Chaotic behaviour

In fact, even in the solar system, despite the relative strengths of planetary attraction, there are constituents, the asteroids, with very irregular, chaotic behaviour. The issue of chaotic motion in molecules is an issue that will appear later with great salience.)... 

Graduate-level text giving detailed summary of status of chaotic behaviour in chemical systems to 1990. 

Shaw R 1981 Strange attractors, chaotic behaviour, and information flow Z. Naturf. a 36 80-112... 

If tlie diffusion coefficients of tlie chemical species are sufficiently different, new types of chemical instability arise which can lead to tlie fonnation of chemical patterns and ultimately to spatio-temporal chaotic behaviour. 

Model instability is demonstrated by many of the simulation examples and leads to very interesting phenomena, such as multiple steady states, naturally occurring oscillations, and chaotic behaviour. In the case of a model which is inherently unstable, nothing can be done except to completely reformulate the model into a more stable form... 

Walas (1991) has shown that the following set of ordinary differential equations gives random chaotic behaviour. 

Figure 5.259, The chaotic behaviour is Figure 5.260. The run of Fig. 5.259 is apparent. plotted as a phase-plane of XI versus X2.

General conditions for transition to irregular and chaotic behaviour in an oscillator under wave action have been derived using the notion about the Melnikov distance ... 

Irregular (chaotic) behaviour occurs for the areas where D(t0,t0) passes through zero. 

Under the condition D(ta, t0) = 0 and taking into account that sin(z/f0 + 0 < I and that 6d. > 0, the general condition for transmission to irregular (chaotic) behaviour in nonlinear oscillator under the wave action takes the form 25dsh( v w) < ttu2 Fg ev. ... 

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, United Kingdom (1997). 

Naturally, A, B and C as well as the constants fe have a completely different meaning than the ones we are used to from chemical kinetics they are not species with a certain concentration. Chaotic behaviour is restricted to certain ranges of initial values and parameters. It is up to the reader to play with these options. 

M. Perez and P. Albertos. Self-osciUating and chaotic behaviour of a PI-controlled CSTR with control valve saturation. J. Process Control, 14 51-57, 2004. 

L. Pellegrini and G. Biardi. Chaotic behaviour of a controlled CSTR. Comput. Chem. Eng., 14 1237-1247, 1990. 

This approach is of course limited by the availability of reliable data and the resolution of the data. An inherent problem in the np-scaling process is the interaction between variance in input parameters and non-linearity in models. This may prodnce chaotic behaviour, van Bodegom et al. (2002) discuss this in relation to CH4 emission from rice. The point at which inpnt data are averaged before making model runs may also be limited by the available computing... 

G. Casati (ed), Chaotic Behaviour in Quantum Systems, Plenum Press, New York, 1985. 

Marek, M., and Schreiber, I., Chaotic Behaviour of Deterministic Dissipative Systems . 2nd edn., Cambrige University Press, Cambridge (1995). 

The title chaos is an unfortunate misnomer. As shown above, there is considerable structure displayed in the onset and interruption of chaotic behaviour. Even within the chaotic regime the value of x evolves according to completely defined rules—one value explicitly determines the next, with 100 per cent certainty. There is no randomness, no element of chance uncertainty or irregularity. If we know the rules and can measure a given starting condition exactly, even a chaotic pattern can be predicted exactly. 

Fig. 13.16. A typical set of concentric period-doubling loci at higher forcing amplitudes, leading ultimately to a region of chaotic behaviour.

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

Even without this additional step, the scheme is capable of supporting Hopf bifurcation as the various rate constants and concentrations of P and Q are varied. In order to introduce the chaotic behaviour of the full model, however, it is more convenient to imagine a situation where the reversible step (13.42) is included but the concentration of species C is somehow maintained constant rather than allowed to vary in response to A. 

Arrowsmith, D. K., and C. M. Place, Dynamical Systems - Differential Equations, Maps and Chaotic Behaviour, Chapman and Hall, London, 1992. 

P. Schuster (ed.), Stochastic Phenomena and Chaotic Behaviour in Complex Systems (Springer, Berlin, 1984). 

Chaotic behaviour of two counter-currently cooled reactors (with K.S. Chang). Lat. Am. Appl. Res. 18,1 (1988). 

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

K.S. Jensen, E. Mosekilde, and N.-H. Holstein-Rathlou, Self Sustained Oscillations and Chaotic Behaviour in Kidney Pressure Regulation, Mondes en Develop. 54/55,91-109 (1986). 

There have been a few interesting fundamental papers on ergodicity 42 45 The distinction was emphasized between true equilibrium and quasiequilibrium phenomena, which now are frequently observed in NMR. An isolated finite system should not be expected to become ergodic. Two other fundamentally interesting processes that have been demonstrated with solid-state NMR are dephasing caused by randomization of geometric phase,46 47 and the possibility of the chaotic behaviour of spin systems.48,49... 

The chaotic behaviour of box C shows that questions of measurement theory and the concept of predictabifity are not just at the foundations of quantum mechanics, but enter in an equally profound way already on the classical level. This was recently emphasized by Sommerer and Ott in an article by Naeye (1994). They argue that in addition to the problem of predictability the problem of reproducability of measurements in classically chaotic systems has to be discussed. The results of Fig. 1.9 indicate that the logistic map displays similar complexity. In fact, regions which act sensitively to initial conditions, intertwined with regions where prediction is possible, are generic in classical particle dynamics. 

Over the past decade quantized chaos has become quite an industry . It has been realized that except for the fleld-free hydrogen atom and related two-body atomic systems, all atoms and molecules, starting with the helium atom, can exhibit chaotic behaviour when treated as classical systems. Although the quantum dynamics of these systems do not show... 

Shepelyansky, D.L. (1985). Quantum diffusion limitation at excitation of Rydberg atom in variable field, in Chaotic Behaviour in Quantum Systems, ed. G. Casati (Plenum, New York). 

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