Remarks on Power Spectra of Chaotic Dynamical Systems

2014

In this paper, power spectral analysis of deterministic multiscale chaotic dynamical system is presented. The system is obtained by coupling two versions of the well-known Lorenz (1963) model with distinct time scales that differ by a certain time-scale factor. This system is commonly used for exploring various aspects of atmospheric and climate dynamics, and also for estimating the computational effectiveness of numerical schemes and algorithms used in numerical weather prediction, data assimilation and climate simulation. The influence of the coupling strength parameter on power spectral densities and spectrogram is discussed. Key-Words: Dynamical System, Deterministic Chaos, Power Spectral Density, Climate Modeling

Physical Review Letters, 1993

The signature of chaos in the spectral autocorrelation function and in its Fourier transform, the survival probability, is shown to be in good agreement with the predictions of random matrix theory. An expression is proposed for the survival probability of an experimentally prepared nonstationary state when the dynamics are intermediate between chaotic and regular. Its validity is tested through the study of a model Hamiltonian. Two parameters can be extracted from the above observable, one which characterizes the level statistics and one which characterizes the distribution of transition intensities.

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.

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.

Physica D: Nonlinear Phenomena, 1993

We investigate anomalous diffusion and the corresponding power spectra generated by iterated maps and analyze the motion in terms of the probabilistic continuous-time random walk approach. Both stationary and non-stationary conditions are considered demonstrating the dependence of the mean-squared displacement and of the power-spectra on the initial conditions. The theoretical results are corroborated by numerical calculations and excellent agreement is obtained.

Physical Review E, 2005

The existence of a formal analogy between quantum energy spectra and discrete time series has been recently pointed out. When the energy level fluctuations are described by means of the ␦ n statistic, it is found that chaotic quantum systems are characterized by 1 / f noise, while regular systems are characterized by 1 / f 2. In order to investigate the correlation structure of the ␦ n statistic, we study the qth-order height-height correlation function C q ͑͒, which measures the momentum of order q, i.e., the average qth power of the signal change after a time delay. It is shown that this function has a logarithmic behavior for the spectra of chaotic quantum systems, modeled by means of random matrix theory. On the other hand, since the power spectrum of chaotic energy spectra considered as time series exhibit 1 / f noise, we investigate whether the qth-order height-height correlation function of other time series with 1 / f noise exhibits the same properties. A time series of this kind can be generated as a linear combination of cosine functions with arbitrary phases. We find that the logarithmic behavior arises with great accuracy for time series generated with random phases.

Chaos, Solitons & Fractals, 2009

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.

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.

Physical Review Letters, 2002

We explore the influence of external perturbations on the energy levels of a Hamiltonian drawn at random from the Gaussian unitary distribution of Hermitian matrices. By deriving the joint distribution function of eigenvalues, we obtain the (n,m)-point parametric correlation function of the initial and final density of states for perturbations of arbitrary rank and strength. A further generalization of these results allows for the incorporation of short-range spatial correlations in diffusive as well as ballistic chaotic structures.

Physical Review E, 2002

We study local and global correlations between the naturally invariant measure of a chaotic one-dimensional map f and the conditionally invariant measure of the transiently chaotic map f H. The two maps differ only within a narrow interval H, while the two measures significantly differ within the images f l (H), where l is smaller than some critical number l c. We point out two different types of correlations. Typically, the critical number l c is small. The 2 value, which characterizes the global discrepancy between the two measures, typically obeys a power-law dependence on the width ⑀ of the interval H, with the exponent identical to the information dimension. If H is centered on an image of the critical point, then l c increases indefinitely with the decrease of ⑀, and the 2 value obeys a modulated power-law dependence on ⑀.

Journal of Physics A: Mathematical and Theoretical, 2010

We apply the maximum entropy principle to construct the natural invariant density and Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique that is based on the solution of Hausdorff moment problem via maximizing Shannon entropy, we estimate the invariant density and the Lyapunov exponent of nonlinear maps in one-dimension from a knowledge of finite number of moments. The accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available. Furthermore, we also consider a very complex example for which no exact analytical result for invariant density is available. A comparison of our results to those available in the literature is also discussed.

International Journal of Bifurcation and Chaos, 2011

The question of spectral analysis for deterministic chaos is not well understood in the literature. In this paper, using iterates of chaotic interval maps as time series, we analyze the mathematical properties of the Fourier series of these iterates. The key idea is the connection between the total variation and the topological entropy of the iterates of the interval map, from where special properties of the Fourier coefficients are obtained. Various examples are given to illustrate the applications of the main theorems.

Journal of Physics A: Mathematical and General, 1996

We study a simple nonlinear mapping with a strange nonchaotic attractor characterized by a singular continuous power spectrum. We show that the symbolic dynamics is exactly described by a language generated from a suitable inflation rule. We derive renormalization transformations for both the power spectrum and the autocorrelation function, thus obtaining a quantitative description of the scaling properties. The multifractal nature of the spectrum is also discussed. † A mixture of periodic and chaotic motion eventually gives a periodic correlation function, never returning to the value 1.

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.