In Matlab:

BifurcDiagramDriver.m:

%This m-file will run through a loop to create a bifurcation diagram %derivs = 'saddlenode'; saddlenodehandle = @saddlenode; check_function = isa(saddlenodehandle, 'function_handle'); if ~check_function fprintf('ERROR: Did not get handle to function. '); end dim=1; y_init = -0.1; T0 = 0; Tspan = [0 20]; count=0; rrange=[-5:0.1:0.0]; %Change this as appropriate for given bifurcation type %Try to eliminate ranges where solution shoots off %to infinity ysave=[ ]; for r=-5:0.1:0.0 %set this equal to the rrange values above clear t; clear y; [t,y]=ode45(@(t,y)saddlenodehandle(t,y,r),Tspan,y_init); count = count+1; ysave(count,:) = y(end,:); end plot(rrange,ysave); %Next see if we can reverse time and find the unstable branches saddlenodehandle2 = @saddlenode_revT; check_function2 = isa(saddlenodehandle2, 'function_handle'); if ~check_function2 fprintf('ERROR: Did not get handle to function. '); end count=0; y_init=0; rrange2=[-5:0.1:0.0]; ysave_unstable = [ ]; for r=-5.0:0.1:0.0 clear t; clear y; [t,y]=ode45(@(t,y)saddlenodehandle2(t,y,r),Tspan,y_init); count = count+1; ysave_unstable(count,:) = y(end,:); end if (length(rrange2)==length(ysave_unstable)) hold on,plot(rrange2,ysave_unstable,'r-'); else LL=length(ysave_unstable) figure,plot(ysave_unstable) end

return

saddlenode.m:

function ydot = saddlenode(t,y,r) %This m-file contains the derivs for a first order DE which undergoes a saddle- %node bifurcation as r varies. ydot = [y.*y+r];

2. Now modify these programs to produce the other types of one-dimensional bifurcation diagrams: transcritical, subcritical pitchfork and supercritical pitchfork. You may need to change y _init to see each branch, as well as the range of the control parameter. Print these out and label each branch with its stability