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Labyrinthine pathways towards supercycle attractors in unimodal maps

100%
Moyano L., Silva D., Robledo A.
Open Physics
|
2009
|
vol. 7
|
issue 3
591-600
EN
As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.
2
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Application of Fractional Scaling in Modelling of Magnetic Power Losses

100%
Najgebauer M.
Acta Physica Polonica A
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2015
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vol. 128
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issue 1
107-110
EN
The paper presents a new approach to the Widom-based scaling procedure, in which additional fractional exponents were introduced into the Maclaurin series. The modified scaling procedure was proposed in order to obtain more universal descriptions in a form of the power law series with fractional exponents. The proposed procedure was examined for the power losses scaling of commercial grain-oriented electrical steel.
3
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Spatially extended populations reproducing logistic map

100%
Dzwinel W.
Open Physics
|
2010
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vol. 8
|
issue 1
33-41
EN
We discuss here the conditions that the spatially extended systems (SES) must satisfy to reproduce the logistic map. To address this dilemma we define a 2-D coupled map lattice with a local rule mimicking the logistic formula. We show that for growth rates of k⩽k ∞ (k ∞ is the accumulation point) the global evolution of the system exactly reproduces the cascade of period doubling bifurcations. However, for k > k ∞, instead of chaotic modes, the cascade of period halving bifurcations is observed. Consequently, the microscopic states at the lattice nodes resynchronize producing dynamically changing spatial patterns. By downscaling the system and assuming intense mobility of individuals over the lattice, the spatial correlations can be destroyed and the local rule remains the only factor deciding the evolution of the whole colony. We found the class of “atomistic” rules for which uncorrelated spatially extended population matches the logistic map both for pre-chaotic and chaotic modes. We concluded that the global logistic behavior can be expected for a spatially extended colony with high mobility of individuals whose microscopic behavior is governed by a specific semi-logistic rule in the closest neighborhood. Conversely, the populations forming dynamically changing spatial clusters behave in a different way than the logistic model and reproduce at least the steady-state fragment of the logistic map.
4
Content available remote

Can Ising model and/or QKPZ equation properly describe reactive-wetting interface dynamics?

84%
Efraim Y., Taitelbaum H.
Open Physics
|
2009
|
vol. 7
|
issue 3
503-508
EN
The reactive-wetting process, e.g. spreading of a liquid droplet on a reactive substrate is known as a complex, non-linear process with high sensitivity to minor fluctuations. The dynamics and geometry of the interface (triple line) between the materials is supposed to shed light on the main mechanisms of the process. We recently studied a room temperature reactive-wetting system of a small (∼ 150 μm) Hg droplet that spreads on a thin (∼ 4000 Å) Ag substrate. We calculated the kinetic roughening exponents (growth and roughness), as well as the persistence exponent of points on the advancing interface. In this paper we address the question whether there exists a well-defined model to describe the interface dynamics of this system, by performing two sets of numerical simulations. The first one is a simulation of an interface propagating according to the QKPZ equation, and the second one is a landscape of an Ising chain with ferromagnetic interactions in zero temperature. We show that none of these models gives a full description of the dynamics of the experimental reactivewetting system, but each one of them has certain common growth properties with it. We conjecture that this results from a microscopic behavior different from the macroscopic one. The microscopic mechanism, reflected by the persistence exponent, resembles the Ising behavior, while in the macroscopic scale, exemplified by the growth exponent, the dynamics looks more like the QKPZ dynamics.
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