), 5}, FlowField -> True, 
        FieldLogScale -> 6, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], ShowEquilibria -> True, 
		
        Nullclines -> True, NullclineShading -> True, 
        NullclinePlotPoints -> 250, NullclineShadingMethod -> SumOfSigns, 
	
        AxesLabel -> {y, v}, PlotStyle -> Thickness[0.006], 
        ShowInitialValues -> True, 
	
        Ticks -> {{\(-3\) π, \(-2\) π, \(-π\), 0, π, 
              2  π, 3  π}, {\(-4\), \(-2\), 0, 2, 4}}, 
	
        PlotPoints -> 100, 
        NullclineStyle
           -> {{Thickness[.006], Yellow}, {Thickness[.006], Orange}}]; \)\)

Display["pic10.gif", pic10, "GIF", ImageSize -> {500, 309}];

\(\(pic11 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {t, 0, 20}, {y, π - 1, π + 1}, {v, \(-2\), 2}, 
        FlowField -> True, FieldLogScale -> 6, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], ShowEquilibria -> True, 
		
        Nullclines -> True, NullclineShading -> True, 
        NullclinePlotPoints -> 80, 
	AxesLabel -> {y, v}, 
        PlotStyle -> Thickness[0.006], ShowInitialValues -> True, 
	
        Ticks -> {{π - 1/2, π, π + 1/2}, {\(-1\), 0, 1}}, 
        InitialValues -> {{π - 1, 1}, {π - 1, 1.2}}, 
	
        PlotPoints -> 100, 
        NullclineStyle
           -> {{Thickness[.006], Yellow}, {Thickness[.006], Orange}}]; \)\)

\(TraditionalForm
    \`\(Display["pic11.gif", pic11, "GIF", ImageSize -> {500, 309}]; 
    \)\)

\(\(pic12 = 
      PhasePlot[{\(y'\)[t] == v[t], \(v'\)[t] == \(-.2\)\ v[t] - Sin[y[t]]}, {
          y[t], v[t]}, {t, 0, 30}, {y, \(-1\), \(+1\)}, {v, \(-2\), 2}, 
        FlowField -> True, FieldLogScale -> 6, FieldColor -> Hue, 
	
        FieldThickness -> Thickness[0.006], ShowEquilibria -> True, 
        InitialValues -> {{\(-1\), 1}}, 
		Nullclines -> True, 
        NullclineShading -> True, NullclinePlotPoints -> 80, 
	
        AxesLabel -> {y, v}, PlotStyle -> Thickness[0.006], 
        ShowInitialValues -> True, 
	
        Ticks -> {{\(-1\)/2, 0, 1/2}, {\(-1\), 0, 1}}, 
	PlotPoints -> 100, 
        NullclineStyle
           -> {{Thickness[.006], Yellow}, {Thickness[.006], Orange}}]; \)\)

\(TraditionalForm
    \`\(Display["pic12.gif", pic12, "GIF", ImageSize -> {500, 309}]; 
    \)\)

\(counter = 1; 

    Do[{Print[
          PaddedForm[counter, 2, NumberPadding -> {"0", ""}]]; 

        	picture = 
          PhasePlot[{\(y'\)[t] == v[t], 
              \(v'\)[t] == \(-.2\)\ y[t] + 1.5 \((1 - y[t]^2)\)\ v[t]}, {
              y[t], v[t]}, {t, 0, T}, {y, \(-3\), 3}, {v, \(-3\), 3}, 
            FlowField -> True, FieldLogScale -> 6, FieldColor -> Hue, 
	
            FieldThickness -> Thickness[0.006], ShowEquilibria -> True, 
		
            Nullclines -> True, NullclineShading -> True, 
            NullclinePlotPoints -> 100, 
	AxesLabel -> {y, v}, 
            PlotStyle -> Thickness[0.004], ShowInitialValues -> True, 

            InitialValues -> {{\(-2\), \(-2\)}, {0, .3}}, 
	
            PlotPoints -> 100, 
            NullclineStyle
               -> {{Thickness[.006], Yellow}, {Thickness[.006], Orange}}]; 

        	Display[
			
          StringJoin[
				
            ToString[
					
              PaddedForm[
						counter, \ 2, 
                NumberPadding -> {"0", ""}]], ".gif"], 
		
          picture, "GIF", ImageSize -> {500, 309}], \(counter++\)}, 

      	{T, 1, 50, .5}	]\)