<< Graphics`PlotField`;

    
<< Graphics`Arrow`

Must know vector fields

\(\(PlotVectorField[{x, 
          y}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 30, 
\ Axes -> True, \ 
        Frame -> True, 
	
        PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}];\)\)

As you can see the default setting of this command does not give satisfactory results. Below is a modified version in which we specify scaling of vectors in a vector field.

\(\(PlotVectorField[{x, 
          y}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 35, 
\ 
        ScaleFunction -> \((# &)\), \ ScaleFactor\  -> \ 2.5, \ 
        Axes -> True, \ 
	
        PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}];\)\)

\(PlotVectorField[{\(-x\), \(-y\)}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 25, 
\ 
      ScaleFunction -> \((# &)\), \ ScaleFactor\  -> \ 1.75, \ 
      Axes -> True, \ 
      Frame -> 
        True, 
		PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)

\(PlotVectorField[{x\/\@\(x\^2 + y\^2\), 
        y\/\@\(x\^2 + y\^2\)}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 30, 
\ 
      ScaleFunction -> \((# &)\), \ ScaleFactor\  -> \ 1, \ 
      Axes -> True, \ Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)

\(PlotVectorField[{y, 
        x}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 30, 
\ 
      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \ 2, \ 
      Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)

\(PlotVectorField[{\(-y\), 
        x}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 30, 
\ 
      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \ 2, \ 
      Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)

\(PlotVectorField[{\(-y\)/Sqrt[x^2 + y^2], 
        x/Sqrt[\((x^2 + 
                y^2)\)]}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-3\), \ \ 3}, \ {y, \ \(-3\), \ 3}, \ 
	PlotPoints\  -> 36, 

      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \  .2, \ 
      Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-2\), 2}, {\(-2\), 2}}]\)

\(PlotVectorField[{y, \(-x\)}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-5\), \ \ 5}, \ {y, \ \(-5\), 5}, \ 
	
      PlotPoints\  -> \ 
        30, \ \ \ \ ScaleFunction -> \((# &)\), 
\ \ \ \ ScaleFactor\  -> \ 3, \ Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-3\), 3}, {\(-3\), 3}}]\)

More vector fields

\(PlotVectorField[{Sin[y], 
        Sin[x]}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-2\) π, \ 
        2\ π}, \ {y, \ \(-2\) π, \ 2  π}, \ 
	
      PlotPoints\  -> \ 30, 

      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \  .75, \ 
      Axes -> True, \ \ Frame -> True, 	
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\) π, 
            2  π}}]\)

\(PlotVectorField[{2 \((y - 1)\), 
        1}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-π\), \ \ π}, \ {y, \ \(-π\), \ π}, \ 
	
      PlotPoints\  -> \ 
        30, 
\ \ \ \ ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \ 1.5, \ Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-π\), π}, {\(-π\), π}}]\)

\(PlotVectorField[{Sin[y*π], 
        Sin[x*π]}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-2\), \ \ 2}, \ {y, \ \(-2\), \ 2}, \ 
	PlotPoints\  -> \ 40, 

      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \  .2, \ 
      Axes -> True, \ \ Frame -> True, \ 
	
      PlotRange -> {{\(-2\), 2}, {\(-2\), 2}}]\)

\(PlotVectorField[{\(-y\), 
        x}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 30, 
\ 
      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \ 2, \ 
      Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}, 
      Epilog -> {{Thickness[0.01], Blue, Line[{{2, 2}, {\(-2\), 2}}], 
            Arrow[{2, 2}, {0, 2}]}, 
{Thickness[0.01], 
            Green, Line[{{1, \(-1\)}, {3, \(-3\)}}], 
            Arrow[{1, \(-1\)}, {2, \(-2\)}]}}]\)

\(PlotVectorField[{x\^2 - y\^2, 
        2  x\ y}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 40, 
\ 
      ScaleFunction -> \((# &)\), \ \ \ \ ScaleFactor\  -> \ 2, \ 
      Axes -> True, \ \ Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)

{a, b}[[1]]

Clear[pP, n];

    
pP[0] = {2, 2};

    
pP[n_] := pP[n] = 
        pP[n - 1] + 
          0.005*{pP[n - 1][[1]]^2 - pP[n - 1][[2]]^2, 
              2  pP[n - 
                      1][[1]] pP[
                    n - 1][[2]]}

pP[0]

pP[2]

Table[pP[k], {k, 0, 10}]

\(PlotVectorField[{x\^2 - y\^2, 
        2  x\ y}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ {x, \ \(-6\), \ \ 6}, \ {y, \ \(-6\), \ 6}, \ 
	PlotPoints\  -> \ 40, 
\ 
      ScaleFunction -> \((# &)\), ScaleFactor\  -> \ 2, \ 
      Axes -> True, \ \ 
Epilog -> \ {{Red, 
            Thickness[0.01], Line[Table[pP[k], {k, 0, 300}]]}}, 
      Frame -> True, 
	
      PlotRange -> {{\(-5\), 5}, {\(-5\), 5}}]\)