unsw ceic3000 w6 process modeling and analysis

# unsw ceic3000 w6 process modeling and analysis - CEIC3000...

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Unformatted text preview: CEIC3000 Process Modelling and Analysis Session 1, 2012 Week 6 Lecturer: A/Prof. J. Bao Contents 1 Bifurcation Behaviour (continued) 1 1.1 Saddle-node Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Dynamic responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Dynamic responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Bifurcation Behaviour (continued) 1.1 Saddle-node Bifurcation Consider the following system: ˙ x = f ( x,μ ) = μ − x 2 (1) The equilibrium is: f ( x,μ ) = 0 = μ − x 2 e (2) The two solutions are: x e 1 = √ μ (3) x e 2 = − √ μ (4) The Jacobian (and eigenvalue) is ∂f ∂x x e ,μ e = − 2 x e = λ (5) The bifurcation conditions are satisfied for μ e = x e = 0 (6) The second derivative is: ∂ 2 f ∂x 2 = − 2 ̸ = 0 (7) which indicates that there are two solutions in the vicinity of the bifurcation point. We can nowwhich indicates that there are two solutions in the vicinity of the bifurcation point....
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unsw ceic3000 w6 process modeling and analysis - CEIC3000...

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