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Chemical reaction chaotic state

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. [Pg.171]

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. [Pg.270]

The U-sequence has been found in experiments on the Belousov—Zhabotinsky chemical reaction. Simoyi et al. (1982) studied the reaction in a continuously stirred flow reactor and found a regime in which periodic and chaotic states alternate as the flow rate is increased. Within the experimental resolution, the periodic states occurred in the exact order predicted by the U-sequence. See Section 12.4 for more details of these experiments. [Pg.372]

A particular class of problems whose solutions are sensitive to initial conditions is known as chaotic problems. The phenomenon of chaos has been observed in fluid mechanics, chemical reactions, and biological systems (Cvitanocic, 1987). A special feature of these systems is unpredictability. The chaotic solutions are so sensitive to initial conditions that two systems with minute differences in their initial states can eventually diverge from each other. Thus their long-term dynamics are unpredictable. [Pg.40]

At the end of this chapter we will now briefly discuss a theoretical approach to the description of chaotic processes encountered in chemical kinetics, see Section 1.3 and Sections 6.2.2.4, 6.3.2.4. In Section 6.2.2.4 we described the method of generation and physical meaning of a chaotic state of the Belousov-Zhabotinskii reaction carried out in a flow reactor. [Pg.271]

On the other hand, the abundance of experimental material stimulates an evolution of the theories explaining non-linear phenomena. For example, as shown above, the transition in a chemical reaction from the stationary state to the state of periodical oscillations, the so-called Hopf bifurcation, is a certain elementary catastrophe. The transition in a chemical reaction to the chaotic state may be explained in terms of catastrophes associated with a loss of stability of a certain iterative process or by using the notion of a strange attractor (anyway, it turns out that both the systems are closely related). The equations of a chemical reaction with diffusion have been extensively studied lately. Based on the progress being made in this area, further interesting achievements in theory may be anticipated, particularly for the phenomena associated with catastrophes — the loss of stability by a non-linear system. [Pg.278]

Another important electronic structure principle is the maximum hardness principle " (MHP) which may be stated as, There seems to be a rule of nature that molecules arrange themselves to be as hard as possible . Numerical verification of this principle has been made in several physico-chemical problems such as molecular vibrations , internal rotations , chemical reactions" , isomer stability , pericyclic reactions and Woodward-Hoffmann rules , stability of magic clusters , stability of super atoms ", atomic shell structure" , aromaticity , electronic excitations , chaotic ionization, time-dependent problems like ion-atom collision and atom-field interaction " etc. [Pg.71]

Although there are many definitions of chaos (Gleick, 1987), for our purposes a chaotic system may be defined as one having three properties deterministic dynamics, aperiodicity, and sensitivity to initial conditions. Our first requirement implies that there exists a set of laws, in the case of homogeneous chemical reactions, rate laws, that is, first-order ordinary differential equations, that govern the time evolution of the system. It is not necessary that we be able to write down these laws, but they must be specifiable, at least in principle, and they must be complete, that is, the system cannot be subject to hidden and/or random influences. The requirement of aperiodicity means that the behavior of a chaotic system in time never repeats. A truly chaotic system neither reaches a stationary state nor behaves periodically in its phase space, it traverses an infinite path, never passing more than once through the same point. [Pg.173]

Roux, J. C., Rossi, A., Bachelart, S., Vidal, C. (1981) Experimental observations of complex djmamical behavior during a chemical reaction. Physica 2D, 395 Satsuma, J. (1981) Exact solutions of a nonlinear diffusion equation. J. Phys. Soc. Japan 50, 1423 Schmitz, R. A., Graziani, K. R., Hudson, J. L. (1977) Experimental evidence of chaotic states in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 67, 3040 Segel, L. A., Jackson, J. L. (1972) Dissipative structure An explanation and an ecological example. J. Theor. Biol. 37, 545... [Pg.152]

Each chaotic state C, consists primarily of what at first glance appears to be a stochastic mixture of the adjacent periodic states and thus, for example, consists of a mixture of Po and P, as Fig. 6 of ROUX et al. [23] illustrates. However, maps constructed from the time series clearly yield smooth curves, not a scatter of points. These maps indicate that behavior is deterministic, not stochastic. Moreover, it is difficult to imagine stochastic processes that would lead to period doubling, the universal sequence, and tangent bifurcations, yet all of these phenomena associated with chaos are found in one-dimensional maps and in experiments on nonequilibrium chemical reactions, as will now be described. [Pg.133]

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. [Pg.3054]

The Belousov-Zhabotinskii (BZ) reaction has been selected as an example illustrating diverse dynamical states observable in chemical systems. The BZ reagent is very convenient both for experimental and theoretical investigations, since the BZ reaction has many dynamical states of interest, which will be described below. In the BZ reaction one may observe the steady state, the time periodic state (concentration oscillations), the spatially periodic state, the stationary state (dissipative structures), the time and spatially periodic state (propagating chemical waves) and turbulent states (chaotic oscillations, stochastic spatial structures, stochastic chemical waves). [Pg.220]

In this section the whole field of exotic dynamics is considered this term includes not merely oscillating reactions but also oligo-oscillatory reactions, multiple steady states, spatial phenomena such as travelling reaction waves, and chaotic systems. All of these have common roots in autocatalytic processes. This area has continued to expand, and there is a case for treatment in future volumes by a specialist reviewer. An entry into the literature can be gained from a recent series of articles in a chemical education joumal, and in a festschrift issue in honor of Professor R. M. Noyes. Other useful sources are a volume of conference proceedings, and a volume of lecture preprints of a 1989 conference. The present summary is concerned with the chemical rather than the mathematical aspects of the topic. [Pg.96]

We summarize our findings. It is suggested the criterion of critical condition, that is, the extremal behavior of the reaction species concentration, may be applied to reveal the critical conditions of nonlinear chemical dynamic systems. This is with the changeover of different dynamic modes of the reactions, such as the quasi-periodic and chaotic oscillations of the intermediate concentrations, as well as the steady-state mode. At the same time the Hamiltonian formalism makes it possible not only to have a successful numerical identification of the critical reaction conditions, but also to specify the role of individual steps of the reaction mechanism under different conditions. [Pg.185]

In chemical reactors an enormous variety of possible regimes, both steady state and non-steady state (transient) can be observed. Steady-state reaction rates can be characterized by maxima and hystereses. Non-steady-state kinetic dependences may exhibit many phenomena of complex behavior, such as fast and slow domains, ignition and extinction, oscillations and chaotic behavior. These phenomena can be even more complex when taking into account transport processes in three-dimensional media. In this case, waves and different spatial structures can be generated. For explaining these features and applying this knowledge to industrial or biochemical processes, models of complex chemical processes have to be simplified. [Pg.83]


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