Invariant structures dynamics

This behavior reminds us of chaotic itinerancy found in dynamical systems with many degrees of freedom [18,19,21,38]. Chaotic itinerancy is the behavior where orbits repetitively approach and leave invariant structures of the phase space. Such behavior has been found in coupled maps [19], turbulence [18], neural networks [38], and Hamilton systems [21]. The mechanism of chaotic itinerancy is not yet fully understood. The study of NHIMs and how their stable and unstable manifolds intersect could offer some clues in revealing its mechanism [20]. 

The conventional theory of reaction processes relies on equilibrium statistical physics where the equi-energy surface is uniformly covered by orbits as shown in Fig. 34. To the contrary, the phase space in multidimensional chaos has various invariant structures, and orbits wander around these structures as shown in Fig. 35. In these processes, those degrees of freedom that constitute the movement along stable or unstable manifolds vary from NHIM to NHIM. Their variance reveals how reaction coordinates change during successive processes in reaction dynamics. 

Nonlinear resonances are important factors in reaction processes of systems with many degrees of freedom. The contributions of Konishi and of Honjo and Kaneko discuss this problem. Konishi analyzes, by elaborate numerical calculations, the so-called Arnold diffusion, a slow movement along a single resonance under the influence of other resonances. Here, he casts doubt on the usage of the term diffusion. In other words, Arnold diffusion is a dynamics completely different from random behavior in fully chaotic regions where most of the invariant structures are lost. Hence, understanding Arnold diffusion is essential when we go beyond the conventional statistical theory of reaction dynamics. The contribution of Honjo and Kaneko discusses dynamics on the network of nonlinear resonances (i.e., the Arnold web), and stresses the importance of resonance intersections since they play the role of the hub there. 

MD simulations with a constant energy is nothing but Hamiltonian dynamics. Recent accumulation of MD simulations will certainly contribute to our further understanding of Hamiltonian systems, especially in higher dimensions. The purpose of this section is to sketch briefly how the slow relaxation process emerges in the Hamiltonian dynamics, and especially to show that transport properties of phase-space trajectories reflect various underlying invariant structures. 

It is true that the hyperbolic system is an ideal dynamical system to understand from where randomness comes into the completely deterministic law and why the loss of memory is inevitable in the chaotic system, but generic physical and chemical systems do not belong strictly to such ideal systems. They are not uniformly hyperbolic, meaning that invariant structures are heterogeneously distributed in phase space, and there may not exist a lower bound of instability. It is believed that dynamical systems of such classes are certainly to be explored for our understanding of dynamical aspects of all relevant physical and chemical phenomena. 

The location of the saddle point in phase space is specified by and Pi = 0, where qi is the reaction coordinate. On top of the saddle point, the reaction coordinate is completely separated from the rest of the degrees of freedom. Therefore, a set of orbits where Pi) is fixed on the saddle point while the rest are arbitrary is invariant under dynamical evolution. Its dimension in phase space is 2n — 2. Such invariant manifolds are considered as the phase-space structure corresponding to transition states, and will play a crucial role in the following discussion. 

Wentzcovitch R (1991) Invariant molecular-dynamics approach to structural phase transitions. Phys Rev B 44 2358-2361... 

Scale-invariant structures originating from growth processes have been found to be extremely widespread in nature. This observation have led to a number of careful experiments, and various growth models have been suggested to describe the fractal outcome but why did they become fractal in the first place To answer this question we must understand the spatio-temporal evolution. Dynamically, the interface is observed to be unstable, and the system eventually reaches a statistically stationary state where a rich ramified pattern is created. A major observation is that this state can be described by power laws - the pattern becomes scale invariant. 

For instance, even mutually exclusive ideologies with - necessarily -opposed intentions and content may lead to comparable effects in terms of certain quantifiable attitudes and material structures in and of a society. In a semi-quantitative model such ideology invariant structures and their dynamics may be identified and verified (or falsified) on an objective basis. 

Another well-established area of mechanical finite-element analysis is in the motion of the structures of the human middle ear (Figure 9.3). Of particular interest are comparisons between the vibration pattern of the eardrum, and the mode of vibration of the middle-ear bones under normal and diseased conditions. Serious middle-ear infections and blows to the head can cause partial or complete detachment of the bones, and can restrict their motion. Draining of the middle ear, to remove these products, is usually achieved by cutting a hole in the eardrum. This invariably results in the formation of scar tissue. Finite-element models of the dynamic motion of the eardrum can help in the determination of the best ways of achieving drainage without affecting significantly the motion of the eardrum. Finite-element models can also be used to optimise prostheses when replacement of the middle-ear bones is necessary. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

The complex island structure in Fig. 7 is a consequence of the complicated dynamics of the activated complex. When a trajectory approaches a barrier, it can either escape or be deflected by the barrier. In the latter case, it will return into the well and approach one of the barriers again later, until it finally escapes. If this interpretation is correct, the boundaries of the islands should be given by the separatrices between escaping and nonescaping trajectories, that is, by the time-dependent invariant manifolds described in the previous section. To test this hypothesis, Kawai et al. [40] calculated those separatrices in the vicinity of each saddle point through a normal form expansion. Whenever a trajectory approaches a barrier, the value of the reactive-mode action I is calculated. If the trajectory escapes, it is assigned this value of the action as its escape action . 

Analytical investigations usually concern samples which are temporally and locally invariant. This kind of analysis is denoted as bulk analysis (average analysis). On the other hand, analytical investigations can particularly be directed to characterize temporal or local dependences of the composition or structure of samples. One has to perform dynamic analysis or process analysis on the one hand and distribution analysis, local analysis, micro analysis, and nano analysis on the other. 

The key idea of the fast torsion angle dynamics algorithm in Dyana is to exploit the fact that a chain molecule such as a protein or nucleic acid can be represented in a natural way as a tree structure consisting of n+1 rigid bodies that are connected by n rotatable bonds (Fig. 2.1) [74, 83]. Each rigid body is made up of one or several mass points (atoms) with invariable relative positions. The tree structure starts from a base, typically... 

The proposed structure of the complex does not assume a static distribution of the sequences. The system is of course a dynamic one, but we study it at equilibrium. A given COOH group, involved in a complex at the moment t, may be free or in the carboxylate from at t + dt. However the average number of complexed sequences remains invariant with time for a fixed composition of the system. The situation can be compared with the behaviour of macromolecules adsorbed at a solid-liquid interface their mean conformation is stable even if locally an adsorption/desorption equilibrium occurs. 

Knowing the three-dimensional structure of a protein is an important part of understanding how the protein functions. However, the structure shown in two dimensions on a page is deceptively static. Proteins are dynamic molecules whose functions almost invariably depend on interactions with other molecules, and these interactions are affected in physiologically important ways by sometimes subtle, sometimes striking changes in protein conformation. In this chapter, we explore how... 

The [(s-trans-diene)ZrCp2] complex (s-trans-1) equilibrates with the [(s-cA-diene)ZrCp2] isomer (x-cA-l) via a reactive high lying (r 2-butadiene) metallocene intermediate (2) [A(s-trans-1 s-cis-l, 283 K) = 22.7 0.3 kcal mol-1]. Syntheses of the (butadiene)zirconocene system carried out under kinetic control invariably led to pure s-trans-1, whereas a ca. 1 1 equilibrium of s-trans-1 and. v-ci.v-l was obtained under conditions of thermodynamic control.5,6 The cr,7i-structured s-cis-l isomer undergoes a dynamic ring-flip automerization process (see Scheme 2) that is rapid on the NMR time scale [AG futom = 12.6 0.5 kcal mol ].5... 

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