Bifurcation and Stability

Micromixing effects in the Nicolis-Puhl reaction Numerical bifurcation and stability analysis of the IEM model. Chemical Engineering Science 46, 1829-1847. 

Fox, R. O., G. Erjaee, and Q. Zou (1994). Bifurcation and stability analysis of micromixing effects in the chlorite-iodide reaction. Chemical Engineering Science 49, 3465-3484. 

R. Seydel. Practical Bifurcation and Stability Analysis From Equilibrium, to Chaos. Springer-Verlag, New York, 2nd. edition, 1994. 

The interest in periodically forced systems extends beyond performance considerations for a single reactor. Stability of structures and control characteristics of chemical plants are determined by their responses to oscillating loads. Epidemics and harvests are governed by the cycle of seasons. Bifurcation and stability analysis of periodically forced systems is especially important in the... 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

Seydel Practical Bifurcation and Stability Analysis From Equilibrium to Chaos... 

Modern developments in bifurcation and stability theory, which lead on to chaos or turbulence in dynamical systems, only serve to show why it is worth embedding as much complexity as possible within the constitutive description of material properties. There is a sound intellectual case for trying to subsume fine-scale motions of polymer melt molecules within an Oldroyd... 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

Mohl et al. [21, 22] implemented a dynamic EQ model (with Murphree type efficiencies) in the DIVA simulator and carried out a numerical bifurcation and stability analysis on the MTBE and TAME processes. They also show that the window of opportunity for MSS to actually occur in the MTBE process is quite small. 

Seydel, R. 1994. Practical Bifurcation and Stability Analysis., 2nd ed. Springer-Verlag New York. 

The bifurcation of new solutions at exactly the point where one solution loses stability is not a coincidence, it is a general property of the solutions of nonlinear equations. (This general relation between bifurcation and stability of solutions of nonlinear equations can be explained using topological degree theory, which is beyond the scope of this discussion.)... 

Methods are being explored for enabling microscopic simulators to perform system-level analysis—mainly numerical bifurcation and stability analy-... 

Burrougbs, E., 2003. Convection in a Thermosypbon, Bifurcation and Stability Analysis (Pb.D. dissertation). Tbe University of New Mexico, Albuquerque, New Mexico. 

Newhouse, S., Palis, J. and Takens, F. [1983] Bifurcations and stability of families of diffeomorphisms, Publ Math. IHES 57, 5-71. 

Figure 11 Relationship between structure bifurcation and stability condition in gas-solid systems. Reprinted from Chen et al (2012) with permission from Elsevier.

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