Local and global statistical properties of deterministic chaos
D. Kernan1995, Il Nuovo Cimento D
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Local and global statistical properties of a class of one dimensional dissipative chaotic maps and a class of 2 dimensional conservative hyperbolic maps are investigated. This is achieved by considering the spectral properties of the Perron-Frobenius operator (the evolution operator for probability densities) acting on two different types of function space. In the first case, the function space is piecewise analytic, and includes functions having support over local regions of phase space. In the second case, the function space essentially consists of functions which are "globally" analytic, i.e analytic over the given systems entire phase space. Each function space defines a space of measurable functions or observables, whose statistical moments and corresponding characteristic times can be exactly determined.
Related papers
Statistical properties of chaos demonstrated in a class of one-dimensional maps
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993
One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.
Local and global statistical dynamical properties of chaotic Markov analytic maps and repellers: A coarse grained and spectral perspective
Chaos, Solitons & Fractals, 2009
Statistical properties of dynamical chaos
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.
Characterization of chaos in random maps
Physica A: Statistical Mechanics and its Applications, 1996
We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operator.
A description of stochastic systems using chaotic maps
Journal of Applied Mathematics and Stochastic Analysis, 2004
Letρ(x,t)denote a family of probability density functions parameterized by timet. We show the existence of a family{τ1:t>0}of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are preciselyρ(x,t). In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.
SPECTRAL ANALYSIS OF EXPANDING ONE-DIMENSIONAL CHAOTIC TRANSFORMATIONS
The purpose of this paper is to show explicitly the spectral distribution function of some stationary stochastic processes as
Chaos in high-dimensional dissipative dynamical systems
For dissipative dynamical systems described by a system of ordinary differential equations, we address the question of how the probability of chaotic dynamics increases with the dimensionality of the phase space. We find that for a system of d globally coupled ODE's with quadratic and cubic nonlinearities with randomly chosen coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from ~10 −5 − 10 −4 for d = 3 to essentially one for d ~ 50. In the limit of large d, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity, but not on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling, universality, and for the probability of chaos.
Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps
Nonlinearity, 2011
We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated in terms of generalised Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behavior. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.
Characterization of chaotic maps using the permutation Bandt-Pompe probability distribution
By appealing to a long list of different nonlinear maps we review the characterization of time series arising from chaotic maps. The main tool for this characterization is the permutation Bandt-Pompe probability distribution function. We focus attention on both local and global characteristics of the components of this probability distribution function. We show that forbidden ordinal patterns (local quantifiers) exhibit an exponential growth for pattern-length range 3 ≤ D ≤ 8, in the case of finite time series data. Indeed, there is a minimum Dmin-value such that forbidden patterns cannot appear for D < Dmin. The system's localization in an entropy-complexity plane (global quantifier) displays typical specific features associated with its dynamics' nature. We conclude that a more "robust" distinction between deterministic and stochastic dynamics is achieved via the present time series' treatment based on the global characteristics of the permutation Bandt-Pompe probability distribution function.
Statistics of eigenfunctions in open chaotic systems: a perturbative approach
Physical Review E - PHYS REV E, 2009
We investigate the statistical properties of the complexness parameter which characterizes uniquely complexness (biorthogonality) of resonance eigenstates of open chaotic systems. Specifying to the regime of isolated resonances, we apply the random matrix theory to the effective Hamiltonian formalism and derive analytically the probability distribution of the complexness parameter for two statistical ensembles describing the systems invariant under time reversal. For those with rigid spectra, we consider a Hamiltonian characterized by a picket-fence spectrum without spectral fluctuations. Then, in the more realistic case of a Hamiltonian described by the Gaussian Orthogonal Ensemble, we reveal and discuss the r\^ole of spectral fluctuations.
Related papers
Spectral statistics: From disordered to chaotic systems
Physical Review Letters, 1995
The relation between disordered and chaotic systems is investigated. It is obtained by identifying the diffusion operator of the disordered systems with the Perron-Frobenius operator in the general case. This association enables us to extend results obtained in the diffusive regime to general chaotic systems. In particular, the two--point level density correlator and the structure factor for general chaotic systems are calculated and characterized. The behavior of the structure factor around the Heisenberg time is quantitatively described in terms of short periodic orbits.
Statistical approach to nonhyperbolic chaotic systems
Spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable, and the Lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant. The method is applied to the logistic map both for parameter values at which finite Markov partitions can be found as well as for more typical parameter values. A universal behavior is found for the spectral gap in the period-doubling inverse cascade of chaotic band-merging bifurcations. Full agreement with numerical simulation is obtained.
Statistical properties of chaos in Chebyshev maps
Physics Letters A, 1984
Statistical properties of fully developed chaotic maps in the form of Chebyshev polynomials are calculated exactly. We derive analytic expressions for characteristic functions, moments, and moment functions and mention a number of other properties. We also determine higher-orde~ moment functions, which are important for a characterization of the nongaussian processes exhibited by many maps.
Power spectra for deterministic chaotic dynamical systems
Nonlinearity, 2008
We present results on the broadband nature of power spectra for large classes of discrete chaotic dynamical systems, including uniformly hyperbolic (Axiom A) diffeomorphisms and certain nonuniformly hyperbolic diffeomorphisms (such as the Hénon map). Our results also apply to noninvertible maps, including Collet-Eckmann maps. For such maps (even the nonmixing ones) and Hölder continuous observables, we prove that the power spectrum is analytic except for finitely many removable singularities, and that for typical observables the spectrum is nowhere zero. Indeed, we show that the power spectrum is bounded away from zero except for infinitely degenerate observables.
Entropy decay as a measure of stochasticity in chaotic systems
Physical review. A, 1986
The series of truncated entropies, approaching the Kolmogorov-Sinai entropy when correlations between increasingly distant signals are taken into account, is considered. A general expression for the related characteristic time is obtained for one-dimensional maps. Evaluating it analytically near the state of maxi- rnal entropy and at intermittency, we find it reflecting faithfully the degree of stochasticity. Recently there has been a rising interest in stochastic properties of chaotic systems. ' In such systems the degree of randomness is particularized globally by the Kolmogorov-Sinai entropy's (see Refs. 4 and 5 for re- views). In addition, Lyapunov characteristic exponents measure, on the average, the divergence rate of nearby tra-
Statistical descriptions of nonlinear systems at the onset of chaos
Physica A: Statistical Mechanics and its Applications, 2006
Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to initial conditions λ. The statistical formalism and the equality K = λ can be extended to weakly chaotic systems by suitable and corresponding generalizations of the logarithm and of the entropy. Using the logistic map as a test case we consider a wide class of deformed statistical description which includes Tsallis, Abe and Kaniadakis proposals. The physical criterion of finite-entropy growth K strongly restricts the suitable entropies. We study how large is the region in parameter space where the generalized description is useful.
Statistical properties of chaotic wavefunctions in two and more dimensions
The European Physical Journal Special Topics, 2007
Starting with Berry's hypothesis for fixed energy waves in a classically chaotic system, and casting it in a Green function form, we derive wavefunction correlations and density matrices for few or many particles. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N → ∞. This arises through a little known asymptotic limit of Bessel functions. The Berry random wave hypothesis in many dimensions may be viewed as an alternative approach to quantum statistical mechanics, when extended to include constraints and potentials.
Properties of fully developed chaos in one-dimensional maps
Journal of Statistical Physics, 1984
We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.
Sensitive Response to Time Dependent Perturbations and Spectral Properties of Deterministic Chaos
When a probabilistic description of deterministic chaos is feasible, it can describe the dynamical evolution of a given system beyond the Lyapunov horizon where a point-like evolutionary description fails. In the one dimensional examples that we have studied a probabilistic description is very informative about the spectral properties of a given system. One example is an intermittent map whose behaviour is similar to that of a bistable stochastic system. When weakly perturbed by an oscillating kick the system responds sensitivelyat forcing periods predicted by the system's spectra.
Spectral properties of classically chaotic systems
AIP Conference Proceedings, 1999
We describe the connection between quantum systems which have a chaotic classical counterpart and random matrix theory. As is well-known, it consists in the fact that the statistical properties of the spectra of such systems in the semiclassical limit are equivalent to those of random matrix theory. Here, we first briefly review some properties of random matrices, and then proceed to justify the above-mentioned connection in two different ways: First, according to the classic work of Berry, we show how the result can be derived from periodic orbit theory, of which we give a rapid overview; second, we show how the same result can be obtained with greater generality but in a more speculative manner using the concept of structural invariance.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.