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... 

Since the speed of information propagation is, as we shall see in chapter 4, related to the Lyapunov exponent for the CA evolution, and is a direct measure of the sensitivity to initial conditions, it should not be surprising to learn that various rules can also be distinguished by the degree of predictability for the outcome of... 

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

Lorentz found that when r = 28, cr = 10 and b = 8/.1, (i) the trajectories are attracted to a bounded region in space, (ii) the motion appears chaotic, and (iii) there is a great sensitivity to initial conditions. 

Figure 12.3 uses a four-center electrocydic reaction [29] to illustrate the sensitivity to initial conditions. There are two simple yes/no questions. One is Did or did not... 

The simulation result (Figure 4) shows that when two initial conditions are very close, after a dimensionless time of 40 units the concentration of reactant A and the reactor temperature are completely different. This means that the system has a chaotic behavior and their d3mamical states diverge from each other very quickly, i.e. the system has high sensitivity to initial conditions. This separation increases with time and the exponential divergence of adjacent phase points has a very important consequence for the chaotic attractor, i.e. 

Fig. 4. Sensitivity to initial conditions for the model Eq.(l)-(4) with two very close initial conditions...

The Lyapunov exponents provide a computable measure of the sensitivity to initial conditions, i.e. characterize the mean exponential rate of divergence of two nearby trajectories if there is at least one positive Lyapunov exponent, or convergence when all Lyapunov exponents are negative. The Lyapunov exponents are defined for autonomous dynamical systems and can be described by ... 

Figure 9. Two trajectories of the periodic hard-disk Lorentz gas. They start from the same position but have velocities that differ by one part in a million, (a) Both trajectories depicted on large spatial scales, (b) Initial segments of both trajectories showing the sensitivity to initial conditions.

LMS adaptive and Neural Large-scale solid rocket motor [14] 1. Sensitive to initial conditions and gradient dynamic parameters... 

A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. 

Chaotic solutions are those which are neither periodic nor asymptotic to a periodic solution but are characterized by extreme sensitivity to initial conditions. A solution that is asymptotic to a stable periodic solution is not sensitive to starting point, for, if we start from two nearby values, the trajectories will both converge on the same periodic solution and get closer and closer together. With a chaotic solution, the trajectories starting at two nearby points ultimately diverge no matter how close they may have been at the beginning. If /( )( ) denotes the nth iterate,... 

You missed the good part about sensitivity to initial conditions, said C. 

As we have already mentioned (cf. Chapter 3), one of the most important features of nonlinear dynamics is the sensitivity to initial conditions. A measure to verify the chaotic nature of a dynamic system is the Lyapunov exponent [32],... 

Several extensions to the sensitivity analysis have been made, e.g. sensitivities to initial conditions, spatial parameters in reactive flow [9], higher order and derived sensitivities [7], but for the purpose of this study only the linear sensitivity problem with respect to the rate parameters will be considered. 

Owing to its gradient-descent nature, back-propagation is very sensitive to initial conditions. The choice of initial weights will influence whether the net reaches a global (or only a local) minimum of the error and, if so, how quickly it converges. In practice, the weights are usually initialized to small zero-mean random values between -0.5 and 0.5 (or between -1 and 1 or some other suitable interval). 

The fact that some, although clearly not all, regions of (xq, r) parameter space indicate extraordinary sensitivity to initial conditions clearly precludes many types of prediction taking the example of Lorenz (1963), prediction of the time and location of an individual tornado is not possible, even though meteorological models exist which successfully predict the onset of tornado season. 

The question of predictability within the deterministic structure of classical mechanics was clearly appreciated by many eminent researchers in nonlinear systems theory and theoretical physics (see, e.g., Brillouin (I960)). Borel (1914) adds an additional twist to the predictability discussion. He argues that the displacement of a lump of matter with mass on the order of 1 g by as little as 1 cm and as far away as, e.g., the star Sirius is enough to preclude any prediction of the motion of the molecules of a volume of a classical gas for any longer than a firaction of a second, even if the initial conditions of the gas molecules are known with mathematical precision. Borel s example shows that many physical systems are not only sensitive to initial conditions, but also to miniscule changes in system parameters. The sensitivity to system parameters is a fundamental additional handicap for accurate long-time predictions. In the face of Borel s example, Brillouin (1960) points out that the prediction of the motion of gas molecules is not only very diflficult , as pointed... 

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. 

In order to allow for the largest possible class of chaotic systems, the degree of sensitivity is not specified in Devaney s definition of chaos. It turns out that many chaotic systems of practical importance are exponentially sensitive to initial conditions. In this case the sensitivity can be characterized quantitatively with the help of Lyapunov exponents. 

None of the classically chaotic quantum systems so far investigated in the atomic and molecular physics literature exhibits type III quantum chaos. On the other hand, atomic and molecular physics systems provide excellent examples for quantized chaos, the topic of this section. The attractive feature of the term quantized chaos is that it does not imply anything about what happens to the classical chaos when it is quantized. Usually, especially in bounded time independent quantum systems, classical chaos does not survive the quantization process. The quantized system does not exhibit any instabilities, or sensitivity to initial conditions, e.g. sensitivity to small variations in the wave function at time t = 0. 

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