1,581 research outputs found
Generalized dimensions of Feigenbaum's attractor from renormalization-group functional equations
A method is suggested for the computation of the generalized dimensions of
fractal attractors at the period-doubling transition to chaos. The approach is
based on an eigenvalue problem formulated in terms of functional equations,
with a coefficient expressed in terms of Feigenbaum's universal fixed-point
function. The accuracy of the results is determined only by precision of the
representation of the universal function.Comment: 6 pages, 2 table
Two stories outside Boltzmann-Gibbs statistics: Mori's q-phase transitions and glassy dynamics at the onset of chaos
First, we analyze trajectories inside the Feigenbaum attractor and obtain the
atypical weak sensitivity to initial conditions and loss of information
associated to their dynamics. We identify the Mori singularities in its
Lyapunov spectrum with the appearance of a special value for the entropic index
q of the Tsallis statistics. Secondly, the dynamics of iterates at the
noise-perturbed transition to chaos is shown to exhibit the characteristic
elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion
and arrest. The properties of the bifurcation gap induced by the noise are seen
to be comparable to those of a supercooled liquid above a glass transition
temperature.Comment: Proceedings of: 31st Workshop of the International School of Solid
State Physics, Complexity, Metastability and Nonextensivity, Erice (Sicily)
20-26 July 2004 World Scientific in the special series of the E. Majorana
conferences, in pres
Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part one: Sunspot dynamics
In this study, the nonlinear analysis of the sunspot index is embedded in the
non-extensive statistical theory of Tsallis. The triplet of Tsallis, as well as
the correlation dimension and the Lyapunov exponent spectrum were estimated for
the SVD components of the sunspot index timeseries. Also the multifractal
scaling exponent spectrum, the generalized Renyi dimension spectrum and the
spectrum of the structure function exponents were estimated experimentally and
theoretically by using the entropy principle included in Tsallis non extensive
statistical theory, following Arimitsu and Arimitsu. Our analysis showed
clearly the following: a) a phase transition process in the solar dynamics from
high dimensional non Gaussian SOC state to a low dimensional non Gaussian
chaotic state, b) strong intermittent solar turbulence and anomalous
(multifractal) diffusion solar process, which is strengthened as the solar
dynamics makes phase transition to low dimensional chaos in accordance to
Ruzmaikin, Zeleny and Milovanov studies c) faithful agreement of Tsallis non
equilibrium statistical theory with the experimental estimations of i)
non-Gaussian probability distribution function, ii) multifractal scaling
exponent spectrum and generalized Renyi dimension spectrum, iii) exponent
spectrum of the structure functions estimated for the sunspot index and its
underlying non equilibrium solar dynamics.Comment: 40 pages, 11 figure
Fluctuating dynamics at the quasiperiodic onset of chaos, Tsallis q-statistics and Mori's q-phase thermodynamics
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
has the form of families of intertwined q-exponentials, of which we determine
the q-indexes and the generalized Lyapunov coefficient spectra. 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 and/or the ocurrence and
characterization of dynamical phase transitions.Comment: Resubmitted to Physical Review E. The title has been changed slightly
and the abstract has been extended. There is a new subsection following the
conclusions that explains the role and usefulness of the q-statistics in the
problem studied. Other minor changes througout the tex
Visualization of Chaos for Finance Majors
Efforts to simulate turbulence in the financial markets include experiments with the logistic equation: x(t)=kappa x(t-1)[1-x(t-1)], with 0Logistic Equation, Visualization, Strange Attractor, Chaos, Hurst Exponent
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