Bifurcation static

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

The kind of bifurcation that we have just encountered is called static bifurcation (SB). This behavior is very common in many chemical/biological processes. A limited number of industrial examples of this behavior include ... 

The current section has covered numerical techniques and MATLAB codes for investigating the static bifurcation behavior of nonadiabatic lumped systems. 

The static bifurcation behavior and its practical implications have been investigated. We have also formulated the unsteady-state dynamic model and we have used it to study the dynamic behavior of the system by solving the associated IVP numerically. Both the controlled and the uncontrolled cases have been investigated. Two particular reactions have been studied, one with three steady states, and one with five steady states. 

The hysteresis phenomenon, i.e., static bifurcation as described earlier for the nonisother-mal CSTR that is associated with multiple steady states plays an important role in start-... 

Industrial verification of our steady-state model and further static-bifurcation studies of industrial FCC units follow. 

Industrial Verification of the Steady State Model and Static Bifurcation of Industrial Units... 

Results for the Dynamic Behavior of FCC Units and their Relation to the Static Bifurcation Characteristics... 

The control loop affects both the static behavior and the dynamic behavior of the system. Our main objective is to stabilize the unstable saddle-type steady state of the system. In the SISO control law (7.72) we use the steady-state values Yfass = 0.872 and Yrdss = 1.5627 as was done in Figures 7.14(a) to (c). A new bifurcation diagram corresponding to this closed-loop case is constructed in Figure 7.20. 

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

The static bifurcation characteristics of the resulting closed loop system have also been discussed in the previous section and we have seen that the bifurcation diagram of the reactor dense-phase dimensionless temperature, namely a plot of Yrd versus the controller gain Kc is a pitchfork. Such bifurcations are generally structurally unstable when any of the system parameters are altered, even very slightly. 

Develop an algorithm to construct the static bifurcation diagram for an industrial type IV FCC unit. 

Develop the mathematical model of a modern riser reactor FCC unit and construct its static bifurcation diagram numerically. 

Using the simple model of this section, find the static bifurcation characteristics for a typical industrial UNIPOI unit. 

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. 

The reader will have noticed by now that we have introduced many new terms such as Hopf bifurcation point, static limit point, Feigenbaum sequence, Poincare diagram and so forth in this section. These terms have never occurred before in this book and they were left undefined in this section. 

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. 

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

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. 

The software package Auto 97 [95] is useful for computing bifurcation diagrams. It uses an efficient continuation technique for both static and periodic branches of bifurcation diagrams. 

Schmahmann JD (2003). Vascular syndromes of the thalamus. Stroke 34 2264-2278 Schulz UG, Rothwell PM (2001). Major variation in carotid bifurcation anatomy a possible risk factor for plaque development Stroke 32 2522-2529 Scott BL, Jankovic J (1996). Delayed-onset progressive movement disorders after static brain lesions. Neurology 46 68-74 Wardlaw JM, Merrick MV, Ferrington CM et al. (1996). Comparison of a simple isotope method of predicting likely middle cerebral artery occlusion with transcranial Doppler ultrasound in acute ischaemic stroke. Cerebrovascular Diseases 6 32-39 Wardlaw JM, Lewsi SC, Dennis MS etal. (1999). Is it reasonable to assume a particular embolic source from the type of stroke Cerebrovascular Diseases 9(Supp 1) 14... 

Static bifurcation can be studied by means of singular perturbation methodp 1.6. taking c - 0. The problem of finding static bifurcation points turns to the steady state problem similar to the CSTR where is expressed by (16)-From this and from the monotonicity of the fvmction G follows that for a given value of bifurcation parameter Da the maximal number of static bifurcation points corresponds to that of the systems with = 1 and Da <0 Da >. 

We want to explain where this chaos comes from, and to understand the bifurcations that cause the wheel to go from static equilibrium to steady rotation to irregular reversals. 

---