Symbolic Dynamics

We propose a theory of deterministic chaos for discrete systems, based on their representations in binary state spaces $ \Omega $, homeomorphic to the space of symbolic dynamics. This formalism is applied to neural networks and cellular automata; it is found that such systems cannot be viewed as chaotic when one uses the Hamming distance as the metric for the

The pressure was studied in a rather abstract theory as an important,notion of the thermodynamic,formalism. The present paper gives a more,concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function 8 to its long-term time averages through,the Hausdorff and packing,dimensions,of the subsets on which,8 has prescribed long-term time-average values. Functions 8 with values in R, are considered. For those 8 depending only on finitely many symbols, we get complete results, unifying and completing,many,partial results. KEY WORDS: Symbolic dynamics; Dimensions; Gibbs measures.

This paper describes a design of BIST system for single channel 2.5-Gb/s serializer/deserializer (SerDes) macros, including PRBS generator and checker. In order to fit the word width of DUT, a 10-bit parallel PRBS data generator is made. And for PRBS checker, a new method, called dynamic detecting is used to obtain the head symbol dynamically. After that, an referenced PRBS generated, and compared with signals received by PRBS checker. Finally, BER will be calculated to make sure if it is satisfied with the target.

Los autómatas celulares, inventados por John von Neumann en la década de 1940, son importantes modelos computacionales discretos que se han convertido en una herramienta fundamental en diversas áreas de la ciencia. Por mucho, el ejemplo más famoso es el Juego de la Vida inventado por John H. Conway. En este capítulo presentamos los fundamentos modernos de la teoría de autómatas celulares, los cuales provienen principalmente de la teoría de grupos y la topología.

Nuestro objetivo principal es demostrar el famoso Teorema de Curtis-
Hedlund, el cual caracteriza a los autómatas celulares sobre alfabetos finitos como funciones G-equivariantes y continuas en la topología prodiscreta del espacio de configuraciones. Para llegar a esto, primero revisamos temas de acciones de grupos y topología, y luego presentamos la definición de autómatas celulares sobre grupos junto con algunos ejemplos importantes.

Unimodal maps have been broadly used as a base of new encryption strategies. Recently, a stream cipher has been proposed in the literature, whose keystream is basically a symbolic sequence of the (one-parameter) logistic map or of the tent map. In the present work a thorough analysis of the keystream is made which reveals the existence of some serious security problems

The discovery of semantic information from complex signals is a task concerned with connecting humans' perceptions and/or intentions with the signals content. In the case of audio signals, complex perceptions are appraised in a listener's mind, that trigger affective responses that may be relevant for well-being and survival. In this paper we are interested in the broader question of relations between uncertainty in data as measured using various information criteria and emotions, and we propose a novel method that combines nonlinear dynamics analysis with a method of adaptive time series symbolization that finds the meaningful audio structure in terms of symbolized recurrence properties. In a first phase we obtain symbolic recurrence quantification measures from symbolic recurrence plots, without the need to reconstruct the phase space with embedding. Then we estimate symbolic dynamical invariants from symbolized time series, after embedding. The invariants are: correlation dimension, correlation entropy and Lyapunov exponent. Through their application for the logistic map, we show that our measures are in agreement with known methods from literature. We further show that one symbolic recurrence measure, namely the symbolic Shannon entropy, correlates positively with the positive Lyapunov exponents. Finally we evaluate the performance of our measures in emotion recognition through the implementation of classification tasks for different types of audio signals, and show that in some cases, they perform better than state-of-the-art methods that rely on low-level acoustic features.

Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then shown to be the Haar measure of a certain transfer operator. A general proof is given that any transfer operator can be understood to be nothing more nor less than a push-forward on a Banach space; such push-forwards induce an invariant measure, the Haar measure, of which the Minkowski measure can serve as a prototypical example. Some minor notes pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made.

Since 1990s chaotic dynamical systems have been widely used to design new strategies to encrypt information. Indeed, the dependency to initial conditions and control parameters, along with the ergodicity of their temporal evolution allow the establishment of chaos as the base of new cryptosystems, i.e., of new schemes of confusion and diffusion of information. However, an optimum design in the context of chaos-based cryptography demands a thorough knowledge not only of the foundations of cryptography, but also of the dynamics and inner structure of chaos. Therefore, any proposal to use chaos in the context of cryptography must respect a series of design rules, in order to avoid the reconstruction of the dynamics of the underlying chaotic system, and to determine an optimum use of the virtues of the chaotic dynamics. Although it is possible to use chaos to design analog cryptosystems based on synchronization techniques, this Thesis is focused on the application of chaotic maps, i.e., chaotic dynamical systems defined in discrete time to cryptography. In this sense, a set of mathematical tools are defined to establish the adequacy of a chaotic map as the base of a cryptosystem, and the requirements that an encryption architecture must satisfy to avoid the dynamical reconstruction of the underlying chaotic map. More precisely, this Thesis provides an extension and systematization of the results derived from the cryptanalysis of chaos-based cryptosystems. The above goal comprises three different stages: 1.- Definition of a set of mathematical tools that allow the selection of the adequate configurations of a dynamical system to implement strategies of confusion and diffusion of information. 2.- Study of the most popular chaotic maps in the field of chaos-based cryptography to determine whether these maps can be used to design new cryptosystems without incurring in security problems. 3.- Summary and conclusions of the first two stages. The aim is to define a set of rules or recommendations as a guide for the design of chaos-based cryptosystems. Recalling the first stage, its main purpose is the search of procedures to infer or estimate the initial conditions and/or the control parameters from the orbits of a chaotic map. Different scenarios are considered depending on whether complete orbits are accesible or it is only possible to work with sampled or discretized versions of the orbits. In all scenarios the goal consist in building bijective functions with respect to the initial conditions and/or the control parameters. The requirements to build these bijective functions are clarified, along with the procedures to guide the estimation of the initial conditions and/or the control parameters. In order to test the set of mathematical tools and the estimation methods, the logistic map and its associated topological conjugate maps are thoroughly studied, since these maps are the most widely used in the design of new digital chaotic cryptosystems. Specially relevant is the study of the symbolic dynamics and order patterns of unimodal maps. The study of this family of chaotic maps leads to a series of very useful results to define a set of recommendations for both the evaluation of the security of chaos-based cryptosystems and the design of encryption schemes based on chaos.

In this paper we study the ergodic theory of a class of symbolic dynamical systems $(\O, T, \mu)$ where $T:{\O}\to \O$ the left shift transformation on $\O=\prod_0^\infty\{0,1\}$ and $\mu$ is a $\s$-finite $T$-invariant measure having the property that one can find a real number $d$ so that $\mu(\tau^d)=\infty$ but $\mu(\tau^{d-\epsilon}) 0$, where $\tau$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $\mu$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d>0$ then $\mu$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $\mu$ is an infinite measure with scaling rate of order $n^d$. Moreove...

Discovering semantic information from complexsignals is a task concerned with connecting humans'perceptions and/or intentions with the signal's content. In thecase of audio textures from environmental sounds produced bya cheering crowd, laughter, crackling fire, car crash orexplosion, complex meanings of the events are inferred andappraised in a listener's mind, which triggers an affectiveresponse that is relevant for well-being and survival. In thispaper we contribute to the ongoing research on affectivesemantics from sound by proposing a novel learningframework of affective auditory scene analysis using arecently developed method of non-linear dynamic signalanalysis. Using an adaptive symbolization process that findsthe best audio structure representation in terms of thesymbolized sequence recurrence properties, we show thatmeasures of periodicity and complexity derived from ourmodel are relevant for the characterization of affect in auditoryscenes, and that they will ...

Resume. Lesecoulements a†eurants presentent generalement des regimes d'oscillations entretenues resultant d'un processus de retro-action complexe, en particulier pour des nombres de Reynolds moderes. Des travaux anterieurs ont revele un phenomµene d'¶echange de modes qui restait µa caracteriser d'un point de vue dynamique. L'intermittence entre les deux modes dominants est ici caracterisee par dynamique symbolique. La premiµereetape consiste µa reconstruire, µa