Synchronous chaos in the coupled system of two logistic maps

Synchronous chaos is investigated in the coupled system of two Logistic maps. Although the diffusive coupling admits all synchronized motions, the stabilities of their configurations are dependent on the transverse Lyapunov exponents while independent of the longitudinal Lyapunov exponents. It is shown that synchronous chaos is structurally stable with respect to the system parameters. The mean motion is the pseudo-orbit of an individual local map so that its dynamics can be described by the local map. (C) 2004 Elsevier Ltd. All rights reserved.

Solving the Levins' paradox in the logistic model to the population growth

We introduce a new method to improve Markov maps by means of a Bayesian approach. The method starts from an initial map model, wherefrom a likelihood function is defined which is regulated by a temperature-like parameter. Then, the new constraints are added by the use of Bayes rule in the prior distribution. We applied the method to the logistic map of population growth of a single species. We show that the population size is limited for all ranges of parameters, allowing thus to overcome difficulties in interpretation of the concept of carrying capacity known as the Levins paradox. © Published under licence by IOP Publishing Ltd.

Consistency in approximate entropy given by a volumetric estimate

Non-linear methods for estimating variability in time-series are currently of widespread use. Among such methods are approximate entropy (ApEn) and sample approximate entropy (SampEn). The applicability of ApEn and SampEn in analyzing data is evident and their use is increasing. However, consistency is a point of concern in these tools, i.e., the classification of the temporal organization of a data set might indicate a relative less ordered series in relation to another when the opposite is true. As highlighted by their proponents themselves, ApEn and SampEn might present incorrect results due to this lack of consistency. In this study, we present a method which gains consistency by using ApEn repeatedly in a wide range of combinations of window lengths and matching error tolerance. The tool is called volumetric approximate entropy, vApEn. We analyze nine artificially generated prototypical time-series with different degrees of temporal order (combinations of sine waves, logistic maps with different control parameter values, random noises). While ApEn/SampEn clearly fail to consistently identify the temporal order of the sequences, vApEn correctly do. In order to validate the tool we performed shuffled and surrogate data analysis. Statistical analysis confirmed the consistency of the method. (C) 2008 Elsevier Ltd. All rights reserved.

Cifrado de imágenes digitales basado en teoría del caos: mapas logísticos

Las nuevas tendencias de compartir archivos multimedia a través de redes abiertas, demanda el uso de mejores técnicas de encriptación que garanticen la integridad, disponibilidad y confidencialidad, manteniendo y/o mejorando la eficiencia del proceso de cifrado sobre estos archivos. Hoy en día es frecuente la transferencia de imágenes a través de medios tecnológicos, siendo necesario la actualización de las técnicas de encriptación existentes y mejor aún, la búsqueda de nuevas alternativas. Actualmente los algoritmos criptográficos clásicos son altamente conocidos en medio de la sociedad informática lo que provoca mayor vulnerabilidad, sin contar los altos tiempos de procesamiento al momento de ser utilizados, elevando la probabilidad de ser descifrados y minimizando la disponibilidad inmediata de los recursos. Para disminuir estas probabilidades, el uso de la teoría de caos surge como una buena opción para ser aplicada en un algoritmo que tome partida del comportamiento caótico de los sistemas dinámicos, y aproveche las propiedades de los mapas logísticos para elevar el nivel de robustez en el cifrado. Es por eso que este trabajo propone la creación de un sistema criptográfico basado sobre una arquitectura dividida en dos etapas de confusión y difusión. Cada una de ellas utiliza una ecuación logística para generar números pseudoaleatorios que permitan desordenar la posición del píxel y cambiar su intensidad en la escala de grises. Este proceso iterativo es determinado por la cantidad total de píxeles de una imagen. Finalmente, toda la lógica de cifrado es ejecutada sobre la tecnología CUDA que permite el procesamiento en paralelo. Como aporte sustancial, se propone una nueva técnica de encriptación vanguardista de alta sensibilidad ante ruidos externos manteniendo no solo la confidencialidad de la imagen, sino también la disponibilidad y la eficiencia en los tiempos de proceso.---ABSTRACT---New trends to share multimedia files over open networks, demand the best use of encryption techniques to ensure the integrity, availability and confidentiality, keeping and/or improving the efficiency of the encryption process on these files. Today it is common to transfer pictures through technological networks, thus, it is necessary to update existing techniques encryption, and even better, the searching of new alternatives. Nowadays, classic cryptographic algorithms are highly known in the midst of the information society which not only causes greater vulnerability, but high processing times when this algorithms are used. It raise the probability of being deciphered and minimizes the immediate availability of resources. To reduce these odds, the use of chaos theory emerged as a good option to be applied on an algorithm that takes advantage of chaotic behavior of dynamic systems, and take logistic maps’ properties to raise the level of robustness in the encryption. That is why this paper proposes the creation of a cryptographic system based on an architecture divided into two stages: confusion and diffusion. Each stage uses a logistic equation to generate pseudorandom numbers that allow mess pixel position and change their intensity in grayscale. This iterative process is determined by the total number of pixels of an image. Finally, the entire encryption logic is executed on the CUDA technology that enables parallel processing. As a substantial contribution, it propose a new encryption technique with high sensitivity on external noise not only keeping the confidentiality of the image, but also the availability and efficiency in processing times.

Investigations on the Dynamics of Combination Maps and the Analysis of Bifurcation in a Discontinuous Logistic Map

This thesis is a study of discrete nonlinear systems represented by one dimensional mappings.As one dimensional interative maps represent Poincarre sections of higher dimensional flows,they offer a convenient means to understand the dynamical evolution of many physical systems.It highlighting the basic ideas of deterministic chaos.Qualitative and quantitative measures for the detection and characterization of chaos in nonlinear systems are discussed.Some simple mathematical models exhibiting chaos are presented.The bifurcation scenario and the possible routes to chaos are explained.It present the results of the numerical computational of the Lyapunov exponents (λ) of one dimensional maps.This thesis focuses on the results obtained by our investigations on combinations maps,scaling behaviour of the Lyapunov characteristic exponents of one dimensional maps and the nature of bifurcations in a discontinous logistic map.It gives a review of the major routes to chaos in dissipative systems,namely, Period-doubling ,Intermittency and Crises.This study gives a theoretical understanding of the route to chaos in discontinous systems.A detailed analysis of the dynamics of a discontinous logistic map is carried out, both analytically and numerically ,to understand the route it follows to chaos.The present analysis deals only with the case of the discontinuity parameter applied to the right half of the interval of mapping.A detailed analysis for the n –furcations of various periodicities can be made and a more general theory for the map with discontinuities applied at different positions can be on a similar footing

Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation

Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map.

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of an n-cycle (n > 1) approach the discontinuities of the nth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.

Chaos in modulated logistic systems

This paper is a review of the work done on the dynamics of modulated logistic systems. Three different problems are treated, viz, the modulated logistic map, the parametrically perturbed logistic map and the combination map obtained by combining two maps of the quadratic family. Many of the interesting features displayed by these systems are discussed.

Obtaining environmental favourability functions from logistic regression

Logistic regression is a statistical tool widely used for predicting species’ potential distributions starting from presence/absence data and a set of independent variables. However, logistic regression equations compute probability values based not only on the values of the predictor variables but also on the relative proportion of presences and absences in the dataset, which does not adequately describe the environmental favourability for or against species presence. A few strategies have been used to circumvent this, but they usually imply an alteration of the original data or the discarding of potentially valuable information. We propose a way to obtain from logistic regression an environmental favourability function whose results are not affected by an uneven proportion of presences and absences. We tested the method on the distribution of virtual species in an imaginary territory. The favourability models yielded similar values regardless of the variation in the presence/absence ratio. We also illustrate with the example of the Pyrenean desman’s (Galemys pyrenaicus) distribution in Spain. The favourability model yielded more realistic potential distribution maps than the logistic regression model. Favourability values can be regarded as the degree of membership of the fuzzy set of sites whose environmental conditions are favourable to the species, which enables applying the rules of fuzzy logic to distribution modelling. They also allow for direct comparisons between models for species with different presence/absence ratios in the study area. This makes themmore useful to estimate the conservation value of areas, to design ecological corridors, or to select appropriate areas for species reintroductions.