Low-dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to initial conditions at the edge of chaos which can be naturally formulated within the generalized Tsallis statistics prescription which i s c haracterized by the entropic index q. General scaling arguments provide a direct relation between the entropic index q and the scaling exponents associated with the extremal sets of the multifractal critical attractor. The above result comes in favor of recent conjectures that Tsallis statistics is the natural frame for studying systems with a fractal-like structure in the phase-space. Power-law sensitivity i n high-dimensional dissipative and Hamiltonian systems are also discussed within the present picture. I IntroductionLow-dimensional non-linear maps are the prototype models to study the emergence of complex behavior in dynamical systems. Their typical behavior include the occurrence of bifurcation instabilities, long-range correlated sequences, fractal structures and chaos, which are commonly observed in a great variety of systems ranging from uids, magnetism, biology, social sciences and many others 1 .The study of the sensitivity to initial conditions of non-linear systems is one of the most important tools used to investigate the nature of the phase-space attractor. It is usually characterized by the Liapunov exponent , de ned for the simple case of a one-dimensional dynamical variable x as xt x0e t x0 ! 0; t ! 1 1 where x0 is the distance between two initially nearby orbits in an equivalent point of view, it is the uncertainty on the precise initial condition. If 0 the system is said to be strongly sensitive to the initial condition with the uncertainty on the dynamical variable growing exponentially in time and this characterizes a chaotic motion in the phase-space. On the other hand, if 0 the system becomes strongly insensitive to the initial condition which is expected for any state whose dynamical attractor is an orbit with a nite period. The problem of the sensitivity to initial conditions can be reformulated in an entropic language as a process of information loss in the case of chaotic behavior or recovery for periodic attractors. Within this context, it is useful to introduce the Kolmogorov-Sinai entropy K. It is basically the rate of variation of the Boltzmann- where S0 and SN are the entropies of the system evaluated at times t = 0 and t = N for maps = 1. With the simplifying assumption that at time t there are Wt occupied cells with the same occupation number, we h a ve from equation 2 that Wt = W 0e Kt ;3 which is equivalent to equation 1 for the sensitivity to initial condition and provides the well-known Pesin
The role of the interlayer magnetic coupling between CuO& planes on bilayer-group high-T, superconductors is studied within a simple randomly decorated bilayer Ising model and a Bethe-lattice approach.We show that for weak interlayer coupling an enhancement of the intralayer hole-hole correlations is developed at finite temperatures. This phenomenon is related to the loss of interlayer coherence between defects and the consequent strengthening of the antiferromagnetic background.In the last few years, much attention has been given to the study of the properties of disordered magnetic systems since frustration effects were proposed to be relevant to explaining the behavior of the doped copperoxide high-T, superconductors. ' In these compounds the supercurrent is supposed to flow in the CuOz planes which develop intraplane antiferromagnetic order when undoped. Doping introduces charge carriers which, for a large range of doping concentrations, are localized basically on the oxygen ions of such planes. The net -, ' spin of each charge carrier interacts with its neighboring -, spin Cu atoms strongly enough to change the original antiferromagnetic coupling into an effective ferromagnetic one. The resulting frustration effect due to the presence of these competing interactions induces an attraction between the charge carriers that may lead to Cooper pairing. ' It has also been observed, as for example in Tl compounds, that the superconducting transition temperature is an increasing function of the number of closely spaced interacting Cu02 layers present in the material.Although this phenomenon has been related with a quantum-well confinement effect, the role played by the coupling between these Cu02 planes is still to be clarified. Furthermore, since a new fractional quantum Hall effect was predicted to occur, and was experimentally observed, on coupled two-dimensional electron systems, a better understanding of the interplay between inplane and interplane correlations on systems with mobile particles are clearly desirable.Recently, within a simple randomly decorated Ising model and a Bethe lattice approach, it was shown that two distinct mechanisms of attraction actually appear in magnetically disordered systems '" a strong one due to frustration effects, and another due to the enhancement of magnetic fluctuations as the antiferromagnetic order is broken down. Although rather strong, the frustration component is not enough to promote a phase separation instability. ' It was also pointed out that the transition to the superconducting phase appears to occur experimentally at a line of constant hole-hole correlation in the temperature versus hole-concentration space. "In this paper we show how the coupling between disordered Ising models affects the intralayer hole-hole correlations. We obtain that in the ground state the intralayer correlations are quite insensitive to interlayer coupling. Nevertheless, an enhancement of the hole-hole correlations is developed at finite temperatures for weakly coupled layers. We relate ...
We study both analytically and numerically the effects of including refractory periods in the Hopfield model for associative memory. These periods are introduced in the dynamics of the network as thresholds that depend on the state of the neuron at the previous time. Both the retrieval properties and the dynamical behaviour are analized, and we found that depending on the value of the thresholds and on the the ratio α between the number of stored memories (p) and the total number of neurons (N ), the system presents not only fixed points but also chaotic or ciclic orbits.
We study an extreme and asymmetrically diluted version of the Hopfield model when the refractory period is taken into account in the dynamics of the neurons through a time dependent threshold. We present an analytical approach that allows one to preserve, in an approximate way, the dependence of the system on its whole history. In particular, we obtain a recurrent equation for the overlap from which one can analyze the retrieval capacity. We also perform numerical simulations that are well fitted by our analytical results. Depending on the amplitude of the potential that mimics the effect of the refractory period and on the ratio ␣ between the number of stored patterns p and the mean connectivity per neuron C, the system presents different dynamical behaviors and retrieval abilities.
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