Bifurcation dynamic

Ezra G S 1996 Periodic orbit analysis of molecular vibrational spectra-spectral patterns and dynamical bifurcations in Fermi resonant systems J. Chem. Phys. 104 26... 

An early example of dynamical bifurcation is seen for SN2/ET borderline reaction shown in Scheme 7.65 68,70 75... 

Combined Static and Dynamic Bifurcation Behavior of Industrial FCC Units... 

We have studied the dynamic behavior of FCC units in Section 7.2.3. Here we explain the dynamic bifurcation behavior of FCC type IV units. The dynamic model that we use will be more general than the earlier one. Specifically, we will relax the assumption of negligible mass capacity of gas oil and gasoline in the dense catalyst phase. This relaxation is based upon considering the catalyst chemisorption capacities of the components. 

In this fermentation process, sustained oscillations have been reported frequently in experimental fermentors and several mathematical models have been proposed. Our approach in this section shows the rich static and dynamic bifurcation behavior of fermentation systems by solving and analyzing the corresponding nonlinear mathematical models. The results of this section show that these oscillations can be complex leading to chaotic behavior and that the periodic and chaotic attractors of the system can be exploited for increasing the yield and productivity of ethanol. The readers are advised to investigate the system further. 

We will use the model to explore the complex static/dynamic bifurcation behavior of this system in the two-dimensional D — Cso parameter space and show the implications of bifurcation phenomena on substrate conversion and ethanol yield and productivity. The system parameters for the specific fermentation unit under consideration are given below. 

In the design problem, the dilution rate D = q/V is generally unknown and all other input and output variables are known. In simulation, usually D is known and we want to find the output numerically from the steady-state equations. For this we can use the dynamic model to simulate the dynamic behavior of the system output. Specifically, in this section we use the model for simulation purposes to find the static and dynamic output characteristics, i.e., static and dynamic bifurcation diagrams, as well as dynamic time traces. 

Figure 7.25 compares the experimental results for this data, drawn in a solid curve, with the simulated results in dashed form. Further details of the static and dynamic bifurcation behavior of this system are shown in Figure 7.27. 

This phenomenon of increased conversion, yield and productivity through deliberate unsteady-state operation of a fermentor has been known for some time. Deliberate unsteady-state operation is associated with nonautonomous or externally forced systems. The unsteady-state operation of the system (periodic operation) is an intrinsic characteristic of this system in certain regions of the parameters. Moreover, this system shows not only periodic attractors but also chaotic attractors. This static and dynamic bifurcation and chaotic behavior is due to the nonlinear coupling of the system which causes all of these phenomena. And this in turn gives us the ability to achieve higher conversion, yield and productivity rates. 

In this case the feed sugar concentration is very high. Figures 7.32(A) to (D) show the static and dynamic bifurcation diagrams with the dilution rate D as the bifurcation parameter and an enlargement of the chaotic region. 

We have used a model for anaerobic fermentation in this section to simulate the oscillatory behavior of an experimental fermentor. Both the steady state and the dynamic behavior of the fermentor with Zymomonas mobilis were investigated. The four ODE model simulates the fermentor quite well. Further studies have shown that this model is suitable for scaling-up and for the design of commercial fermentors. Our model has shown the rich static and dynamic bifurcation characteristics of the system, as well as its chaotic ones. All these characteristics have been confirmed experimentally and the oscillatory/chaotic fermentor model is highly suitable for design, optimization and control purposes. 

The most important dynamic bifurcation is Hopf bifurcation. This occurs when Ai and A2 cross the imaginary axis into the right half-plane of C as the bifurcation parameter g changes. At the crossing point both roots are purely imaginary with det(A) > 0 and tr(A) = 0, making Ay2 = i y/det(A). At this value of g, periodic solutions (stable limit cycles) start to exist as depicted in Figures 10 and 11 (A-2). 

This limit cycle represents a trajectory that starts at the static saddle point and ends after one period at the same saddle point. This trajectory is called the homoclinical orbit and will occur at some critical value jiuc- It has an infinite period and therefore this bifurcation point is called infinite period bifurcation . For p < hc the limit cycle disappears. This is the second most important type of dynamic bifurcation after Hopf bifurcation. 

Research Areas Modeling, Simulation and Optimization of Chemical and Biological Processes, Clean Fuels (Hydrogen, Biodiesel and Ethanol), Fixed and Fluidized Bed Catalytic Reactors, Nonlinear Dynamics, Bifurcation and Chaos,... 

A dynamic bifurcation occurs when the dynamic behavior of the solution to a system undergoes a qualitative change. For example, a subcritical Hopf bifurcation occurs when a dynamic system changes from a stable node to a limit cycle. Again, AUTO can be used to determine parameter changes that cause this bifurcation to occur. 

The most important dynamic bifurcation point is the Hopf bifurca-... 

Palsson, B.O. T.M. Groshans. 1988. Mathematical modelling of dynamics and control in metabolic networks. VI. Dynamic bifurcations in single biochemical control loops. J. Theor. Biol. 131 43-53. 

See, for example, the following and references contained therein E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, /. Chem. Phys., 81, 1115 (1984). Intramolecular Vibrational Relaxation and Spectra of CH and CD Overtones in Benzene and Perdeuterobenzene. S. P. Neshyba and N. De Leon,. Chem. Phys., 86, 6295 (1987). Qassical Resonances, Fermi Resonances, and Canonical Transformations for Three Nonlinearly Coupled Oscillators. S. P. Neshyba and N. De Leon,. Chem. Phys., 91, 7772 (1989). Projection Operator Formalism for the Characterization of Molecular Eigenstates Application to a 3 4 R nant System. G. S. Ezra, ]. Chem. Phys., 104, 26 (1996). Periodic Orbit Analysis of Molecular Vibrational Spectra Spectral Patterns and Dynamical Bifurcations in Fermi Resonant Systems. Also see Ref. 6. 

Plaschko P, Brod K (1995) Nichtlineare dynamik, bifuikation und chaotische systeme [Non-linear Dynamics, Bifurcation and Chaotic Systems], Vieweg Verlag, Braunschweig... 

A variety of bifurcation phenomena in chemical systems have been discovered during the last few decades [3]. Results have been interpreted in terms of methodology considerably improved and refined by developments in non-linear dynamics. Bifurcation is also quite common in transition from one state to another in biological systems indicated by small change in timing or electrical conductivity. 

Koper MTM (1996) Oscillations and complex dynamical bifurcations in electrochemical systems. In Prigogine I, Rice SA (eds) Advances in chemical physics, vol 92. Wiley, Hoboken, pp 161-298... 

Garhyan P, Elnashaie S. Static/dynamic bifurcation and chaotic behavior of an ethanol fermentor. Ind Eng Chem Res 2004 43(5) 1260—73. 

Recent developments of plastic stents have aimed to improve the resistance of plastic stents to external compression forces. Therefore, metal has been incorporated into the plastic material of the stent. One of the latest developments is the dynamic bifurcation stent made of silicone (Freitag et al. 1994). This Dynamic stent (Riisch, Kernen, Germany) is reinforced with horseshoe-shaped steel struts. A posteriorly located flexible membrane allows dynamic compression of the stent during coughing, whereas the steel struts prevent airway compression from external forces. Theoretically, this stent mimics the mechanical dynamics of the normal trachea. The distal end is a Y shape which rides on the carina to prevent distal migration. 

Garhyan, P. and Elnashaie, S. S. E. H. Exploitation of static/dynamic bifurcation and chaotic behavior of fermentor for higher productivity of fuel ethanol. AIChE Annual Meeting, 2001. 

The most important dynamic bifurcation is the Hopf bifurcation point, when A] and X2 cross the imaginary axis into positive parts of X and A.2-This is the point where both roots are purely imaginary and at which... 

Sheintuch, M. and Luss, D., 1987, Chem. Engng Sci., vol. 42, 41 Identification of observed dynamic bifurcations and development of qualitative models. 

In Section 2 the formalism of the Master equation, our main tool in the microscopic approach developed in this chapter, is laid down. This formalism, which constitutes a convenient intermediate between purely microscopic and macroscopic theories, accounts for microscopic dynamics through the fluctuations of the macrovariables. We review the main assumptions at the basis of this description, the formal properties of its solutions, and some results established in the early literature on this subject in connection with bifurcations leading to steady-state solutions. We subsequently focus on dynamical bifurcation phenomena and discuss, successively, thermodynamic fluctuations near Hopf bifurcation (Section 3) and in the regime of deterministic chaos (Section 4). A summary and suggestions for further study are given in the final Section 5. 

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