Complex dynamic behavior

Whereas there is extensive Hterature on design methods for azeotropic and extractive distillation, much less has been pubUshed on operabiUty and control. It is, however, widely recognized that azeotropic distillation columns are difficult to operate and control because these columns exhibit complex dynamic behavior and parametric sensitivity (2—11). In contrast, extractive distillations do not exhibit such complex behavior and even highly optimized columns are no more difficult to control than ordinary distillation columns producing high purity products (12). 

Note that while a system s static complexity certainly influences its dynamical complexity, the two measures are clearly not equivalent. A system may be structurally rather simple (i.e. have a low static complexity), but have a complex dynamical behavior. (Think of the chaotic behavior of Feigenbaum s logistic equation, for example). 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

Calculations of alkali metal allyl derivatives involving all alkali metals (Li-Cs) indicate a preferred geometry with the metal symmetrically bound in a predominantly electrostatic manner to all three carbon atoms.143 Solution studies of allyllithium in ether indicate the compounds to be highly aggregated in THF complex dynamic behavior is observed. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Besides the analytical techniques, the theoretical description of polymer brushes allows a deeper understanding of the complex dynamic behavior of polymers on surfaces and is useful for future developments. Here, Roland Netz gives - also for the non-expert - a very helpful theoretical background on the theoretical approaches for the description of neutral and charged polymer brushes. 

A complex dynamical behavior was experimentally and numerically found in a system of spin- atoms in an optical resonator with near-resonant cw laser light and external static magnetic field [69]. Three-dimensional Bloch equations were solved, and a chaotic motions was found and compared with experiment. 

Accordingly, we conclude that the dual-site MR approach is compatible with the ammonia inhibition effects observed during unsteady SCR experiments, as well as with the oxygen dependence of the SCR kinetics at low temperatures, and can be successfully applied to simulate the complex dynamic behavior of... 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

The dynamic behavior of fixed-bed reactors has not been extensively investigated in the literature. Apparently the only reaction which has received close attention is the oxidation over platinum catalysts. The investigations reveal interesting and complex dynamic behavior and show the occurrence of oscillatory and chaotic behavior [88-90], It is easy to speculate that further studies of the dynamic behavior of catalytic/biocatalytic reactions will reveal similar complex dynamics like those discovered for the CO oxidation over a Pt catalyst, since most of these phenomena are due to nonmonotonicity of the rate process which is widespread in catalytic systems. 

Whereas the operation of batch reactors is intrinsically unsteady, the continuous reactors, as any open system, allow for at least one reacting steady-state. Thus, the control problem consists in approaching the design steady-state with a proper startup procedure and in maintaining it, irrespective of the unavoidable changes in the operating conditions (typically, flow rate and composition of the feed streams) and/or of the possible failures of the control devices. When the reaction scheme is complex enough, the continuous reactors behave as a nonlinear dynamic system and show a complex dynamic behavior. In particular, the steady-state operation can be hindered by limit cycles, which can result in a marked decrease of the reactor performance. The analysis of the above problem is outside the purpose of the present text ... 

II. Complex dynamical behavior in clusters and proteins, and data mining to extract information on dynamics. 

COMPLEX DYNAMICAL BEHAVIOR IN CLUSTERS AND PROTEINS, AND DATA MINING TO EXTRACT INFORMATION ON DYNAMICS... 

This inherent complexity of ABC systems is reflected by a more complex dynamic behavior in the following respects ... 

Some ABC systems can give rise to unusual and complex dynamic behavior such as photochemical bistability, corresponding to the possible presence of two different photostationary states for the same irradiation and initial concentration conditions. The system ABC, 2cj)s, 2kc (see Table 3) is bistable under conditions in which ba/ ca 1 and AbsJ, > 5. We assume that the molar extinction coefficients of A and B are equivalent (eA eB) and that C does not absorb (e = 0). 

EXAMPLES OF COMPLEX DYNAMIC BEHAVIOR USING EMPIRICAUDIRECT MODELS... 

In the last section we focused on mechanistic requirements that give rise to simple periodic oscillations. The statement that more complex dynamic behaviors have been observed for all electrochemical oscillators is hardly exaggerated, however. The expression more complex dynamics includes all phenomena whose mathematical description requires at least three variables. Perhaps the most popular complex behavior is deterministic chaos, of which there are numerous clear-cut examples for oscillating electrochemical systems in the literature. More unusual... 

HIE) Rathousky, J., Hlavacek, V. Theoretical Investigation of Complex Dynamic Behavior... 

---