Chaos, degree

It is obvious that the structure of amorphous polymers (or the amorphous phase of semi-crystalline polymers) gives grounds to assume the availability in them of a definite chaos degree. It is also quite obvious that the chaos degree of an amorphous phase structure represents an important parameter determining structural characteristics and, consequently, a polymer s properties. That is why the question about interconnection of these parameters arises. The second important problem is the physical nature of chaos in polymers is it random (and unpredictable) or deterministic chaos ... 

As defined above, the Lyapunov exponents effectively determine the degree of chaos that exists in a dynamical system by measuring the rate of the exponential divergence of initially closely neighboring trajectories. An alternative, and, from the point of view of CA theory, perhaps more fundamental interpretation of their numeric content, is an information-theoretic one. It is, in fact, not hard to see that Lyapunov exponents are very closely related to the rate of information loss in a dynamical system (this point will be made more precise during our discussion of entropy in the next section). 

It was pointed out in Chap. 8, Sect. 2.1 that there are primarily two reasons for the failure of the diffusion equation to describe molecular motion on short times. They are connected with each other. A molecule moving in a solvent does not forget entirely the direction it was travelling prior to a collision [271, 502]. The velocity after the collision is, to some degree, correlated with its velocity before the collision. In essence, the Boltzmann assumption of molecular chaos is unsatisfactory in liquids [453, 490, 511—513]. The second consideration relates to the structure of the solvent (discussed in Chap. 8, Sects. 2.5 and 2.6). Because the solvent molecules interact with each other, despite the motion of solvent molecules, some structure develops and persists over several molecular diameters [451,452a]. Furthermore, as two reactants approach each other, the solvent molecules between them have to be squeezed-out of the way before the reactants can collide [70, 456]. These effects have been considered in a rather heuristic fashion earlier. While the potential of mean force has little overall effect on the rate of reaction, its effect on the probability of recombination or escape is rather more significant (Chap. 8, Sect. 2.6). Hydrodynamic repulsion can lead to a reduction in the rate of reaction by as much as 30-40% under the most favourable circumstances (see Chap. 8, Sect. 2.5 and Chap. 9, Sect. 3) [70, 71]. 

That which proves that the dark Abyss, the Chaos, or the World s First Matter, was an aqueous and humid mass, is that, besides the reasons which we have brought forward, we have a palpable instance under our eyes. The property of water is to run, to flow, so long as heat animates and holds it in its fluid state. The continuity of bodies, the adhesion of their parts, is due to the aqueous humour. It is the ciment which unites and binds the elementary parts of bodies. So long as it is not separated from them entirely, they preserve the solidity of their mass. But if fire warms these bodies beyond the degree necessary for their preservation in their state of actual being, it drives away, rarefies this humour, makes it evaporate, and the body is reduced to powder, because the bond which united its parts no longer exists. 

M 84] [P 73] The degree of chaos was determined by calculating the Lyapunov exponent, a measure for material line stretching [48], Within the parametric limits of the simulation study made, the highest Lyapunov exponent was 0.1. On adding a further adjacent channel, this parameter can be increased to 0.4. 

Zhou, C and Kurths,). Hierarchical synchronization in complex networks with heterogeneous degrees. Chaos 2006, 16 015104. 

In multidimensional systems, these intersections would exhibit much more variety than they do in the lower-degrees-of-freedom systems that have been traditionally studied in nonlinear physics. One of the new aspects is tangency , which was found in the predissociation of a van der Waals complex of three bodies [11,12]. The tangency gives birth to transition in chaos [12], which is called a crisis [13]. Then, the extention of the concept of reaction rates to multidimensional chaos, which was first proposed in Ref. 9, breaks down [11,14]. 

The third stage of our strategy is discussed in Sections IX and X. Our discussion is speculative, since quantitative analysis is lacking at present. In Section IX, we point out that, in reaction dynamics, breakdown of normal hyperbolicity would also play an important role. Such cases would include phase transitions in systems with a finite number of degrees of freedom. In Section X, we will discuss the possibility of bifurcation in the skeleton of reaction paths, and we point out that it corresponds to crisis in multidimensional chaos. This approach offers an interesting mechanism for chemical evolution. 

The Melnikov integral also offers a method to estimate the reaction rates for systems with two degrees of freedom. This idea comes from the work of Davis and Gray [9]. However, their idea breaks down for systems with more than two degrees of freedom because of tangency [11,14]. This breakdown requires a new conceptual structure to describe the reaction dynamics from the viewpoint of multiple-dimensional chaos. What we propose for this new concept is the skeleton of reaction paths, where the connections among NHIMs are the focus of our study. 

Chaos in systems with N degrees of freedom with N >3 has characteristics that are not shared by chaos in systems with two degrees of freedom. In this section, we show that the Melnikov integral reveals one of these characteristics. They are exhibited in the intersections between the stable and unstable manifolds of whiskered tori with different action values. 

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