Chaotic evolution

Much of the regularity in classical systems can often be best discerned directly by observing their spatial power spectra (see section 6.3). We recall that in the simplest cases, the spectra consist of few isolated discrete peaks in more complex chaotic evolutions, we might get white noise patterns (such as for elementary additive rules). A discrete fourier transform (/ ) of a typical quantum state is defined in the most straightforward manner ... 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

Chaotic Evolution and Strange Attractors D. Ruelle Introduction to Polymer Dynamics P. de Gennes The Geometry and Physics of Knots M. Atiyah Attractors for Semigroups and Evolution Equations ... 

One may wonder if this expansion is possible for chaotic orbits, since the distance between the original orbit and the one under the perturbation increases exponentially in time for chaotic evolution. We will present an intuitive answer to this question. For example, the distance between XQ t) and a (f) increases exponentially for f oo. Then, the perturbative analysis will break down for this time interval. On the other hand, their distance shrinks exponentially for t —cxD, since they go to the same saddle. Similar reasoning would also hold for the stable orbit a (f). Thus, for some time to, the perturbative analysis will be applicable to Xu for f S [—oo, fo] and to Xg for f [fo, co], respectively. 

D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989. 

Fig. 4.9. Chaotic evolution of the biochemical model with multiple regulation. The cmve is obtained by numerical integration of eqns (4.1) in the domain of chaos shown in fig. 4.2 (Goldbeter Decroly, 1983).

Faybishenko, B Babchin, A. J. Frenkel, A. L. Halpem, D. Sivashinsky, G. I. A Model of Chaotic Evolution of an Ultrathin Liquid Film Flowing Down an Inclined Plane. Colloids Surf. A lOdi. 192 (1-3), 377-385. 

In contrast to FeCls hydrolysis, the hydrolysis of Fe(N03)3 leads to a more chaotic evolution of Fe polymers and Fe nuclei. Two possible mechanisms have been suggested to explain this observation. The first consists of crystal growth or precipitation based on ion diffusion and includes the following steps [59] (i) diffusion of Fe(in) ions to the surface of the nucleus (ii) dehydration of Fe at the surface (iii) adsorption of dehydrated Fe (iv) diffusion of Fe on the surface to a more energetically favorable position. In contrast to the ion diffusion mechanism, the growth of Fe species appears to be explained better by the aggregation of primary polymers. This second... 

Class 3 Evolution leads to chaotic nonperiodic patterns. 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

Fluid advection—be it regular or chaotic—forms a template for the evolution of breakup, coalescence, fragmentation, and aggregation processes. Let v(x, t) represent the Eulerian velocity field (typically we assume that V v = 0). The solution of... 

Recent studies have indicated that fluidized beds may be deterministic chaotic systems (Daw etal.,1990 Daw and Harlow, 1991 Schouten and van den Bleek, 1991 van den Bleek and Schouten, 1993). Such systems are characterized by a limited ability to predict their evolution with time. If fluidized beds are deterministic chaotic systems, the scaling laws should reflect the restricted predictability associated with such systems. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

Figure 4 shows a pattern of the concentration when the chaotic motion is established as well as the evolution of the deviation from two very close initial conditions. Note that nowadays it is very difficult to prove rigorously that a strange attractor is chaotic. In accordance with [35], a nonlinear system has chaotic dynamics if ... 

Essentially, MLE is a measure on time-evolution of the distance between orbits in an attractor. When the dynamics are chaotic, a positive MLE occurs which quantifies the rate of separation of neighboring (initial) states and give the period of time where predictions are possible. Due to the uncertain nature of experimental data, positive MLE is not sufficient to conclude the existence of chaotic behavior in experimental systems. However, it can be seen as a good evidence. In [50] an algorithm to compute the MLE form time series was proposed. Many authors have made improvements to the Wolf et al. s algorithm (see for instance [38]). However, in this work we use the original algorithm to compute the MLE values. 

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