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, ) = x 2 (1) The equilibrium is: f ( x, ) = 0 = x 2 e (2) The two solutions are: x e 1 = (3) x e 2 = (4) The Jacobian (and eigenvalue) is f x x e , e = 2 x e = (5) The bifurcation conditions are satisfied for 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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