Safe bifurcation

The area of reactor design has been widely studied, and there are many excellent textbooks that cover this subject. Most of the emphasis in these books is on steady-state operation. Dynamics are also considered, but mostly from the mathematical standpoint (openloop instability, multiple steady states, and bifurcation analysis). The subject of developing effective stable closedloop control systems for chemical reactors is treated only very lightly in these textbooks. The important practical issues involved in providing reactor control systems that achieve safe, economic, and consistent operation of these complex units are seldom understood by both students and practicing chemical engineers. 

For the case of the safe/dangerous points on a stability boundary where periodic trajectories do not exist (the case of global bifurcations), the situation becomes less definite and caimot yet be well specified in general. However, it is well understood in the main cases (see below). 

For the remainder of this section we consider only safe and dangerous stability boundaries of codimension one. This allows us to use only one bifurcation parameter. We therefore assume that at 5 = 0, the system... 

Fig. 14.2.1. Super-critical Andronov-Hopf bifurcation occurs on the safe boundary.

Summary The set of principal stability boundaries of equilibrium states consists of surfaces of three kinds Si, Sr and Ss. Only the Si-like boundaries are safe. As for periodic orbits, there are nine types of principal stability boundaries among them Se, Sg, Sio, Sn are dangerous, while S2, S3, S4 S5 and Si, S2 2ire safe (the latter two correspond to the subcritical Andronov-Hopf and flip bifurcations, respectively). 

Bautin, N. N. and Shilnikov, L. P. [1980] Suplement I Safe and dangerous boundaries of stability regions, The Hopf Bifurcation and Its Applications Russian translation of the book by Marsden, J. E. and McCracken, M. (Mir Moscow). 

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