Order in chaos

Order in chaos Presence of only long- or occasionally short-range order... 

Passardi F, Zamocky M, Favet J et al (2007) Phylogenetic distribution of catalase-peroxidases are there patches of order in chaos Gene 397 101-113... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

Motion near an energetic barrier of a potential energy surface is typically (but not always) chaotic at energies above the barrier height. If chaotic were synonymous with statistical, then reaction dynamics could in such cases be completely understood using standard statistical methods and theories. However, modern work in the theory of dynamical systems has shown that chaotic motion has an underlying structure there is order in chaos. The underlying structure within chaotic motion is the source of nonstatistical behavior. This structure needs to be understood in order to predict how the system will behave. 

Despite the fact that the motion is now ergodic, there is a considerable amount of structure in the chaotic regions of phase space, which may be referred to as order in chaos. In particular, separatrix structures still exist and can be reconstructed numerically. These structures determine the rate at which chaotic trajectories travel from one part of phase space to another. Moreover, the separatrix structures can be present if the dynamics has any measure of chaos whatsoever, regardless of whether or not the system is ergodic. The periodic orbits of the system admitted by the system are intimately connected to these structures, as we will see. 

N. De Leon and C. C. Marston, J. Chem. Phys., 91, 3405 (1989). Order in Chaos and the Dynamics and Kinetics of Unimolecular Conformational Isomerization. 

The ability to see order in chaos has won the mathematician Yakov G. Sinai the 2014 Abel Prize. The Prize, named after the Norwegian mathematician Niels Henrik Abel, is awarded annually by the Norwegian Academy of Science and Letters and is viewed by many as on par with the Nobel Prize. 

Not chaos-like crush d and bruis d But, as the world, harmoniously confus d. Where order in variety we see. 

Two limiting cases chaos and order are determined here, but in addition one can also consider chaos with some correlation to particles localization (type 3, Table 9.3) type 4 assumes presence of some order, for example long-range order in silicate mesophases, or platinum particles in a xerogel, etc. One can also consider division of these types, which allow or disallow overlapping of particles. 

The first part of the manual explains the running of the CHAOS program and the second, the CHAOSBASE program. We believe the best way to read the manual is in the order in which it is printed. 

A more complicated behavior of the MLE is observed for higher values of 7j. Varying the length of the pulse 7j, we observe regions of order and chaos. By way of an example, the phase portrait Reoti versus Imai for a chaotic attractor is shown in Fig. 15. 

The emergence of order and chaos in the system of two oscillators depends on the value of the Kerr coupling constant 2- F°r the fixed parameters of damping... 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

The reduction of chaos for 9 = 1.45 is presented in the intensity portraits of Fig. 39. However, as is seen in Fig. 38a, there is a small region (1.68 < 9 < 1.80) where the system behaves orderly in the classical case but the quantum correction leads to chaos. By way of an example for 9=1.75, the classical system, after quantum correction, loses its orderly features and the limit cycle settles into a chaotic trajectory. Generally, Lyapunov analysis shows that the transition from classical chaos to quantum order is very common. For example, this kind of transition appears for 9 = 3.5 where chaos is reduced to periodic motion on a limit cycle. Therefore a global reduction of chaos can be said to take place in the whole region of the parameter 0 < 9 < 7. 

C. Vidal, Chaos and Order in Nature, in Proc. Int. Symp. on Synergetics (Springer, Berlin, 1981) p. 68. 

New experiences can bring adventure, but also uncertainty and chaos. That applies to new product development just as well. These first four lessons will help you to get started, and to bring some order in the chaos. 

Moore, M. and Bliimel, R. (1993). Quantum manifestations of order and chaos in the Paul trap, Phys. Rev. A48, 3082-3091. 

Kolish who helped preserve some order in the midst of chaos. We also thank our colleagues at Johns Hopkins University and Carleton College who supported, advised, instructed, and simply bore with us during this arduous task. One of us (JLT) was graciously awarded a grant from Carleton College to relieve him of some of his academic tasks so that he could focus more fully on the book. 

Who is your audience, the person to whom you are writing As discussed in detail on pages 19-23 in Chapter 1, effective writers select the kinds of information, the level of complexity, and even the appropriate voice in response to their readers needs, knowledge, and attitudes. Remember that no matter who your letter-reader happens to be, all readers want clarity, not confusion order, not chaos and useful information, not irrelevant chitchat. Put yourself in the reader s place what should she or he know, understand, or decide to do after reading this letter ... 

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