omegax  =  Pi
omegay  =  2 Pi
Plot[{Cos[omegax t] ,   .5 Cos[omegay t]}, {t, 1, 4},  
      PlotStyle -> {{Thickness[0.005], 
            RGBColor[1, 0, 0]},  {Thickness[0.005], 
            RGBColor[0, 0, 1]}}]

omegax  =  Pi

    omegay  =  2.3 Pi

    Plot[{Cos[omegax t] ,   .5 Cos[omegay t]}, {t, 1, 4}, 
      PlotStyle -> {{Thickness[0.005], 
            RGBColor[1, 0, 0]},  {Thickness[0.005], 
            RGBColor[0, 0, 
              1]}}]

x[t_] = Cos[Pi t]
dx[t_]  =  D[x[t], t]

ParametricPlot[{x[t], dx[t]} , {t, 1, 50}]

y[t_]  =  Cos[2.1 Pi t]
dy[t_] =  D[y[t], t]
ParametricPlot[{x[t], y[t]}, {t, 1, 5}, 
          PlotStyle -> {{RGBColor[1, 0, 0]}}]

ParametricPlot[{dx[t], dy[t]}, {t, 1, 50}, 
          PlotStyle -> {{RGBColor[0, 1, 0]}}]

ParametricPlot[{x[t], y[t]}, {t, 1, 500}, 
          PlotStyle -> {{RGBColor[1, 0, 0]}}]
x[t_] = Cos[Pi t]
dx[t_]  =  D[x[t], t]

ParametricPlot[{x[t], dx[t]} , {t, 1, 50}]

y[t_]  =  Cos[Sqrt[2] Pi t]
dy[t_] =  D[y[t], t]
ParametricPlot[{x[t], y[t]}, {t, 1, 5500}, 
          PlotStyle -> {{RGBColor[1, 0, 0]}}]

x[t_] = Cos[Pi t] Exp[- .1 t]
dx[t_]  =  D[x[t], t]

ParametricPlot[{x[t], dx[t]} , {t, 1, 10}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]

\(\(\(
\)\(
\)\( (*THE\ AXION*) \)\( (*Frank\ \(Wilczek\ --\)\ Today' s\ Nobel\ Prize; \ also\ S . \ Weinberg, \ 
      M . \ Dine, \ W . \ Fischler, \ 
      and\ M . \ Srednicki*) \)\(
\)\(sol1 = 
      NDSolve[{\(a''\)[t] +  .1\ \(a'\)[t]\  + \ Pi\ a[t] == 0, \ 
          a[0] == 1, \ \(a'\)[0] == 0}, a, {t, 0, 15}, \ 
        MaxSteps -> 1000000]

    \(Plot[Evaluate[a[t]/t /. sol1], {t, 0, 10}, 
        PlotStyle -> {{Thickness[0.005], RGBColor[1, 0, 0]}}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}}, 
        AxesLabel -> \ {"t", "s"}];\)\)\)\)

sol1 = 
      NDSolve[{a''[t] + 2/t a'[t]  +  Pi a[t] == 0,  
          a[1] == 1,  a'[1] == 0}, a, {t, 1, 50},  
        MaxSteps -> 1000000]
Plot[Evaluate[a[t]/t /. sol1], {t, 1, 10}, 
        PlotStyle -> {{Thickness[0.005], RGBColor[1, 0, 0]}}, 
        AxesLabel ->  {"t", "s"}];
Plot[Evaluate[a[t]/t /. sol1], {t, 30, 50}, 
        PlotStyle -> {{Thickness[0.005], RGBColor[0, 1, 0]}}, 
        AxesLabel ->  {"t", "s"}];

omega  =  .7
c =  .01
F =  .02
Clear[
      x]
sol1 = 
      NDSolve[{x''[t] + c  x'[t]  + Sin[x[t]] - 
              F Cos[omega t] == 0,  
          x[0] == 1,  x'[0] == 0}, x, {t, 0, 1500},  
        MaxSteps -> 
          1000000]
Plot[{Evaluate[x[t]]/t /. 
            sol1, Evaluate[x'[t]]/t /. sol1}, {t, 0, 20}, 
        PlotStyle -> {{Thickness[0.010], 
              RGBColor[1, 0, 0]}, {Thickness[0.010], RGBColor[0, 0, 1]}}, 
        AxesLabel ->  {"t", "s"}];
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1, 20}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1400, 1500}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]

c =  .03
F =  .2
Clear[x]
sol1 = 
      NDSolve[{x''[t] + c  x'[t]  + Sin[x[t]] - 
              F Cos[omega t] == 0,  
          x[0] == 1,  x'[0] == 0}, x, {t, 0, 1500},  
        MaxSteps -> 
          1000000]
Plot[{Evaluate[x[t]]/t /. 
            sol1, Evaluate[x'[t]]/t /. sol1}, {t, 0, 20}, 
        PlotStyle -> {{Thickness[0.010], 
              RGBColor[1, 0, 0]}, {Thickness[0.010], RGBColor[0, 0, 1]}}, 
        AxesLabel ->  {"t", "s"}];
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1, 20}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1400, 1500}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]

c =  .01
F =  .6
Clear[x]
sol1 = 
      NDSolve[{x''[t] + c  x'[t]  + Sin[x[t]] - 
              F Cos[omega t] == 0,  
          x[0] == 1,  x'[0] == 0}, x, {t, 0, 1500},  
        MaxSteps -> 
          1000000]
Plot[{Evaluate[x[t]]/t /. 
            sol1, Evaluate[x'[t]]/t /. sol1}, {t, 0, 100}, 
        PlotStyle -> {{Thickness[0.010], 
              RGBColor[1, 0, 0]}, {Thickness[0.010], RGBColor[0, 0, 1]}}, 
        AxesLabel ->  {"t", "s"}];
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1, 100}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1400, 1500}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]

c =  .03
F =  .6
Clear[x]
sol1 = 
      NDSolve[{x''[t] + c  x'[t]  + Sin[x[t]] - 
              F Cos[omega t] == 0,  
          x[0] == 1,  x'[0] == 0}, x, {t, 0, 15000},  
        MaxSteps -> 
          1000000]
Plot[{Evaluate[x[t]]/t /. 
            sol1, Evaluate[x'[t]]/t /. sol1}, {t, 0, 500}, 
        PlotStyle -> {{Thickness[0.010], 
              RGBColor[1, 0, 0]}, {Thickness[0.010], RGBColor[0, 0, 1]}}, 
        AxesLabel ->  {"t", "s"}];
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 1, 100}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]
ParametricPlot[{x[t], x'[t]} /. sol1 , {t, 10000, 15000}, 
        PlotStyle -> {{RGBColor[1, 0, 0]}}]

\(\(\( (*Here\ we\ very\ the\ initial\ conditions . \ \ The\ transition\ to\ Chaos\ is\ Clearly\ Seen*) \)\(
\)\(
\)\(\(omega\  =  .7;\)

    \(c =  .01;\)

    \(F =  .02;\)

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    x[1500] /. sol1

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1.001, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    \(x[1500] /. sol1;\)

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(omega\  =  .7;\)

    \(c =  .03;\)

    \(F =  .40930;\)

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    \(x[1500] /. sol1;\)

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1.001, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    \(x[1500] /. sol1;\)

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(F =  .40931;\)

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1.001, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    x[1500] /. sol1

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(omega\  =  .7;\)

    \(c =  .03;\)

    \(F =  .6;\)

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    x[1500] /. sol1

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]

    \(Clear[x];\)

    \(sol1 = 
        NDSolve[{\(x''\)[t] + c\ \ \(x'\)[t]\  + Sin[x[t]] - 
                F\ Cos[omega\ t] == 0, \ 
            x[0] == 1.001, \ \(x'\)[0] == 0}, x, {t, 0, 1500}, \ 
          MaxSteps -> 1000000];\)

    \(x[1500] /. sol1;\)

    \(xxx = x[1500] /. sol1;\)

    Print["F= ", F, \ " x[0]= ", x[0] /. sol1, " x[1500]= ", 
      xxx]\)\)\)