Statistical characterization of the standard map

Physical Review E, 1997

We analytically establish the role of a spectrum of Lyapunov exponents in the evolution of phase-space distributions ρ(p, q). Of particular interest is λ2, an exponent which quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and which is generally larger than the maximal Lyapunov exponent λ for trajectories. The approach is trajectory-independent and is therefore applicable to both classical and quantum mechanics. In the latter case we show that theh → 0 limit yields the classical, fully chaotic, result for the quantum cat map.

Physical Review E, 2009

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. PACS numbers: 05.20.-y, 05.45.Ac, 05.45.Pq

Physics Letters A, 2001

We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form Sq ≡ [1 − W i=1 p q i ]/[q − 1] (with S1 = − W i=1 pi ln pi) for two families of one-dimensional dissipative maps, namely a logistic-and a periodic-like with arbitrary inflexion z at their maximum. At t = 0 we choose N initial conditions inside one of the W small windows in which the accessible phase space is partitioned; to neutralize large fluctuations we conveniently average over a large amount of initial windows. We verify that one and only one value q * < 1 exists such that the limt→∞ limW →∞ limN→∞ Sq(t)/t is finite, thus generalizing the (ensemble version of) Kolmogorov-Sinai entropy (which corresponds to q * = 1 in the present formalism). This special, z-dependent, value q * numerically coincides, for both families of maps and all z, with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal f (α) function).

Physical Review E, 2015

We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finitetime Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semi-ordered (or semi-chaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase-space associated to them. Applying our methodology to a chain of coupled standard maps we obtain: (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; (iii) the dependence of the Lyapunov exponents with the coupling strength.

Physica A: Statistical Mechanics and its Applications, 2006

We analyze the fluctuating dynamics at the golden-mean transition to chaos in the critical circle map and find that trajectories within the critical attractor consist of infinite sets of power laws mixed together. We elucidate this structure assisted by known renormalization group (RG) results. Next we proceed to weigh the new findings against Tsallis' entropic and Mori's thermodynamic theoretical schemes and observe behavior to a large extent richer than previously reported. We find that the sensitivity to initial conditions ξt has the form of families of intertwined q-exponentials, of which we determine the q-indexes and the generalized Lyapunov coefficient spectra λq. Further, the dynamics within the critical attractor is found to consist of not one but a collection of Mori's q-phase transitions with a hierarchical structure. The value of Mori's 'thermodynamic field' variable q at each transition corresponds to the same special value for the entropic index q. We discuss the relationship between the two formalisms and indicate the usefulness of the methods involved to determine the universal trajectory scaling function σ and/or the ocurrence and characterization of dynamical phase transitions.

Europhysics Letters (EPL), 2002

The dissipation associated with nonequilibrium flow processes is reflected by the formation of strange attractor distributions in phase space. The information dimension of these attractors is less than that of the equilibrium phase space, corresponding to the extreme rarity of nonequilibrium states. Here we take advantage of a simple model for heat conduction to demonstrate that the nonequilibrium dimensionality loss can definitely exceed the number of phase-space dimensions required to thermostat an otherwise Hamiltonian system. PACS: 05.20, 5.45.D, 05.70.L Nonequilibrium molecular dynamics has been used to establish a close link between microscopic dynamical phase-space instabilities and the macroscopic irreversible dissipation associated with the Second Law of Thermodynamics [1,2]. Both non-Hamiltonian and Hamiltonian methods have been used . The simplest such connection between microscopic dynamics and macroscopic dissipation results when one or more Nosé-Hoover thermostats [4] are used to control nonequilibrium steady states. In the absence of nonequilibrium fluxes and with sufficient phase-space mixing, these thermostats generate Gibbs' and Boltzmann's canonical distribution. In cases which include nonequilibrium driving the instantaneous external entropy production rate (due to heat transfer with the thermostats) is proportional to the sum of the instantaneous Lyapunov exponents [5]:

PHYSICAL REVIEW A, 1989

It is shown that a dynamical phase transition is typically present in chaotic Hamiltonian systems due to the intermittent motion near regular islands. The transition shows up in the spectrum of the Renyi entropies K~ as a jump from a finite K& to zero at q =1+0. Numerical measurements have been carried out for the correlation entropy K2.

Entropy, 2013

Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann-Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.

Mathematical Biosciences and Engineering, 2004

This study presents a survey of the results obtained by the authors on statistical description of dynamical chaos and the effect of noise on dynamical regimes. We deal with nearly hyperbolic and nonhyperbolic chaotic attractors and discuss methods of diagnosing the type of an attractor. We consider regularities of the relaxation to an invariant probability measure for different types of attractors. We explore peculiarities of autocorrelation decay and of power spectrum shape and their interconnection with Lyapunov exponents, instantaneous phase diffusion and the intensity of external noise. Numeric results are compared with experimental data.

Physica A: Statistical Mechanics and its Applications, 2008

We study the connection between the appearance of a 'metastable' behavior of weakly chaotic orbits, characterized by a constant rate of increase of the Tsallis q-entropy [J. of Stat. Phys. Vol. 52 (1988)], and the solutions of the variational equations of motion for the same orbits. We demonstrate that the variational equations yield transient solutions, lasting for long time intervals, during which the length of deviation vectors of nearby orbits grows in time almost as a power-law. The associated power exponent can be simply related to the entropic exponent for which the q-entropy exhibits a constant rate of increase. This analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits, that we call 'Average Power Law Exponent' (APLE). We compare the APLE with other established indicators of the literature. In particular, we give examples of application of the APLE in a) a thin separatrix layer of the standard map, b) the stickiness region around an island of stability in the same map, and c) the web of resonances of a 4D symplectic map. In all these cases we identify weakly chaotic orbits exhibiting the 'metastable' behavior associated with the Tsallis q-entropy.