MultipleSteadyStatesSurfaceReactions

Nb in51 contourplot 1 f 0005 0 03 0 1

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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 Α-Γ 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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