Improved Computation of the Bifurcation Diagram 
for the Three-Variable Autocatalator
Authors' Data: Housam Binous
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :
Bo Peng, Stephen K. Scott, and Kenneth Showalter, "Period Doubling and Chaos in a Three-Variable Autocatalator", The Journal of Physical Chemistry. Vol. 94, No. 13, 1990.

Bifurcation Diagram - Computation takes less than ten minutes using a Core 2 Duo 2.4 GHz Computer

sol[μ_]:=NDSolve[{
D[α[t],{t,1}]== κ+μ γ[t]-α[t] β[t]^2-α[t],
σ D[β[t],{t,1}]== α[t] β[t]^2+α[t]-β[t],
δ D[γ[t],{t,1}]== β[t]-γ[t],
α[0] == 1,β[0]== 1,γ[0]== 1},{α[t],β[t],γ[t]},{t,0,20},MaxSteps->1000000,AccuracyGoal-> 10]

κ=10;δ=2 10^-2;σ=5 10^-3;

iterate=Compile[{μ},{f=sol[μ];Map[{μ,#}&,FindMinValue[Log[10,γ[t]/.f],{t,#,#+0.01}]&/@Table[i,{i,4,10.0,0.01}]]}]

pts=Flatten[Table[iterate[μ],{μ,0.10,0.18,0.0001}],1];

ListPlot[pts,PlotStyle->Table[{PointSize[0.004],RGBColor[0,0,1]},{i,1,Length[pts]}],Frame-> True,ImageSize-> {450,450},FrameLabel->{Style["μ",Italic],Style["Log[10,γ]",Italic]},ImageSize-> {500,500},AspectRatio-> 1]