Control of chaotic behavior of a non-isothermal 
chemical system
Authors' Data: Housam Binous & Nasri Zakia
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :
S. K. Scott and A. S. Tomlin, Period doubling and other complex bifurcations in non-isothermal chemical systems, Phil. Trans. R. Soc. Lond. A (1990) 332, 51-68.

Governing equations. α, β and ϕ are dimensionless concentrations of species A and B and dimensionless temperature. μ is a bifurcation parameter.

sol[μ_]:=NDSolve[{D[α[t],{t,1}]== μ Exp[δ ϕ[t]]-α[t] β[t]^2-(Subscript[κ, u])α[t],
 D[β[t],{t,1}]== α[t] β[t]^2+(Subscript[κ, u])α[t]-β[t],
D[ϕ[t],{t,1}]== β[t]-γ ϕ[t],α[0]== 1,β[0]== 0.5,ϕ[0]== 0},{α[t],β[t],ϕ[t]},{t,0,10000},MaxSteps-> 1000000]

(Subscript[κ, u])=10^-3;γ=0.05;δ=0.025;

Time series for μ = 0.005. Stable behavior is obtained.

fstab=sol[0.005]

Plot[β[t]/.fstab,{t,10,10000},PlotRange-> All]

Time series for μ = 0.5586. Chaotic behavior is obtained.

fchaotic=sol[0.5586]

Plot[β[t]/.fchaotic,{t,500,2000},PlotRange-> All]

Chaos Control
		
		(dX/dt)=f(X)-(Subscript[K, u]) (X-(Subscript[X, stable]))
		
		with (Subscript[K, u])=0.1 and X=(α, β, ϕ)

stab[μ_]:=NDSolve[{D[α[t],{t,1}]== μ Exp[δ ϕ[t]]-α[t] β[t]^2-(Subscript[κ, u])α[t]-Ku(α[t]-First[α[t]/.fstab]),
 D[β[t],{t,1}]== α[t] β[t]^2+(Subscript[κ, u])α[t]-β[t]-Ku(β[t]-First[β[t]/.fstab]),
D[ϕ[t],{t,1}]== β[t]-γ ϕ[t]-Ku (ϕ[t]-First[ϕ[t]/.fstab]),α[0]== 1,β[0]== 0.5,ϕ[0]== 0},{α[t],β[t],ϕ[t]},{t,0,10000},MaxSteps-> 10000]

Ku=0.1;

fo=stab[0.5586]

Plot[β[t]/.fo,{t,10,10000},PlotRange-> All]