Fázové portréty nelineárních rovnic

<< Matematika`FazPort`

Příklad 1 :   x'=-2x-y-x y
                   y'=-x-2 y-x y

?NFazPort

?LFazPort

\(f1[x_, y_] = \(-2\) x - y - x\ y; 
f2[x_, y_] = \(-x\) - 2\ y - x\ y; 
    \)

Solve[{f1[x, y] == 0, f2[x, y] == 0}, {x, y}]

\(\(	A1 = 
      Outer[D, {f1[x, y], f2[x, y]}, {x, y}] /. {x -> 0, y -> 0}\)\)

LFazPort[0, 4, -2, -1, -1, -2, Ticks -> None]

\(\(	A1 = 
      Outer[D, {f1[x, y], f2[x, y]}, {x, y}] /. {x -> \(-3\), y -> \(-3\)}
          \)\)

LFazPort[0, -5, 1, 2, 2, 1, Ticks -> None]

\(\(data1 = {{3, 3}, {5, 1}, {1, 3}, {\(-1\), 3}, {5, \(-1\)}, {0, 
          \(-5\)}, {\(-5\), 0}, 

        	\ \ {\(-2.9\), \(-2.9\)}, {\(-1.905\), \(-5\)}, {\(-5\), 
          \(-1.905\)}, {\(-1.5\), \(-5\)}, {\(-5\), \(-1.5\)}}; 

    data2 = {{\(-3.1\), \(-3.1\)}, {\(-2\), \(-5\)}, {\(-5\), \(-2\)}}; 

    data3 = {{\(-1.9\), \(-5\)}, {\(-5\), \(-1.9\)}}; 
\)\)

\(\(g1 = 
      NFazPort[f1, f2, 2.0, \(-5\), 5, \(-5\), 3, data1, 
        DisplayFunction -> Identity]; 

    g2 = NFazPort[f1, f2, 1, \(-5\), 5, \(-5\), 3, data2, 
        DisplayFunction -> Identity]; 

    g3 = NFazPort[f1, f2, 3.5, \(-5\), 5, \(-5\), 3, data3, 
        DisplayFunction -> Identity]; 
\)\)

Needs["Graphics`PlotField`"]

g5 = 
      PlotVectorField[{f1[x, y], f2[x, y]}, {x, -5, 5, 2}, {y, -5, 3, 
          2}, ScaleFunction -> (1.0&), ScaleFactor -> 0.7, 
        DisplayFunction -> Identity]

\(\(
Show[{g1, g2, g3, g5}, 
      PlotRange -> {{\(-5.5\), 5}, {\(-5.5\), 3.5}}, 
      DisplayFunction -> $DisplayFunction]\)\)

Příklad 3 :   x'=y-x x-x
                   y'=3x-x x-y

\(f1[x_, y_] = y - x\ x - x; 
f2[x_, y_] = 3  x - x\ x - y; \)

Solve[{f1[x, y] == 0, f2[x, y] == 0}, {x, y}]

\(\(	A1 = 
      Outer[D, {f1[x, y], f2[x, y]}, {x, y}] /. {x -> 0, y -> 0}\)\)

LFazPort[0, 4, -2, -1, -1, -2, Ticks -> None]

\(\(	A1 = 
      Outer[D, {f1[x, y], f2[x, y]}, {x, y}] /. {x -> 2, y -> 1}\)\)

LFazPort[0, 6, -3, -3, -2, -4, Ticks -> None]

Needs["Graphics`PlotField`"]

g5 = 
      PlotVectorField[{f1[x, y], f2[x, y]}, {x, -4, 5, 2}, {y, -3, 5, 
          2}, ScaleFunction -> (1.0&), ScaleFactor -> 0.6, 
        DisplayFunction -> Identity]

\(data1 = {{5, 5}, {5, \(-2\)}, {1, 5}}; 

    data2 = {{\(-1\), 5}, {\(-1.8\), 5}, {5, \(-3\)}, {5, 0}}; 

    data3 = {{\(-2\), 5}}; \)

\(\(g1 = 
      NFazPort[f1, f2, 8, \(-4\), 5, \(-3\), 5, data1, 
        DisplayFunction -> Identity]; 

    g2 = NFazPort[f1, f2, 3, \(-4\), 5, \(-3\), 5, data2, 
        DisplayFunction -> Identity]; 

    g3 = NFazPort[f1, f2, 1.1, \(-4\), 5, \(-3\), 5, data3, 
        DisplayFunction -> Identity]; 


\)\)

\(\(
Show[{g1, g2, g3, g5}, PlotRange -> {{\(-4.5\), 5}, {\(-3\), 5}}, 
      DisplayFunction -> $DisplayFunction]\)\)