Initial conditions sensitive dependence

Isolated-molecule dynamics is expected to be a sufficiently elementary process to permit observation of microscopic reversibility in the dynamics and, hence, to display a dependence of the outcome of dynamics on initial conditions. This dependence is desirable since the ability to retain information about initial conditions is necessary in order to achieve the technologically desirable goal of externally influencing chemical reactions. However, a great many experiments, perhaps with insufficiently well-characterized preparation and measurement, have indicated that time-irreversible relaxation is a useful model for many intramolecular processes. Thus, isolated-molecule intramolecular dynamics serves as a laboratory for the study of the inter-relationship between irreversible relaxation behavior in systems that are fundamentally de-scribable by time-reversible equations of motion. It also presents an experimental challenge to prepare sufficiently well-characterized states to observe time reversibility and sensitivity to initial conditions. 

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

It has sensitive dependence on initial conditions, i.e. two very close initial conditions diverge with time. 

According to the Shilnikov s theorem, the reactor presents a chaotic behavior. In order to test the presence of a strange attractor, it is necessary to raise the value of xe ax to introduce a perturbation in the vector field around the homoclinic orbit. Taking xemax = 5, the results of the simulation are shown in Figure 18, where the sensitive dependence on initial conditions has been corroborated. 

Fig. 18. Sensitive dependence of the dimensioniess concentration of R modei with two very close initial conditions. Kt = 0.094 m /h°C f2 = 5 h.

The course of addition reactions of ROH-XeF2 to alkenes has been elucidated using norbomene, 2-methylpent-2-ene and hex-l-ene as model substrates. It turned out that the alkoxyxenon fluoride intermediates (ROXeF) can react either as oxygen electrophiles (initially adding alkoxy substituent) or as apparent fluorine electrophiles (initially adding fluorine), depending on the reaction conditions. Simple addition of poorly nucleophilic alcohols to norbomene was also observed in certain instances. Selectivity between the various reaction pathways (simple fluorination, alkoxyfluorina-tion, or alcohol addition) proved to be sensitive to various reactions conditions, especially solvent, temperature, and catalyst.27... 

Two typical properties of nonlinear dynamical systems are responsible for the realization of controlling chaos. Firstly, nonlinear systems show a sensitive dependence on initial conditions. This is represented in Table 14.1 by the nonlinear equation... 

According to van Damme [201], fractal fingering is in many respects a chaotic phenomenon because it exhibits a sensitive dependence on the initial conditions. Although this kind of performance for a dissolution system is currently unacceptable, it might mirror more realistically the erratic dissolution of drugs with very low extent of absorption. 

A Dynamic Perspective of Variability The model under study here offers an opportunity to refer to some implications of the existence of nonlinear dynamics. Apart from the jagged cortisol concentration profile, elements such as the sensitive dependence from the initial conditions (expressed by the positive Lyapunov exponent), as well as the system s parameters, play an important role and may explain the inter- and intraindividual variability observed in the secretion of cortisol. These implications, together with other features absent from classical models, are demonstrated in Figure 11.12. 

Diffusion measurements under nonequilibrium conditions are more complicated due to the difficulties in ensuring well defined initial and boundary conditions. IR spectroscopy has proved to be a rather sensitive tool for studying simultaneously the intracrystalline concentration of different diffusants, including the occupation density of catalytic sites [28], By choosing appropriate initial conditions, in this way both co- and counterdiffusion phenomena may be followed. Information about molecular transport diffusion under the conditions of multicomponent adsorption may also be deduced from flow measurements [99], As in the case of single-component adsorption, the diffusivities arc determined by matching the experimental data (i.e. the time dependence of the concentration of the effluent or the adsorbent) to the corresponding theoretical expressions. 

While the outcome of an individual trajectory depends in a sensitive way on the details at the surface, the ensemble average is robust. In other words, if an ensemble of initial conditions of the cluster is reused, the yield of dissociation is essentially unchanged even though the new run samples a somewhat different set of conditions at the surface. Similarly, the ensemble-averaged yield is unchanged if a new set of cluster initial conditions is used. We will return to this point later but it is an important practical consideration since it means that a finite number of trajectory computations suffices. 

Sensitivity Studies on 1969 Trajectories with the Expanded Model. Based on the semi-Lagrangian formulation of the photochemical/diffusion model, the computed endpoint composition of the air masses depends on initial conditions, flux from the ground along the trajectories, and reaction rates. For our tests we concentrate on El Monte data because much of the polluted air there comes from somewhere else. This is believed to be a more severe test of the model than that at Huntington Park. The initial conditions are based on measurements insofar as possible. The principal initial values for the 1030 trajectory are as follows for 0730 PST (given in parts per hundred million) ... 

Because of the extreme dependence on initial conditions, our history analysis concentrates on an air mass with relatively well-defined concentrations at the beginning and the end of its travel. Giving it the initial values, we see the concentrations unfold as the air parcel moves through the computed simulation procedure. Because of the sensitivities discovered, the transition of oxidant species O3 and NO2 proceeds better than one might expect. The previously adopted biases on the NO-flux and the propylene oxidation rates were confirmed in this run having different conditions from those in Huntington Park represented by 1968 data. 

If a molecule decays in a mode-specific way, the assessment of the accuracy of classical calculations is much more complicated and depends, we believe, sensitively on the initially prepared resonance state. Considering a micro-canonical ensemble certainly will not be appropriate. The initial conditions of the ensemble of trajectories should mimic the quantum mechanical distribution function of coordinates and/or momenta as closely as possible [20,385]. The gross features of the final state distributions, e.g. the peaking of the CO vibrational distribution in the dissociation of HCO close to the maximum allowed state (Fig. 36), may be qualitatively reproduced. However, more subtle structures are unlikely to be described well, because they often reflect details of the quantum wave function (reflection principle [20]). More work to explore this question is certainly needed. 

The motion on the attractor exhibits sensitive dependence on initial conditions. This means that two trajectories starting very close together will rapidly diverge from each other, and thereafter have totally different futures. Color Plate 2 vividly illustrates this divergence by plotting the evolution of a small red blob of 10,000 nearby initial conditions. The blob eventually spreads over the whole attractor. Hence nearby trajectories can end up anywhere on the attractor The practical implication is that long-term prediction becomes impossible in a system like this, where small uncertainties are amplified enormously fast. 

Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. 

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast, i.e., the system has a positive Liapunov exponent. 

Finally, we define a strange attractor to be an attractor that exhibits sensitive dependence on initial conditions. Strange attractors were originally called strange because they are often fractal sets. Nowadays this geometric property is regarded as less important than the dynamical property of sensitive dependence on initial conditions. The terms chaotic attractor and fractal attractor are used when one wishes to emphasize one or the other of those aspects. 

By our definition, the dynamics in Example 9.5.1 are not chaotic, because the long-term behavior is not aperiodic. On the other hand, the dynamics do exhibit sensitive dependence on initial conditions—if we had chosen a slightly different initial condition, the trajectory could easily have ended up at C instead of C Thus the system s behavior is unpredictable, at least for certain initial conditions. 

We have seen that the logistic map can exhibit aperiodic orbits for certain parameter values, but how do we know that this is really chaos To be called chaotic, a system should also show sensitive dependence on initial conditions, in the sense that neighboring orbits separate exponentially fast, on average. In Section 9.3 we quantified sensitive dependence by defining the Liapunov exponent for a chaotic differental equation. Now we extend the definition to one-dimensional maps. 

Binary shift map) Show that the binary shift map x , = 2x (modi) has sensitive dependence on initial conditions, infinitely many periodic and aperiodic orbits, and a dense orbit. (Hint Redo Exercises 10.3.7 and 10.3.8, but write x as a binary number, not a decimal.)... 

Our work in the previous three chapters has revealed quite a bit about chaotic systems, but something important is missing intuition. We know what happens but not why it happens. For instance, we don t know what causes sensitive dependence on initial conditions, nor how a differential equation can generate a fractal attractor. Our first goal is to understand such things in a simple, geometric way. 

A strange attractor typically arises when the flow contracts the blob in some directions (reflecting the dissipation in the system) and stretches it in others (leading to sensitive dependence on initial conditions). The stretching cannot go on forever— the distorted blob must be folded back on itself to remain in the bounded region. 

The baker s map exhibits sensitive dependence on initial conditions, thanks to the stretching in the x-direction. It has many chaotic orbits—uncountably many, in fact. These and other dynamical properties ofthe baker s map are discussed in the exercises. 

Figure 12.3.3a shows the flow near a typical trajectory. In one direction there s compression toward the attractor, and in the other direction there s divergence along the attractor. Figure 12.3.3b highlights the sheet on which there s sensitive dependence on initial conditions. These are the expanding directions along which stretching takes place. Next the flow folds the wide part of the sheet in two and then bends it around so that it nearly joins the narrow part (Figure 12.3.4a). Overall, the flow has taken the single sheet and produced two...

Now the tangle resolves itself—the points fall on a fractal set, which we interpret as a cross section of a strange attractor for (1). The successive points (x(t),y(ty) are found to hop erratically over the attractor, and the system exhibits sensitive dependence on initial conditions, just as we d expect. 

Even when (1) has no strange attractors, it can still exhibit complicated dynamics (Moon and Li 1985). For i nstance, consider a regime in which two or more stable limit cycles coexist. Then, as shown in the next example, there can be transient chaos before the system settles down. Furthermore the choice of final state depends sensitively on initial conditions (Grebogi et al. 1983b). 

Hoyle s [1] successful prediction of the 7.6 MeV resonance of the carbon-12 nucleus, based on observation of his own carbon-based existence, established the scientific usefulness of anthropic principles. These principles have become common, if not yet standard, tools in cosmology, where theories of initial conditions may not yet exist - or, if they do exist, may admit a range of values [2, 3, 4, 5, 6]. At the same time, anthropic principles have retained a traditional role in religion and philosophy, where sensitive dependence of human existence on laws of nature that could imaginably have been otherwise is interpreted as evidence for human significance in the creation of the universe. 

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