Unformatted text preview: Mathematica code: MultipleSteadyStatesSurfaceReactions.nb In[51]:= ContourPlot 1 Θ f Θ, Α, 0.005 , Α, 0, 0.3 , Θ, 0, 1 ,
Contours
0 , ContourShading False, PlotPoints 100,
ContourStyle
Thick, Blue , FrameLabel
Style "Α", 16 , Style "Θ", 16
Epilog
Style Text "Γ 0.005", 0.05, 0.6 , 16 5 , 1.0 0.8 Γ 0.005 Θ 0.6 Out[51]= 0.4 0.2 0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 Α
It is also instructive to show how the multiplicity of solutions depends on the parameters Α and Γ. The
steady states are defined by a function G(Θ,Α,Γ)=0. Thus we need to determine the values of Α Α and
Γ Γ for the existence of multiple steady states. Inspecting the above ContourPlot, it is apparent that
we will get multiple steady states when Α lies between the two turning points. Hence we need to determine the locus of points in the ΑΓ plane such that G/ Θ=0. Recall that at bifurcation points /turning
points the quantity G/ Θ=0.
Let us define the following function:
ΓΘ Θ 1 Θ
G Θ; Α, Γ 2 1Θ
Α Thus we need to solve when both G Θ ; Α , Γ
0 and when G ' Θ ; Α , Γ
do this is to eliminate Θ from the following two equations
G
G Θ, Α, Γ 0, 0
Θ This can be done using Mathematica’ Eliminate function:
s
ΓΘ
In[64]:= G Θ_ , Α_...
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 Winter '14
 Trigraph, α, θ, steady states, θs, Γ Θs

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