Dynamical fractals, defined

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

A mathematically definable structure which exhibits the property of always appearing to have the same morphology, even when the observer endlessly enlarges portions of it. In general, fractals have three features heterogeneity, setf-similarity, and the absence of a well-defined scale of length. Fractals have become important concepts in modern nonlinear dynamics. See Chaos Theory... 

However, the same difficulty that observers meet in defining a cluster exist for theorists to define clusters in a numerical simulation typical numerical simulations handled several millions dark matter particle and a similar number of gas particle when hydro-dynamical processes are taken into account the actual distribution of dark matter, at least on non linear scales is very much like a fractal, for which the definition of an object is somewhat conventional Different algorithms are commonly used to define clusters. Friend of friend is commonly used because of its simplicity, however its relevance to observations is very questionable, especially for low mass systems. On the analytical side... 

The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices. The fractal dimension defines the self-similar connectivity of hole motions, the relaxation spectrum, and stretched exponential... 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

Finally, we define a strange attractor to be an attractor that exhibits sensitive dependence on initial conditions. Strange attractors were originally called strange because they are often fractal sets. Nowadays this geometric property is regarded as less important than the dynamical property of sensitive dependence on initial conditions. The terms chaotic attractor and fractal attractor are used when one wishes to emphasize one or the other of those aspects. 

Classical mechanics provides the least ambiguous statement of the nature of chaotic motion, with chaos also defined through a heirarchy of ideal model systems. We note, at the outset, that isolated molecule dynamics relates to chaotic motion in conservative Hamiltonian systems. This is distinct from chaotic motion in dissipative systems where considerable simplifications result from the reduction in degrees of freedom during evolution10 and where objects such as strange attractors and fractal dimensions play an important role. 

In Fig. 31, the dependences of D on the value A6= 6 6j are adduced, which characterizes as the first approximation the solvent thermo-dynamical affinity in respect to polymer [16]. As it follows from this figure data, certain laws are observed for these parameters relation. At first, A6 increase or solvent thermo-dynamical affinity in respect to polyaner change for the worse in all cases results in increase or macromolecular coil compactness enhancement, that is defined by the Eq. (8) within the framework of fractal analysis. Thus, the larger A5, is the smaller macromolecular coil gyration radius is at the same polymer molecular weight MM (or polymerization degree N). 

The Eq. (182) at the condition /g=const=0.270 nm and C =const=10 is reduced to a purely fractal form, that is, to the Eq. (8) with 5=0.349. Let us note essential distinctions of the Eqs. (180) and (8). Firstly, if the first from the indicated equations takes into accoimt object mass only, then the second one uses elements number N of macromolecule, that is, takes into account dynamics of molecular structure change. Secondly, the Eq. (8) takes into account real structural state of macromolecule with the aid of its fractal dimension The indicated above factors appreciation defines correct description by the equation (8) the dependence of macromolecular coil gyration radius R on molecular weight MIT of polynner [235]. 

Thus, the fractal model for the description of a macromolecular coil gyration radius change at polymer molecular weight variation is offered. This model takes into account both macromolecule molecular structure change dynamics at chemical composition variation and its structural state, defined, for example, by solvent choice [71]. These factors appreciation allows the concrete theoretical description of the indicated dependence. 

On the other hand, it is striking that fractals of fantastic complexity and shape may be constructed in an amazingly simple way by using the dynamics of the iteration processes described on p. 858. Let us take, for example, the following operation defined on the complex plane let us choose a complex number C, and then let us carry out the iterations... 

Recently there has been considerable interest in the phase transition of polymerized (tethered) membranes with attractive interactions [1-4]. In a pioneer work [1], Abraham and Nelson found by molecular dynamics simulations that the introduction of attractive interactions between monomers leads to a collapsed membrane with fractal dimension 3 at a sufficiently low temperature. Subsequently, Abraham and Kardar [2] showed that for open membranes with attractive interactions, as the temperature decreases, there exists a well-defined sequence of folding transitions and then the membrane ends up in the collapsed phase. They also presented a Landau theory of the transition. Grest and Petsche [4] extensively carried out molecular dynamics simulations of closed membranes. They considered flexible membranes the nodes of the membrane are connected by a linear chain of n monomers. For short monomer chains, n = 4, there occurs a first-order transition from the high-temperature flat phase to... 

Dynamic degradation of a polymer-starch blend is defined as the time-dependent accessibility of starch by micro-organisms A t) [13]. In this chapter, the percolation theory is extended to investigate the time dependence of starch removal in polymer-starch blends as a function of starch concentration p, the invasion mechanism, enzyme diffusion, microbial population, and starch size distribution. In this case, the time dependence of the accessibility on the fractal pathways representing the connected starch network with emphasis on both invasion and diffusion controlled mechanisms is explored. 

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