Chaos reversing

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

Herandez J, Amador L, Amantea M, Chao H, Hawley P, Paradise L (2000) Short-course monotherapy with AG1549, a novel non-nucleoside reverse transcriptase inhibitor (NNRTI), in antiretroviral naive patients. In 7th conference on retroviruses and opportunistic infections. San Francisco, CA, Abstract 669... 

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

J. E. Ferrell, Jr., and W. Xiong, Bistability in cell signaling How to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11(1), 227 236 (2001). 

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

From a standpoint of organic chemistry this representation cannot be really considered as a mechanism of the reaction, but is rather a description of a complicated stepwise process leading to the transformation of the initial chaos of C clusters into the highly ordered structures of Ceo and C70 fullerenes. It is assumed that thermodynamic control serves as the driving force over the sequence of these reversible and kinetically plausible steps. 

By careful fractional distillation they separated the l,2-dichloro-2-methyl-butane from the reaction mixture, and found it to be optically inactive. From this they concluded that the mechanism involving free alkyl radicals, (2a), (3a), is the correct one. This mechanism is accepted without question today, and the work of Brown, Kharasch, and Chao is frequently referred to as evidence of the stereochemical behavior of free radicals, with the original significance of the work exactly reversed. 

We want to explain where this chaos comes from, and to understand the bifurcations that cause the wheel to go from static equilibrium to steady rotation to irregular reversals. 

The second law describes the entire universe. On a more personal level, we all fall victim to the law of increasing disorder. Chaos in our room or workplace is certainly not our intent It happens almost effortlessly. However, reversal of this process requires work and energy. The same is true at the molecular level. The gradual deterioration of our cities infrastructure (roads, bridges, water mains, and so forth) is an all-too-familiar example. Millions of dollars (translated into energy and work) are needed annually just to try to maintain the status quo. 

Theoretical investigations of this model (A. G. Makeev, B. E. Nieuwenhuys, Mathematical modeling of the NO + H2/Pt(100) reaction "Surface explosion," kinetic oscillations, and chaos, Journal of Chemical Physics, 108 (1998) 3740-3749) with 11 reversible and irreversible elementary steps included lateral interactions for only two steps in the forward direction and two steps in the reverse direction, leading to the following rate expressions... 

As would be expected from this affinity sequence, phosphate out-competed Mo for surface sites in multisorbate systems. The work of Balistrieri and Chao (1990) indicated that at pH 7, phosphate has greater affinity for amorphous Fe oxide than does molybdate, whereas the reverse is true for the affinity sequence on manganese dioxide. This difference also was reflected in the abilities of phosphate and molybdate to compete with selenite at pH 7 on the two oxides. 

By doing this the sciences of complexity have opened up space within the social sciences for a different approach to science, one centering around the end of certainties. We are aware that in the last 30 years the Newtonian model of science has been under sustained challenge from within the belly of the beast — physics and mathematics. I shall simply point to the counter slogans of this challenge in place of certainties, probabilities in place of determinism, deterministic chaos, in place of linearity, the tendency to move far from equilibrium and towards bifurcation, in place of integer dimensions, fractals, in place of reversibility, the arrow of time (Paraphrased from Wallerstein, 2005). 

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