Bifurcations dynamical model

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. 

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 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. 

S. Elnashaie, S. Elshishini, Dynamic Modeling, Bifurcation and Chaotic Behaviour of Gas-Solid Catalytic Reactors, Gordon and Breach, 1996, 646 p... 

Aluko, M., and H.-C. Chang, Dynamic modelling of a heterogeneously catalyzed system with stiff Hopf bifurcations, Chem. Eng. Sci., 41, 317-331 (1986). 

Volume 9 DYNAMIC MODELLING, BIFURCATION AND CHAOTIC BEHAVIOUR OF GAS-SOLID CATALYTIC REACTORS by S.S.E.H. Elnashaie and S.S. Elshishini... 

Most of the published literature on reactive distillation (RD) has been focussed mainly on aspects such as conceptual design with the aid of residue curve maps, steady-state multiplicity and bifurcations, and development of equilibrium (EQ) stage and rigorous non-equilibrium (NEQ) steady-state and dynamic models [1, 2]. Relatively little attention has been paid to hardware design. In this chapter the concepts underlying the selection of the appropriate hardware for RD columns are discussed. For RD column design, detailed information on hydrodynamics and mass transfer for the chosen hardware is required, but this information is often lacking. Modern tools such as computational fluid dynamics (CFD) can be invaluable aids in hydrodynamic and mass-transfer studies. 

Because the defining conditions can be solved for the state variables and two parameters e.g. f and q), the above mentioned varieties are said to be of codimension-2. The dynamic model of the reactive flash contains several algebraic, but only one differential equation, when the holdup and pressure are fixed and the phase equilibrium is instantaneous. Such one-dimensional systems cannot exhibit Hopf bifurcations leading to oscillatory behavior. Therefore, dynamic classification is not necessary. 

This chapter covers the basic principles of multiplicity, bifurcation, and chaotic behavior. The industrial and practical relevance of these phenomena is also explained, with referenee to a number of important industrial processes. Chapter 7 eovers the main sources of these phenomena for both isothermal and nonisothermal systems in a rather pragmatic manner and with a minimum of mathematics. One of the authors has published a more detailed book on the subject (S. S. E. H. Elnashaie and S. S. Elshishini, Dynamic Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactors, Gordon Breach, London, 1996) interested readers should eonsult this reference and the other references given at the end of Chapter 7 to further broaden their understanding of these phenomena. 

The catastrophes dynamic model (Fig. 3) determines the interpretation function in the form of bifurcation multitude B(0,t) in the area of control variables multitude GZ(0,t) defining the control dynamic environment integrating impact on external disturbances and peculiarities of the object s dynamics multitude F(W(t),D(t)), determming external environment and stmctural changes in the vessel s behaviour. 

In the following chapters we present the theory of bifurcations of dynamical systems with simple dynamics. It is difficult to over-emphasize the role of bifurcation theory in nonlinear dynamics the reason is quite simple the methods of the theory of bifurcations comprise a working tool kit for the study of dynamical models. Besides, bifurcation theory provides a universal language to communicate and exchange ideas for researchers from different scientific fields, and to understand each other in interdisciplinary discussions. 

Therefore, Andronov s approach (Sec. 8.4) for studying dynamical models has to be corrected in cases where a complete bifurcation analysis may not be possible without moduli. We note, however, that if some fine delicate phenomena may be ignored, or if the problem is restricted to the analysis of non-wandering orbits like equilibrium states, periodic and quasiperiodic motions, a study of the main bifurcations in systems with simple dynamics still remains realistic within the framework of finite-parameter families under certain reasonable requirements (Sec. 8.4). 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

This concludes the mathematical preliminaries needed for our discussion of systems with three or more independent concentrations. The different dynamical responses which arise from the various bifurcations listed above will now be exemplified through a number of model schemes, each with three variables. 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. 

Here, I would like to elaborate further on the theme of bifurcation. Section II describes the present state of bifurcation analysis of nonequilibrium systems and gives some of my personal perspectives on what I consider to be some promising lines of development. Section III reviews a number of physical, chemical, or biological problems which can be modeled by means of this theory and which provide us with illustrations of chemical evolution, the subject of the present volume. A representative case, the origin and selection of chirality, is analyzed in Section IV. Some conclusions regarding the dynamics of self-organizing systems are presented in Section V. 

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