Mixed-mode oscillations in chemical systems
Authors' Data: Housam Binous & Nasri Zakia
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :

V. Petrov, S. K. Scott and K. Showalter, Mixed-mode oscillations in chemical systems, J. Chem. Phys. 97 (9) 1992 pp. 6191-6198.

Three coexisting stable oscillations exist for  μ = 0.29776

Period-2 oscillations for μ=0.29776

κ=2.5;δ=1.0;σ=0.013;

sol[μ_]:=NDSolve[{
D[α[t],{t,1}]== μ  (κ+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,0,200},MaxSteps->1000000,AccuracyGoal-> 10]

f=sol[0.29776]

Plot[Log[β[t]/.f],{t,50,60}]

(1^3) state for μ=0.29776.  We get one large plus three small excursions.

f=sol[0.29795]

Sol[μ_]:=NDSolve[{
D[α[t],{t,1}]== μ  (κ+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] == α[t]/.f/.t-> 200,β[0]== β[t]/.f/.t-> 200,γ[0]== γ[t]/.f/.t-> 200},{α[t],β[t],γ[t]},{t,0,200},MaxSteps->1000000,AccuracyGoal-> 10]

f=Sol[0.29776]

Plot[Log[β[t]/.f],{t,50,60}]

(1^2) state for μ=0.29776.  We get one large plus two small excursions.

f=sol[0.29965]

Sol[μ_]:=NDSolve[{
D[α[t],{t,1}]== μ  (κ+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] == α[t]/.f/.t-> 200,β[0]== β[t]/.f/.t-> 200,γ[0]== γ[t]/.f/.t-> 200},{α[t],β[t],γ[t]},{t,0,200},MaxSteps->1000000,AccuracyGoal-> 10]

f=Sol[0.29776]

Plot[Log[β[t]/.f],{t,50,60}]

Variation of observed maxima in log(β) with increasing μ for κ = 2.5, δ = 1.0 and σ = 0.013.

μ=0.2970;

f=sol[μ]

p=0;

While[p<401,{μ=0.2970+0.000017*p,f=Sol[μ],p+=1,Print[p," ",μ],i=0,k=0,While[i<201,{t0=50+0.1*i,i+=1,solt[i]=Quiet[FindRoot[D[First[Log[10,β[t]/.f]],{t,1}]==0,{t,t0}]],If[50<(t/.solt[i])<70 &&Quiet[(D[First[Log[10,β[t]/.f]],{t,2}]/.solt[i])<0],{pt[k]={μ,First[Log[10,β[t]/.f]/.solt[i]]},k+=1}]}],tbl[p]=Table[pt[i],{i,1,k-1}],Print[k]}]

We get the same diagram as Figure 1 page 6192 in Petrov et al. (1990).

ListPlot[Table[tbl[i],{i,1,390}],PlotRange-> All,AspectRatio->1,AxesOrigin->{0.297,0.0},Frame->True,PlotStyle-> Table[{PointSize[0.0059],RGBColor[1,0,0]},{i,1,390}]]