Off[General::spell];
Off[General::spell1]; Clear["Global`*"]

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 0.1; om = 2.5; per = 2*Pi/om;


    	\(nper = 65;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 1/100, p[0] == 1/100}, {x, p}, 
	{t, 0, 
            tmax}, MaxSteps -> 10000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\((x[t] /. aa)\)[\([1]\)], \((p[t] /. 
                  aa)\)[\([1]\)]}, {t, 0, tmax}, PlotPoints -> 2000, 
          AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\((x[ll*per] /. aa)\)[\([1]\)], \((p[ll*per] /. 
                aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 0.1; om = 2.5; per = 2*Pi/om;


    	\(nper = 65;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 1, p[0] == 1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 10000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\((x[t] /. aa)\)[\([1]\)], \((p[t] /. 
                  aa)\)[\([1]\)]}, {t, 0, tmax}, PlotPoints -> 2000, 
          AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\((x[ll*per] /. aa)\)[\([1]\)], \((p[ll*per] /. 
                aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 0.1; om = 1.3; per = 2*Pi/om;


    	\(nper = 15;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 0, p[0] == 0.1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 100000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\(-Pi\) + 
              Mod[Pi + \((x[t] /. aa)\)[\([1]\)], 
                2*Pi], \((p[t] /. aa)\)[\([1]\)]}, {t, 0, tmax}, 
          PlotPoints -> 2000, AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\(-Pi\) + 
            Mod[Pi + \((x[ll*per] /. aa)\)[\([1]\)], 
              2*Pi], \((p[ll*per] /. aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 0.5; om = 1.1; per = 2*Pi/om;


    	\(nper = 15;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 0, p[0] == 0.1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 100000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\(-Pi\) + 
              Mod[Pi + \((x[t] /. aa)\)[\([1]\)], 
                2*Pi], \((p[t] /. aa)\)[\([1]\)]}, {t, 0, tmax}, 
          PlotPoints -> 2000, AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\(-Pi\) + 
            Mod[Pi + \((x[ll*per] /. aa)\)[\([1]\)], 
              2*Pi], \((p[ll*per] /. aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 2; om = 1.01; per = 2*Pi/om;


    	\(nper = 25;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 0, p[0] == 0.1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 100000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\(-Pi\) + 
              Mod[Pi + \((x[t] /. aa)\)[\([1]\)], 
                2*Pi], \((p[t] /. aa)\)[\([1]\)]}, {t, 0, tmax}, 
          PlotPoints -> 2000, AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\(-Pi\) + 
            Mod[Pi + \((x[ll*per] /. aa)\)[\([1]\)], 
              2*Pi], \((p[ll*per] /. aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        12;\)

    \(precgoal = 15;\)

    epsi = 5; om = 0.5; per = 2*Pi/om;


    	\(nper = 6;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 0, p[0] == 0.1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 100000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\(-Pi\) + 
              Mod[Pi + \((x[t] /. aa)\)[\([1]\)], 
                2*Pi], \((p[t] /. aa)\)[\([1]\)]}, {t, 0, tmax}, 
          PlotPoints -> 2000, AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\(-Pi\) + 
            Mod[Pi + \((x[ll*per] /. aa)\)[\([1]\)], 
              2*Pi], \((p[ll*per] /. aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(acgoal = 
        10;\)

    \(precgoal = 15;\)

    epsi = 10; om = 0.5; per = 2*Pi/om;


    	\(nper = 5;\)

    \(tmax = nper*per;\)

    	\(aa = 
        NDSolve[{\(x'\)[t] == 
              p[t], \(p'\)[t] == \(-\((1 + epsi*Cos[om*t])\)\)*Sin[x[t]], 
            x[0] == 0, p[0] == 0.1}, {x, p}, 
	{t, 0, tmax}, 
          MaxSteps -> 100000, 
	AccuracyGoal -> acgoal, 
          PrecisionGoal -> precgoal];\)

    \(pl1 = 
        ParametricPlot[{\(-Pi\) + 
              Mod[Pi + \((x[t] /. aa)\)[\([1]\)], 
                2*Pi], \((p[t] /. aa)\)[\([1]\)]}, {t, 0, tmax}, 
          PlotPoints -> 2000, AxesLabel -> {x, p}, PlotRange -> All];\)

    	ListPlot[
      Table[{\(-Pi\) + 
            Mod[Pi + \((x[ll*per] /. aa)\)[\([1]\)], 
              2*Pi], \((p[ll*per] /. aa)\)[\([1]\)]}, 
			{ll, 1, nper}], 
      PlotStyle -> PointSize[0.02]]

    \)\)\)