Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity

Moura, F. A. B. F. de; Tirnakli, Uğur; Lyra, M. L.

doi:10.1103/physreve.62.6361

Cited by 83 publications

(114 citation statements)

References 24 publications

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“…(4) with q = 1.68 if 2n is odd and q = 1.70 if 2n is even, for the respective a values given in the Table. It is also evident from this figure that small log-periodic modulations are present on top of the curves. The existence of such modulations is expected due to discrete scale invariance in these types of systems, as demonstrated long ago by de Moura et al [25] (see also [26]); (iii) as we approach a c more closely, and consequently the appropriate value N * increases, the region consistent with the q-Gaussian grows in size. In order to illustrate this statement we include Fig.…”

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confidence: 67%

Closer look at time averages of the logistic map at the edge of chaos

2009

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The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [see U. Tirnakli, C. Beck and C. Tsallis, Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, maximizes the nonadditive entropy Sq, the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular its central part. This is important in view of a recent q-generalization of the Central Limit Theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q-Gaussian both in the central region and in the tail region, and find a scaling law involving the Feigenbaum constant δ. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy Sq and its associated concepts. One of the cornerstones of statistical mechanics and of probability theory is the Central Limit Theorem (CLT). It states that the sum of N independent identically distributed random variables, after appropriate centering and rescaling, approaches a Gaussian distribution as N → ∞. In general, this concept lies at the very heart of the fact that many stochastic processes in nature which consist of a sum of many independent or nearly independent variables converge to a Gaussian process [1,2]. On the other hand, there are also many other occasions in nature for which the limit distribution is not a Gaussian. The common ingredient for such systems is the existence of strong correlations between the random variables, which prevent the limit distribution of the system to end up being a Gaussian. Recently, for certain classes of strong correlations of this kind, it has been proved that the distribution of the rescaled sum approaches a q-Gaussian, which constitutes a q-generalization of the standard CLT [3,4,5,6]. This represents a progress since the q-Gaussians are the distributions that optimize the nonadditive entropy S q (defined to be S q ≡ (1 − i p q i ) / (q − 1)), on which nonextensive statistical mechanics is based [7,8]. A q-generalized CLT was expected for several years since the role of q-Gaussians in nonextensive statistical mechanics is pretty much the same as that of Gaussians in Boltzmann-Gibbs statistical mechanics. Therefore it is not surprising at all to see q-Gaussians replace the usual Gaussian distributions for those systems whose agents exhibit certain types of strong correlations.Immediately after these achievements, an increasing interest developed for checking these ideas and findings in real and model systems whose dynamical properties make them appropriate candidates to be analyzed along these l...

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confidence: 67%

Closer look at time averages of the logistic map at the edge of chaos

2009

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“…In some cases, when the scale invariance is discrete [16,36,14,32,29,18,2,1], the amplitude of the powerlaw acquires a periodicity, often called log-periodic oscillations: see [35,21] for reviews on the topic. These oscillatory amplitudes are usually more difficult to calculate [13,12,7,26] than the critical exponents.…”

Section: Introductionmentioning

confidence: 99%

Log-periodic Critical Amplitudes: A Perturbative Approach

Derrida

Giacomin

2013

J Stat Phys

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Abstract. Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.Dedicated to Herbert Spohn, our colleague, friend and source of inspiration, on the occasion of his 65 th birthday

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“…It can be verified that, for the sequence lim W →∞ lim N→∞ , we asymptotically have W (t) ∝ 1/t 1 q rel (z)−1 , where q eq (z) > 1 (the subindex rel stands for relaxation). The entropic index q rel monotonically increases when z increases from 1 to infinity; also, within some range it is verified [27] …”

Section: One-dimensional Mapsmentioning

confidence: 84%

“…We would then have the generalization of the Pesin identity. Let us next address another aspect [27] concerning the edge of chaos a c (z). We spread now, at t = 0, the N points uniformly within the entire [x min , x max ] interval, i.e., over all the W windows, and follow, as function of time t, the shrinking of the number W (t) of windows which contain at least one point (disappearence of the Lebesgue measure on the x-axis); W (0) = W .…”

Section: One-dimensional Mapsmentioning

confidence: 99%

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Nonextensive statistical mechanics: a brief review of its present status

Tsallis

2002

An. Acad. Bras. Ciênc.

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We briefly review the present status of nonextensive statistical mechanics. We focus on (i) the central equations of the formalism, (ii) the most recent applications in physics and other sciences, (iii) the a priori determination (from microscopic dynamics) of the entropic index q for two important classes of physical systems, namely low-dimensional maps (both dissipative and conservative) and long-range interacting many-body hamiltonian classical systems.

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Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity

Cited by 83 publications

References 24 publications

Closer look at time averages of the logistic map at the edge of chaos

Closer look at time averages of the logistic map at the edge of chaos

Log-periodic Critical Amplitudes: A Perturbative Approach

Nonextensive statistical mechanics: a brief review of its present status

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