Study of the dynamic behavior of 
Three-Variable Autocatalator
Authors' Data: Housam Binous & Nasri Zakia
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 over one hour using a Pentium 3 800 MHz Computer

κ=10;δ=2 10^-2;σ=5 10^-3;

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]

p=0;

While[p<401,{μ=0.1+0.0002*p,f=sol[μ],p+=1,Print[μ],i=0,k=0,While[i<101,{t0=4+0.08*i,i+=1,solt[i]=Quiet[FindRoot[D[First[Log[10,γ[t]/.f]],{t,1}]==0,{t,t0}]],If[4<(t0/.solt[i])<12 &&Quiet[(D[First[Log[10,γ[t]/.f]],{t,2}]/.solt[i])>0],{pt[k]={μ,First[Log[10,γ[t]/.f]/.solt[i]]},k+=1}]}],tbl[p]=Table[pt[i],{i,1,k-1}]}]

We get the same bifurcation diagram as Figure 3 page 5246 in Peng et al. (1990).

ListPlot[Table[tbl[i],{i,1,401}],PlotRange-> All,AspectRatio->1,AxesOrigin->{0.1,0.2},Frame->True,PlotStyle-> Table[{PointSize[0.0059],RGBColor[1,0,0]},{i,1,401}]]