Chaos

As we have seen, there are indeed bound to be many assumptions made during planning that will inevitably lead to some measure of revision and re-work. The issue is not therefore whether to permit any changes, but how to control them so that they do not destroy the manageability of the project. 

Change control has a few components change recognition, change evaluation, change approval or rejection, and change implementation, including the consequent revision to plan. These are discussed below. 

Process plant design and construction is such a heavily interlinked exercise of different activities and disciplines, that the consequences of change are very easy to underestimate. In fact change is invariably one of the most emotive issues in the business. 

Change recognition is mainly dependent on the training and conscientiousness of the work performers and their supervisors. It is 

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In coimection with the energy transfer modes, an important question, to which we now turn, is the significance of classical chaos in the long-time energy flow process, in particnlar the relative importance of chaotic classical dynamics, versus classically forbidden processes involving dynamical tuimelling . 

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

Blumel R and Reinhardt W P 1997 Chaos in Atomic Physics (Cambridge Cambridge University Press)... 

Tabor M 1989 Chaos and Integrability in Nonlinear Dynamics An Introduction (New York Wiley)... 

There are many examples in nature where a system is not in equilibrium and is evolving in time towards a thennodynamic equilibrium state. (There are also instances where non-equilibrium and time variation appear to be a persistent feature. These include chaos, oscillations and strange attractors. Such phenomena are not considered here.)... 

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York Springer)... 

Meagher J F, Chao K J, Barker J R and Rabinovitch B S 1974 Intramolecular vibrational energy relaxation. Decomposition of a series of chemically activated fluoroalkyl cyclopropanes J. Phys. Chem. 78 2535 3... 

Under some conditions, it is observed that complex oscillatory sequences develop even in batch systems, typically towards the end of the oscillatory phase of the reaction. Transient chaos —see section A3.14.3.3— appears to be established [18]. 

Epstein I R and Pojman J A 1999 Overview nonlinear dynamios related to polymerio systems Chaos 9 255-9... 

Winfree A T 1984 The prehistory of the Belousov-Zhabotinsky osoillator J. Chem. Eduo. 61 661-3 [11 ] Zhabotinsky AM 1991 A history of ohemioal osoillations and waves Chaos 1 379-86... 

Swinney H L, Argoul F, Arneodo A, Richetti P and Roux J-C 1987 Chemical chaos from hints to confirmation Acc. Chem. Res. 20 436-42... 

Gyorgyi L and Field R J 1992 A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction Nature 355 808-10... 

Johnson B R and Scott S K 1990 Period doubling and chaos during the oscillatory ignition of the CO + O2 reaction J. Chem. Soc. Faraday Trans. 86 3701-5... 

Ditto W L and Showalter K (eds) 1997 Control and synchronization of chaos focus issue Chaos 7 509-687... 

Winfree A T 1998 Evolving perspectives during 12 years of electrical turbulence Chaos 8 1-19... 

Ouyang Q and Swinney FI L 1991 Transition to chemical turbulence Chaos 1 411-20... 

Scott S K 1994 Oscillations, Waves and Chaos in Chemical Kinetics (Oxford Oxford University Press) A short, final-year undergraduate level introduction to the subject. 

Epstein I R and Pojnian J A 1998 An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns and Chaos (Oxford Oxford University Press)... 

EttI R, Chao I, Diederich F and Whetten R L 1991 Isolation of C-g, a chiral (Dg) allotrope of carbon Nature 353... 

Diederich F, Whetten R L, Thilgen C, EttI R, Chao I and Alvarez M M 1991 Fullerene isomerism—isolation of C2y-... 

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. 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

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... 

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