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plotting - Transcritical Bifurcation phase portraits


An example equation for a Transcritical Bifurcations is given by:


$$\dfrac{dx}{dt} = f(x, r) = r x - x^2$$


In Mathematica, we can define the function as:


  f[x_, r_] := r x - x^2


We can create a grid of plots to show the Transcritical bifurcation as:


  p1 = Plot[f[x, 0], {x, -3, 3}, PlotRange -> {{-3, 3}, {-4, 3}}, Frame ->    True, 
 FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r=0 case"}}, BaseStyle -> 12, 
 RotateLabel -> False, PlotTheme -> "Classic", 
 PlotStyle -> Thick, ImageSize -> 250]; 
   p2 = Plot[f[x, 3], {x, -5, 5}, PlotRange -> {{-5, 5}, {-4, 3}}, Frame -> True, 
 FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r>0 case"}}, BaseStyle -> 12, 
 RotateLabel -> False, PlotTheme -> "Classic", 
 PlotStyle -> Thick, ImageSize -> 250]; 

  p3 = Plot[f[x, -3], {x, -5, 5}, PlotRange -> {{-5, 5}, {-4, 3}}, Frame -> True, 
 FrameLabel -> {{"f(x,\[Lambda]}", None}, {"x", "r<0 case"}}, BaseStyle -> 12, 
 RotateLabel -> False, PlotTheme -> "Classic", 
 PlotStyle -> Thick, ImageSize -> 250]; 
 Grid[{{p1, p2, p3}}, Frame -> True, FrameStyle -> LightGray]

However, what is the best approach to having it look like the grid below by adding the arrows and circles for stability and type of stability?


enter image description here


Is there a way to generalize this for different type of bifurcations (Hopf, Supercritical ...)?



Answer




Code


phasePortrait[f_, {{xmin_, xmax_}, {ymin_, ymax_}}] := Plot[
  f[x], {x, xmin, xmax},
  Frame -> True, PlotStyle -> Directive[Black, Thick],
  ImageSize -> 500, PlotRange -> {{xmin, xmax}, {ymin, ymax}},
  Epilog -> {getMarkers[f], getArrows[f, {xmin, xmax}]}
  ]

right = Triangle[{{2, 0}, {-1, 1}, {-1, -1}}];
left = Triangle[{{-2, 0}, {1, 1}, {1, -1}}];

stable = Disk[];
unstable = {White, Disk[], Black, Thick, Circle[]};
halfStableRight = {White, Disk[], Black, Thick, Circle[], Disk[{0, 0}, {1, 1}, {-Pi/2, Pi/2}]};
halfStableLeft = {White, Disk[], Black, Thick, Circle[], Disk[{0, 0}, {1, 1}, {Pi/2, 3 Pi/2}]};

insetMarker[marker_, x_] := Inset[Graphics[marker], {x, 0}, {0, 0}, Scaled[{0.05, 0.05}]]

getMarkers[f_] := Module[{x},
  Switch[
    {f[x - 0.01], f[x + 0.01]},

    {_?Positive, _?Positive}, insetMarker[halfStableLeft, x],
    {_?Negative, _?Negative}, insetMarker[halfStableRight, x],
    {_?Positive, _?Negative}, insetMarker[stable, x],
    {_?Negative, _?Positive}, insetMarker[unstable, x]
    ] /. Solve[f[x] == 0, x, Reals]
  ]

getArrows[f_, {xmin_, xmax_}] := Module[{x, sols, pos},
  sols = DeleteDuplicates[x /. Solve[f[x] == 0, x, Reals]];
  sols = Select[sols, xmin < # < xmax &];

  sols = Prepend[sols, xmin];
  sols = Append[sols, xmax];
  pos = MovingAverage[sols, 2];
  If[f[#] > 0, insetMarker[right, #], insetMarker[left, #]] & /@ pos
  ]

Usage


A simple usage example is this:


f[r_][x_] := r x - x^2
phasePortrait[f[-1], {{-3, 3}, {-4, 3}}]


Mathematica graphics


Note the way the function is defined, f[r_][x_] = ..., it is imperative to define the function in this way. The function passed to phasePortrait must be dependent on x only. The second argument of phasePortrait is the desired plot range in the form {{xmin, xmax}, {ymin, ymax}}.


Transcritical bifurcation


f[r_][x_] := r x - x^2
Row[{
  phasePortrait[f[-1], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[0], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[1], {{-3, 3}, {-4, 3}}]
  }]


Mathematica graphics


Supercritical pitchfork bifurcation


f[r_][x_] := r x - x^3
Row[{
  phasePortrait[f[-1], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[0], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[1], {{-3, 3}, {-4, 3}}]
  }]


Mathematica graphics


Subcritical pitchfork bifurcation


f[r_][x_] := r x + x^3
Row[{
  phasePortrait[f[-1], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[0], {{-3, 3}, {-4, 3}}],
  phasePortrait[f[1], {{-3, 3}, {-4, 3}}]
  }]

Mathematica graphics



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