Reactions chaotic

T. Komatsuzaki and R. S. Berry, Regularity in chaotic reaction paths. II. Ar6. Energy dependence and visualization of the reaction bottleneck, Phys. Chem. Chem. Phys. 1, 1387 (1999). 

For reactions with non-simple pathways or networks, the formulas and procedures described so far are not valid. Any step involving two or more molecules of intermediates as reactants destroys the linearity of mathematics, and any intermediate that builds up to higher than trace concentrations makes the Bodenstein approximation inapplicable. Such non-simple reactions are quite common and will be examined later in this chapter. The discussion of reactions with abnormal nonsimple behavior—detonations and periodic or chaotic reactions—will be sketched in Chapter 14. However, a discussion of more than only their most elementary facets is beyond the scope of this book. The reader interested in more than this is referred to the literature cited in that chapter. 

Self-acceleration by a product or intermediate is a key factor in periodic or chaotic reactions, a topic to be visited in Chapter 14. 

Figure A3.14.7. Example oscillatory time series for CO + O2 reaction in a flow reactor corresponding to different P-T locations in figure A3,14,6 (a) period-1 (b) period-2 (c) period-4 (d) aperiodic (chaotic) trace (e) period-5 (1) period-3.

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

In tills chapter we shall examine how such temporal and spatial stmctures arise in far-from-equilibrium chemical systems. We first examine spatially unifonn systems and develop tlie tlieoretical tools needed to analyse tlie behaviour of systems driven far from chemical equilibrium. We focus especially on tlie nature of chemical chaos, its characterization and the mechanisms for its onset. We tlien turn to spatially distributed systems and describe how regular and chaotic chemical patterns can fonn as a result of tlie interjilay between reaction and diffusion. 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

Figure C3.6.4(a) shows an experimental chaotic attractor reconstmcted from tire Br electrode potential, i.e. tire logaritlim of tire Br ion concentration, in tlie BZ reaction [F7]. Such reconstmction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements (A (tj) / = 1,...

Figure C3.6.4 Single-handed chaotic attractor and next-amplitude map reconstmcted from experimental data for tire BZ reaction, (a) The reconstmcted attractor projected onto tire + x)) plane (see tire text for a...

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Catalytic reactors can roughly be classified as random and structured reactors. In random reactors, catalyst particles are located in a chaotic way in the reaction zone, no matter how carefully they are packed. It is not surprising that this results in a nonuniform fiow over the cross-section of the reaction zone, leading to a nonuniform access of reactants to the outer catalyst surface and, as a consequence, undesired concentration and temperature profiles. Not surprisingly, this leads, in general, to lower yield and selectivity. In structured reactors, the catalyst is of a well-defined spatial structure, which can be designed in more detail. The hydrodynamics can be simplified to essentially laminar, well-behaved uniform fiow, enabling full access of reactants to the catalytic surface at a low pressure drop. 

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. 

Stimulus molecules approach the receptor area in a random distribution. Therefore, there cannot be a homogeneous distribution of chemical or enzymic processing capabilities over the area, as this would produce a chaotic mass of information. The capabilities of such precise chemoreceptory discrimination that we observe can only arise from an ordered system in such a way that specific reaction-types would be localized, or at least be concentrated in specific areas of the epithelium. ... 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

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. 

We demonstrate the use of Matlab s numerical integration routines (ODE-solvers) and apply them to a representative collection of interesting mechanisms of increasing complexity, such as an autocatalytic reaction, predator-prey kinetics, oscillating reactions and chaotic systems. This section demonstrates the educational usefulness of data modelling. 

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