Network analysis of chaotic dynamics in fixed-precision digital domain

When implemented in the digital domain with time, space and value discretized in the binary form, many good dynamical properties of chaotic systems in continuous domain may be degraded or even diminish. To measure the dynamic complexity of a digital chaotic system, the dynamics can be transformed to the form of a state-mapping network. Then, the parameters of the network are verified by some typical dynamical metrics of the original chaotic system in infinite precision, such as Lyapunov exponent and entropy. This article reviews some representative works on the network-based analysis of digital chaotic dynamics and presents a general framework for such analysis, unveiling some intrinsic relationships between digital chaos and complex networks. As an example for discussion, the dynamics of a state-mapping network of the Logistic map in a fixed-precision computer is analyzed and discussed.Comment: 5 pages, 9 figure

Complex dynamics of elementary cellular automata emerging from chaotic rules

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behaviour. CA are well known computational substrates for studying emergent collective behaviour, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict behaviour of any given function. Examples include mechanical computation, \lambda{} and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behaviour when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behaviour from almost any initial condition. Thus just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions

Decay of Nuclear Giant Resonances: Quantum Self-similar Fragmentation

Scaling analysis of nuclear giant resonance transition probabilities with increasing level of complexity in the background states is performed. It is found that the background characteristics, typical for chaotic systems lead to nontrivial multifractal scaling properties.Comment: 4 pages, LaTeX format, pc96.sty + 2 eps figures, accepted as: talk at the 8th Joint EPS-APS International Conference on Physics Computing (PC'96, 17-21. Sept. 1996), to appear in the Proceeding

Dimension of Scrambled Sets and The Dynamics of Tridiagonal Competitive-Cooperative System

One of the central problems in dynamical systems and differential equations is the analysis of the structures of invariant sets. The structures of the invariant sets of a dynamical system or differential equation reflect the complexity of the system or the equation. For example, any omega-limit set of a finite dimensional differential equation is a singleton implies that each bounded solution of the equation eventually stabilizes at some equilibrium state. In general, a dynamical system or differential equation can have very complicated invariant sets or so called chaotic sets. It is of great importance to classify those systems whose minimal invariant sets have certain simple structures and to characterize the complexity of chaotic type sets in general dynamical systems. In this thesis, we focus on the following two important problems: estimates for the dimension of chaotic sets and stable sets in a finite positive entropy system, and characterizations of minimal sets of nonautonomous tridiagonal competitive-cooperative systems

Quantum chaos, integrability, and late times in the Krylov basis

Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by Random Matrix Theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffices to produce the full long time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra

Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of λ starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map

Chaos and quantum-nondemolition measurements

The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ‘‘deterministic randomness’’ in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity