Pitchfork bifurcations

In this lecture, we're going to take a look at pitchfork bifurcations. In this kind of bifurcation, changing a parameter of the problem leads to the formation or destruction of fixed points. We will begin with the equation $\dot{x} = rx-x^3$.

1;
set(0, "defaultlinelinewidth", 4);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)
graphics_toolkit qt;

error: graphics_toolkit: qt toolkit is not available
error: called from
    graphics_toolkit at line 81 column 5

r_a = linspace(-3, 3,33);

function y = f(x, r)
    y = r*x- x.^3;
end 

function dy = dfdx(x, r)
    dy = r-3*x.^2;
end

for i=1:length(r_a)
    r = r_a(i);
    [sol, fval, info]= fsolve(@(x) f(x, r), [-2 0 2]);
    if info == 1 % Solution has converged
        for j = 1:length(sol)
            root = sol(j);
            slope_at_root = dfdx(root, r);
            if slope_at_root >= 0
                plot(r, root, 'bx');
            else
                plot(r, root, 'ro');
                hold on;
            end
        end
    end
end

xlim([min(r_a) max(r_a)]);
line ([min(r_a) max(r_a)], [0 0], "linestyle", "-", "color", "m");
line ([0 0], [min(r_a) max(r_a)],"linestyle", "-", "color", "m");
xlabel("r");
ylabel("x*");
title("Supercritical pitchfork bifurcation");
hold off;

warning: matrix singular to machine precision, rcond = 7.40149e-17
warning: called from
    fsolve>__dogleg__ at line 532 column 5
    fsolve at line 351 column 11
warning: matrix singular to machine precision, rcond = 7.40149e-17
warning: called from
    fsolve>__dogleg__ at line 532 column 5
    fsolve at line 351 column 11
warning: matrix singular to machine precision, rcond = 9.36751e-17
warning: called from
    fsolve>__dogleg__ at line 532 column 5
    fsolve at line 351 column 11

set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders

function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    end
    
    if (recalc || init)
      r = -2 + 4*get (h.slider1, "value");
      set (h.label1, "string", sprintf ("r: %.1f", r));       
      x = linspace(-2,2);
      xd = r*x - x.^3;
      h.plot = plot(x, xd, x, 0.*x, '-k');
            
      
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
 
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13

set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders

function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    end
    
    if (recalc || init)
      r = -2 + 4*get (h.slider1, "value");
      set (h.label1, "string", sprintf ("r: %.1f", r));       
      x = linspace(-2,2);
      xd = -x + r*tanh(x);
      h.plot = plot(x, xd, x, 0.*x, '-k');
            
      
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
 
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13

In the snippet that follows, we aim at looking at the bifurcation diagram $\dot{x} = -x + r\tanh(x)$

r_a = linspace(-2, 2,33);

function y=f(x, r)
    y=-x + r*tanh(x);
end

function dy=dfdx(x, r)
    dy= -1 + r*1/(cosh(x)).^2;
end

for r = r_a
    [sol, fval, info] = fsolve(@(x) f(x,r), [-2,0, 2]);
    if info == 1 # Solution has converged
        for i=1:length(sol)
            root = sol(i);
            slope_at_root = dfdx(root, r);
            if slope_at_root > 0
                plot(r, root, 'bx');
            else
                plot(r, root, 'ro');
            end
            hold on;
        end
    end
end
xlim([min(r_a), max(r_a)]);
line ([min(r_a) max(r_a)], [0 0], "linestyle", "-", "color", "m");
line ([0 0], [min(r_a) max(r_a)],"linestyle", "-", "color", "m");
xlabel("r");
ylabel("x*");
title("Supercritical pitchfork bifurcation");
hold off;

set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders

function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    end
    
    if (recalc || init)
      r = -2 + 4*get (h.slider1, "value");
      set (h.label1, "string", sprintf ("r: %.1f", r));       
      x = linspace(-2,2);
      xd = r*tanh(x);
      h.plot = plot(x, xd, x, x, x, 0.*x, '-k');
            
      
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
 
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13

set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders

function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    end
    
    if (recalc || init)
      r = -2 + 4*get (h.slider1, "value");
      set (h.label1, "string", sprintf ("r: %.1f", r));       
      x = linspace(-2,2);
      xd = r*x + x.^3;
      h.plot = plot(x, xd,x, 0.*x, '-k');
      h.title = title("\dot{x} = rx + x^3");
      
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
 
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13

In the snippet that follows, we aim at looking at the bifurcation diagram $\dot{x} = -x + r\tanh(x)$

set(0, "defaultlinelinewidth", 2);
set (0, "defaulttextfontname", "TimesNewRoman")
set (0, "defaulttextfontsize", 20)
set (0, "DefaultAxesFontName", "TimesNewRoman")
set(0, 'DefaultAxesFontSize', 20)

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders

function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    end
    
    if (recalc || init)
      r = -2 + 4*get (h.slider1, "value");
      set (h.label1, "string", sprintf ("r: %.1f", r));       
      x = linspace(-2,2);
      xd = r*x + x.^3 - x.^5;
      h.plot = plot(x, xd,x, 0.*x, '-k');
      h.title = title("\\dot{x} = rx + x^3 - x^5");
      h.ylim = ylim([-0.25, 0.25]);
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
 
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

error: value on right hand side of assignment is undefined
error: called from
    update_plot at line 21 column 14

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13

#will take a while to compute
1;
function y=f(x, r)
    y=r*x + x.^3 - x.^5;
end

function dy= dfdx(x, r)
    dy= r + 3*x.^2 - 5*x.^4;
end

x = linspace(-2, 2,50); r_a = linspace(-0.5, 1,100);
xg = linspace(-2,2,20);

for r=r_a
    [sol, fval, info] = fsolve(@(x) f(x,r), xg);
    if info == 1 # Solution has convered
        for i=1:length(sol)
            root = sol(i);
            slope_at_root = dfdx(root, r);
            if slope_at_root > 0
                plot(r, root, 'bx')
            else
                plot(r, root, 'ro')
            end
          hold on;
        end
    end
end
xlim([min(r_a), max(r_a)]);
line ([min(r_a) max(r_a)], [0 0], "linestyle", "-", "color", "m");
line ([0 0], [min(r_a) max(r_a)],"linestyle", "-", "color", "m");
xlabel("r");
ylabel("x*");
title("Subcritical pitchfork bifurcation");
hold off;

clear;
clc; close all;

h.ax = axes ("position", [0.05 0.4 0.5 0.5]);  #reduce plot windows size to accomodate sliders
function y = f(x, t, r)
  y = r*x + x.^3 - x.^5;
end  


function update_plot (obj, init = false)
  ## gcbo holds the handle of the control
  h = guidata (obj);
  replot = false;
  recalc = false;
  switch (gcbo)                       # If we make any change then we replot
    case {h.print_pushbutton}
      fn =  uiputfile ("*.png");
      print (fn);
    case {h.slider1}
      recalc = true;
    case {h.slider2}
      recalc = true;
     case {h.slider3}
      recalc = true;
    end
    
    if (recalc || init)        
      x0 = -0.1+ 0.6*get (h.slider1, "value");
      r = -0.5 + 0.9*get (h.slider2, "value");
      t_end =  10 + 40*get (h.slider3, "value");
      set (h.label1, "string", sprintf ("x0: %.1f", x0));
      set (h.label2, "string", sprintf ("r: %.1f", r));
      set (h.label3, "string", sprintf ("t_end: %.1f", t_end));       
      t=linspace(0,t_end,100)';
      xd = @(x,t) f(x, t, r);
      sol = lsode(xd, x0, t);
      plot(t, sol, "-k");
      xlabel("t"); ylabel("x");               
      ylim([-1.2, 1.2]);
      guidata (obj, h);
    else
      set (h.plot, "ydata", y);
    end
  end
  
  # specify the GUI plot properties like sliders and label formatting
  
  ## print figure
  h.print_pushbutton = uicontrol ("style", "pushbutton",
  "units", "normalized",
  "string", "print plot\n(pushbutton)",
  "callback", @update_plot,
  "position", [0.6 0.45 0.35 0.09]);
  ## guess
  h.label1 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.05 0.25 0.35 0.08]);
  
  h.slider1 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.05 0.20 0.35 0.06]);
  
  h.label2 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Iteration:",
  "horizontalalignment", "left",
  "position", [0.05 0.15 0.35 0.08]);
  
  h.slider2 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.5,
  "position", [0.05 0.10 0.35 0.06]);
  
    h.label3 = uicontrol ("style", "text",
  "units", "normalized",
  "string", "Guess:",
  "horizontalalignment", "left",
  "position", [0.6 0.25 0.35 0.08]);
  
  h.slider3 = uicontrol ("style", "slider",
  "units", "normalized",
  "string", "slider",
  "callback", @update_plot,
  "value", 0.1,
  "position", [0.6 0.20 0.35 0.06]);
  
  set (gcf, "color", get(0, "defaultuicontrolbackgroundcolor"))
  guidata (gcf, h)
  update_plot (gcf, true);

Inline plot failed, consider trying another graphics toolkit
error: __gnuplot_draw_figure__: unknown object class, uicontrol
error: called from
    _make_figures>safe_print at line 125 column 7
    _make_figures at line 49 column 13