2000
DOI: 10.1103/physreve.62.6361
|View full text |Cite
|
Sign up to set email alerts
|

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

Abstract: For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W (t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's f… Show more

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

10
103
0
1

Year Published

2002
2002
2017
2017

Publication Types

Select...
4
4

Relationship

0
8

Authors

Journals

citations
Cited by 83 publications
(114 citation statements)
references
References 24 publications
10
103
0
1
Order By: Relevance
“…(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.…”
mentioning
confidence: 67%
“…(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.…”
mentioning
confidence: 67%
“…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%
“…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%
See 1 more Smart Citation