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Ugur Tirnakli

Publications and source records attributed to Ugur Tirnakli.

At least 55 records · Page 3Linked to original sources
arXiv2006

Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos

It is well known that, for chaotic systems, the production of relevant entropy (Boltzmann-Gibbs) is always linear and the system has strong (exponential) sensitivity to initial conditions. In recent years, various numerical results indicate that basically the same type of behavior emerges at the edge of chaos if a specific generalization of the entropy and the exponential are used. In this work, we contribute to this scenario by numerically analysing some generalized nonextensive entropies and their related exponential definitions using $z$-logistic map family. We also corroborate our findings by testing them at accumulation points of different cycles.

cond-mat.stat-mech↗
arXiv2005

Chaos edges of $z$-logistic maps: Connection between the relaxation and sensitivity entropic indices

Chaos thresholds of the $z$-logistic maps $x_{t+1}=1-a|x_t|^z$ $(z>1; t=0,1,2,...)$ are numerically analysed at accumulation points of cycles 2, 3 and 5. We verify that the nonextensive $q$-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify $\lim_{t \to\infty}< S_{q_{sen}^{av}} >(t)/t= \lim_{t \to\infty}< \ln_{q_{sen}^{av}} ξ>(t)/t \equiv λ_{q_{sen}^{av}}^{av}$, where the entropy $S_{q} \equiv (1- \sum_i p_i^q)/ (q-1)$ ($S_1=-\sum_ip_i \ln p_i$), the sensitivity to the initial conditions $ξ\equiv \lim_{Δx(0) \to 0} Δx(t)/Δx(0)$, and $\ln_q x \equiv (x^{1-q}-1)/ (1-q) $ ($\ln_1 x=\ln x$). The entropic index $q_{sen}^{av}<1$, and the coefficient $λ_{q_{sen}^{av}}^{av}>0$ depend on both $z$ and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as $1/t^{1/(q_{rel}-1)}$ ($q_{rel}>1$). These results led to (i) the first illustration of the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely $q_{rel}-1 \simeq A (1-q_{sen}^{av})^α$, where the positive numbers $(A,α)$ depend on the cycle; (ii) an unexpected and new scaling, namely $q_{sen}^{av}(cycle n)=2.5 q_{sen}^{av}(cycle 2)+ ε$ ($ε=-0.03$ for $n=3$, and $ε= 0.03$ for $n=5$).

cond-mat.stat-mech↗
arXiv2004

Sensitivity to initial conditions in coherent noise models

Sensitivity to initial conditions in the coherent noise model of biological evolution, introduced by Newman, is studied by making use of damage spreading technique. A power-law behavior has been observed, the associated exponent $α$ and the dynamical exponent $z$ are calculated. Using these values a clear data collapse has been obtained.

cond-mat.stat-mech↗
arXiv2004

Aging in coherent noise models and natural time

Event correlation between aftershocks in the coherent noise model is studied by making use of natural time, which has recently been introduced in complex time-series analysis. It is found that the aging phenomenon and the associated scaling property discovered in the observed seismic data are well reproduced by the model. It is also found that the scaling function is given by the $q$-exponential function appearing in nonextensive statistical mechanics, showing power-law decay of event correlation in natural time.

cond-mat.stat-mech↗
arXiv2003

Two-dimensional dissipative maps at chaos threshold: Sensitivity to initial conditions and relaxation dynamics

The sensitivity to initial conditions and relaxation dynamics of two-dimensional maps are analyzed at the edge of chaos, along the lines of nonextensive statistical mechanics. We verify the dual nature of the entropic index for the Henon map, one ($q_{sen}<1$) related to its sensitivity to initial conditions properties, and the other, graining-dependent ($q_{rel}(W)>1$), related to its relaxation dynamics towards its stationary state attractor. We also corroborate a scaling law between these two indexes, previously found for $z$-logistic maps. Finally we perform a preliminary analysis of a linearized version of the Henon map (the smoothed Lozi map). We find that the sensitivity properties of all these $z$-logistic, Henon and Lozi maps are the same, $q_{sen}=0.2445...$

cond-mat.stat-mech↗
arXiv2003

Mixing and relaxation dynamics of the Henon map at the edge of chaos

The mixing properties (or sensitivity to initial conditions) and relaxation dynamics of the Henon map, together with the connection between these concepts, have been explored numerically at the edge of chaos. It is found that the results are consistent with those coming from one-dimensional dissipative maps. This constitutes the first verification of the scenario in two-dimensional cases and obviously reinforces the idea of weak mixing and weak chaos. Keywords: Nonextensive thermostatistics, Henon map, dynamical systems

cond-mat.stat-mech↗
arXiv2002

Two-dimensional maps at the edge of chaos: Numerical results for the Henon map

The mixing properties (or sensitivity to initial conditions) of two-dimensional Henon map have been explored numerically at the edge of chaos. Three independent methods, which have been developed and used so far for the one-dimensional maps, have been used to accomplish this task. These methods are (i)measure of the divergence of initially nearby orbits, (ii)analysis of the multifractal spectrum and (iii)computation of nonextensive entropy increase rates. The obtained results strongly agree with those of the one-dimensional cases and constitute the first verification of this scenario in two-dimensional maps. This obviously makes the idea of weak chaos even more robust.

cond-mat.stat-mech↗
arXiv2002

On the correct entropic form for systems with power-law behaviour: the case of dissipative maps

Maximum entropy principle does not seem to distinguish between the use of Tsallis and Renyi entropies as either of them may be used to derive similar power-law distributions. In this paper, we address the question whether the Renyi entropy is equally suitable to describe those systems with power-law behaviour, where the use of the Tsallis entropy is relevant. We discuss a specific class of dynamical systems, namely, one dimensional dissipative maps at chaos threshold and make our study from two aspects: i) power-law sensitivity to the initial conditions and the rate of entropy increase, ii) generalized bit cumulants. We present evidence that the Tsallis entropy is the more appropriate entropic form for such studies as opposed to Renyi form.

cond-mat.stat-mech↗
arXiv1999

Asymmetric Unimodal Maps: Some Results from q-generalized Bit Cumulants

In this study, using q-generalized bit cumulants (q is the nonextensivity parameter of the recently introduced Tsallis statistics), we investigate the asymmetric unimodal maps. The study of the q-generalized second cumulant of these maps allows us to determine, for the first time, the dependence of the inflexion paremeter pairs (z_1,z_2) to the nonextensivity parameter q. This behaviour is found to be very similar to that of the logistic-like maps (z_1=z_2=z) reported recently by Costa et al. [Phys.Rev.E 56 (1997) 245].

cond-mat.stat-mech↗
arXiv1999

Exact and approximate results of non-extensive quantum statistics

We develop an analytical technique to derive explicit forms of thermodynamical quantities within the asymptotic approach to non-extensive quantum distribution functions. Using it, we find an expression for the number of particles in a boson system which we compare with other approximate scheme (i.e. factorization approach), and with the recently obtained exact result. To do this, we investigate the predictions on Bose-Einstein condensation and the blackbody radiation. We find that both approximation techniques give results similar to (up to ${\cal O}(q-1)$) the exact ones, making them a useful tool for computations. Because of the simplicity of the factorization approach formulae, it appears that this is the easiest way to handle with physical systems which might exhibit slight deviations from extensivity.

cond-mat.stat-mech↗
arXiv1998

Circular-like Maps: Sensitivity to the Initial Conditions, Multifractality and Nonextensivity

We generalize herein the usual circular map by considering inflexions of arbitrary power $z$, and verify that the scaling law which has been recently proposed [Lyra and Tsallis, Phys.Rev.Lett. 80 (1998) 53] holds for a large range of $z$. Since, for this family of maps, the Hausdorff dimension $d_f$ equals unity for all $z$ values in contrast with the nonextensivity parameter $q$ which does depend on $z$, it becomes clear that $d_f$ plays no major role in the sensitivity to the initial conditions.

cond-mat.stat-mech↗
arXiv1998

Some Bounds upon Nonextensivity Parameter Using the Appropriate Generalized Distribution Functions

In this study, approximate generalized quantal distribution functions and their applications, which appeared in the literature so far, have been summarized. Making use of the generalized Planck radiation law, which have been obtained by the authors of the present manuscript [Physica A240 (1997) 657], some alternative bounds for the nonextensivity parameter $q$ has been estimated. It has been shown that these results are similar to those obtained by Tsallis et al. [Phys. Rev. E52 (1995) 1447] and by Plastino et al. [Phys. Lett. A207 (1995) 42].

cond-mat.stat-mech↗
arXiv1996

Comparison of the Standard Statistical Thermodynamics (SST) with the Generalized Statistical Thermodynamics (GST) Results for the Ising Chain

In this study, the internal energy and the specific heat of the one- dimensional Ising model obtained in the frame of the generalized statistical thermodynamics by Andrade [Physica A175 (1991) 285], have been extended somewhat and compared with the internal energy and the specific heat which have been calculated in the standard statistical thermodynamics.

cond-mat.stat-mech↗