Self-similar dynamics

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

If the self-similar spectrum extends over a sufficiently wide time window, approximate solutions for the relaxation modulus G(t) and the dynamic moduli G (co), G"(co) might be explored by neglecting the end effects... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

The success of the ID fluid dynamic model to describe the flow field in the DPF channel (Konstandopoulos and Johnson, 1989 Konstandopoulos et al., 1999, 2003) is an indication for the existence of a (nearly) self-similar flow field. A necessary condition for the application of the ID model for the heat transfer problem as well, is that the wall velocity ww variation must be small along the characteristic channel length required for establishment of a steady heat transfer pattern (i.e. a length of a2ftz/y.lh). In transferring the above to the case of flow and heat transfer in a DPF channel we may formally write the heat balance as... 

Then, for y greater than -y , we find more grey areas, where it seems that the dynamics is chaotic, but there is also a succession of windows where stable periodic solutions are the only ones to be found. The last of these is from y = 3.828 to y = 3.857 where period-three solutons exist. Even the structure of this is beautifully self-similar. And then there s the Charkovsky sequence.. .. But do just let me mention the sensitivity question. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

All the reaction systems considered, despite being greatly different chemically, have been found to have similar dynamic characteristics of the autowave processes occurring therein. Particularly, the linear velocities of the wave-front propagation are in the range of 1 -4 cm/s for all systems. All of them have a certain critical irradiation dose below which the excitation of an autowave process becomes impossible and the system responds to a local disturbance only with local conversion incapable of self-propagating (the situation discussed above and illustrated by Fig. 5). 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

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... 

Available theoretical solutions in dynamic fracture are few, and limited to finite or semi-infinite cracks in an infinite solid for Mode I, self-similar crack extension. Despite the above limitations, short of conducting detailed numerical analysis of the crack tip state of stress, these solutions must be used to deduce the characteristics of the crack tip state of stress, as well as to extract the dynamic stress intensity factor for elastodynamic fracture mechanics. In the following sections, a brief description of available theoretical solutions is presented. 

Recently, some very simple cellular automata models of such randomly driven dissipative systems have been developed and have been studied extensively. It has been shown that the dynamics of such models leads to a critical state characterised by power laws induced by stochastically developed self-similarities in the system. One such popular model, known as the BTW model, introduced by Bak et al (1987,1988), attempts to capture the avalanche dynamics of a sandpile where the sand grains are being added to the pile at a constant rate. The model has been studied extensively, both numerically and analytically, and the existence of the self-organised criticality in the model has been established. 

H. Rabitz and M.D. Smooke, Scaling Relations and Self-Similarity Conditions in Strongly Coupled Dynamical Systems, J. Phys. Chem. 92 (1988) 1110-1119. 

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