1,395 research outputs found
Complexity for extended dynamical systems
We consider dynamical systems for which the spatial extension plays an
important role. For these systems, the notions of attractor, epsilon-entropy
and topological entropy per unit time and volume have been introduced
previously. In this paper we use the notion of Kolmogorov complexity to
introduce, for extended dynamical systems, a notion of complexity per unit time
and volume which plays the same role as the metric entropy for classical
dynamical systems. We introduce this notion as an almost sure limit on orbits
of the system. Moreover we prove a kind of variational principle for this
complexity.Comment: 29 page
Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model
Using a renormalization group method, we calculate 800 high-temperature
coefficients of the magnetic susceptibility of the hierarchical Ising model.
The conventional quantities obtained from differences of ratios of coefficients
show unexpected smooth oscillations with a period growing logarithmically and
can be fitted assuming corrections to the scaling laws with complex exponents.Comment: 10 pages, Latex , uses revtex. 2 figures not included (hard copies
available on request
A "metric" complexity for weakly chaotic systems
We consider the number of Bowen sets which are necessary to cover a large
measure subset of the phase space. This introduce some complexity indicator
characterizing different kind of (weakly) chaotic dynamics. Since in many
systems its value is given by a sort of local entropy, this indicator is quite
simple to be calculated. We give some example of calculation in nontrivial
systems (interval exchanges, piecewise isometries e.g.) and a formula similar
to the Ruelle-Pesin one, relating the complexity indicator to some initial
condition sensitivity indicators playing the role of positive Lyapunov
exponents.Comment: 15 pages, no figures. Articl
Dynamical estimates of chaotic systems from Poincar\'e recurrences
We show that the probability distribution function that best fits the
distribution of return times between two consecutive visits of a chaotic
trajectory to finite size regions in phase space deviates from the exponential
statistics by a small power-law term, a term that represents the deterministic
manifestation of the dynamics, which can be easily experimentally detected and
theoretically estimated. We also provide simpler and faster ways to calculate
the positive Lyapunov exponents and the short-term correlation function by
either realizing observations of higher probable returns or by calculating the
eigenvalues of only one very especial unstable periodic orbit of low-period.
Finally, we discuss how our approaches can be used to treat data coming from
complex systems.Comment: subm. for publication. Accepted fpr publication in Chao
Dynamics of a map with power-law tail
We analyze a one-dimensional piecewise continuous discrete model proposed
originally in studies on population ecology. The map is composed of a linear
part and a power-law decreasing piece, and has three parameters. The system
presents both regular and chaotic behavior. We study numerically and, in part,
analytically different bifurcation structures. Particularly interesting is the
description of the abrupt transition order-to-chaos mediated by an attractor
made of an infinite number of limit cycles with only a finite number of
different periods. It is shown that the power-law piece in the map is at the
origin of this type of bifurcation. The system exhibits interior crises and
crisis-induced intermittency.Comment: 28 pages, 17 figure
A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory
The goal of this article is to provide a practical method to calculate, in a
scalar theory, accurate numerical values of the renormalized quantities which
could be used to test any kind of approximate calculation. We use finite
truncations of the Fourier transform of the recursion formula for Dyson's
hierarchical model in the symmetric phase to perform high-precision
calculations of the unsubtracted Green's functions at zero momentum in
dimension 3, 4, and 5. We use the well-known correspondence between statistical
mechanics and field theory in which the large cut-off limit is obtained by
letting beta reach a critical value beta_c (with up to 16 significant digits in
our actual calculations). We show that the round-off errors on the magnetic
susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the
systematic errors (finite truncations and volume) can be controlled with an
exponential precision and reduced to a level lower than the numerical errors.
We justify the use of the truncation for calculations of the high-temperature
expansion. We calculate the dimensionless renormalized coupling constant
corresponding to the 4-point function and show that when beta -> beta_c, this
quantity tends to a fixed value which can be determined accurately when D=3
(hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure
A solvable model of the genesis of amino-acid sequences via coupled dynamics of folding and slow genetic variation
We study the coupled dynamics of primary and secondary structure formation
(i.e. slow genetic sequence selection and fast folding) in the context of a
solvable microscopic model that includes both short-range steric forces and and
long-range polarity-driven forces. Our solution is based on the diagonalization
of replicated transfer matrices, and leads in the thermodynamic limit to
explicit predictions regarding phase transitions and phase diagrams at genetic
equilibrium. The predicted phenomenology allows for natural physical
interpretations, and finds satisfactory support in numerical simulations.Comment: 51 pages, 13 figures, submitted to J. Phys.
On the spectrum of Farey and Gauss maps
In this paper we introduce Hilbert spaces of holomorphic functions given by
generalized Borel and Laplace transforms which are left invariant by the
transfer operators of the Farey map and its induced version, the Gauss map,
respectively. By means of a suitable operator-valued power series we are able
to study simultaneously the spectrum of both these operators along with the
analytic properties of the associated dynamical zeta functions.Comment: 23 page
- …