Dynamical rules chaotic

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

Since (I-A) is a measure of hardness according to the maximum hardness principle, the stability of a system or the favorable direction of a physicochemical process is often dictated by this quantity. Because aromatic systems are much less reactive, especially toward addition reactions, I -A may be considered to be a proper diagnostic of aromaticity. Moreover, (/ - A) has been used in different other contexts, such as stability of magic clusters, chemical periodicity, molecular vibrations and internal rotations, chemical reactions, electronic excitations, confinement, solvation, dynamics in the presence of external field, atomic and molecular collisions, toxicity and biological activity, chaotic ionization, and Woodward-Hoffmann rules. The concept of absolute hardness as a unifying concept for identifying shells and subshells in nuclei, atoms, molecules, and metallic clusters has also been discussed by Parr and Zhou. ... 

Those negative results did not disprove the hypothesis that computational capability can be correlated with phase transitions in CA rule space they showed only that Packard s results did not provide support for that hypothesis. Relationships between computational capability and phase transitions have been found in other types of dynamical systems. For example, Crutchfield and Young (1989, 1990) looked at the relationship between the dynamics and computational structure of discrete time-series generated by the logistic map at different parameter settings. They found that at the onset of chaos there is an abrupt jump in computational class of the time series, as measured by the formal-language class required to describe the time series. This result demonstrated that a dynamical system s computational capability—in terms of the richness of behavior it produces—is qualitatively increased at a phase transition from ordered to chaotic behavior. 

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