Synchronization Based Approach for Estimating All Model Parameters of Chaotic Systems

The problem of dynamic estimation of all parameters of a model representing chaotic and hyperchaotic systems using information from a scalar measured output is solved. The variational calculus based method is robust in the presence of noise, enables online estimation of the parameters and is also able to rapidly track changes in operating parameters of the experimental system. The method is demonstrated using the Lorenz, Rossler chaos and hyperchaos models. Its possible application in decoding communications using chaos is discussed.Comment: 13 pages, 4 figure

GPU accelerated investigation of a dual-frequency driven nonlinear oscillator

The bifurcation structure of a dual-frequency driven, second order nonlinear oscillator (Keller–Miksis equation) is investigated by exploiting the high computational resources of professional GPUs. The numerical scheme of the applied initial value problem solver was the explicit, adaptive Runge–Kutta–Cash–Karp method with embedded error estimation using solutions of order 4 and 5. The four dimensional parameter space (amplitudes and frequencies of the driving) is explored by means of several high resolution bi-parametric plots with the amplitudes as control parameters at fixed frequencies. The resolution of the control parameter plane is 500 × 500 with 10 initial conditions at each parameter pair (altogether 2.5 million initial value problems in each bi-parametric plot). The program code for one fine parameter scan runs approximately 50 times faster on a Tesla K20 GPU (Kepler architecture) than on an Intel i7-4790 4 core CPU even applying double precision floating point operations

An ensemble framework for time delay synchronisation

Synchronisation theory is based on a method that tries to synchronise a model with the true evolution of a system via the observations. In practice, an extra term is added to the model equations that hampers growth of instabilities transversal to the synchronisation manifold. Therefore, there is a very close connection between synchronisation and data assimilation. Recently, synchronisation with time delayed observations has been proposed, in which observations at future times are used to help synchronise a system that does not synchronise using only present observations, with remarkable successes. Unfortunately, these schemes are limited to small-dimensional problems. In this paper, we lift that restriction by proposing ensemble-based synchronisation scheme. Tests were performed using Lorenz96 model for 20, 100 and 1000-dimension systems. Results show global synchronisation errors stabilising at values of at least an order of magnitude lower than the observation errors, suggesting that the scheme is a promising tool to steer model states to the truth. While this framework is not a complete data assimilation method, we develop this methodology as a potential choice for a proposal density in a more comprehensive data assimilation method, like a fully nonlinear particle filter

A hybrid model for chaotic front dynamics: From semiconductors to water tanks

We present a general method for studying front propagation in nonlinear systems with a global constraint in the language of hybrid tank models. The method is illustrated in the case of semiconductor superlattices, where the dynamics of the electron accumulation and depletion fronts shows complex spatio-temporal patterns, including chaos. We show that this behavior may be elegantly explained by a tank model, for which analytical results on the emergence of chaos are available. In particular, for the case of three tanks the bifurcation scenario is characterized by a modified version of the one-dimensional iterated tent-map.Comment: 4 pages, 4 figure

Transition to Chaotic Phase Synchronization through Random Phase Jumps

Phase synchronization is shown to occur between opposite cells of a ring consisting of chaotic Lorenz oscillators coupled unidirectionally through driving. As the coupling strength is diminished, full phase synchronization cannot be achieved due to random generation of phase jumps. The brownian dynamics underlying this process is studied in terms of a stochastic diffusion model of a particle in a one-dimensional medium.Comment: Accepted for publication in IJBC, 10 pages, 5 jpg figure

Forecasting confined spatiotemporal chaos with genetic algorithms

A technique to forecast spatiotemporal time series is presented. it uses a Proper Ortogonal or Karhunen-Lo\`{e}ve Decomposition to encode large spatiotemporal data sets in a few time-series, and Genetic Algorithms to efficiently extract dynamical rules from the data. The method works very well for confined systems displaying spatiotemporal chaos, as exemplified here by forecasting the evolution of the onedimensional complex Ginzburg-Landau equation in a finite domain.Comment: 4 pages, 5 figure

Generalized synchronization of chaos in autonomous systems

We extend the concept of generalized synchronization of chaos, a phenomenon that occurs in driven dynamical systems, to the context of autonomous spatiotemporal systems. It means a situation where the chaotic state variables in an autonomous system can be synchronized to each other but not to a coupling function defined from them. The form of the coupling function is not crucial; it may not depend on all the state variables nor it needs to be active for all times for achieving generalized synchronization. The procedure is based on the analogy between a response map subject to an external drive acting with a probability p and an autonomous system of coupled maps where a global interaction between the maps takes place with this same probability. It is shown that, under some circumstances, the conditions for stability of generalized synchronized states are equivalent in both types of systems. Our results reveal the existence of similar minimal conditions for the emergence of generalized synchronization of chaos in driven and in autonomous spatiotemporal systems.Comment: 5 pages, 7 figures, accepted in PR

New way to achieve chaotic synchronization in spatially extended systems

We study the spatio-temporal behavior of simple coupled map lattices with periodic boundary conditions. The local dynamics is governed by two maps, namely, the sine circle map and the logistic map respectively. It is found that even though the spatial behavior is irregular for the regularly coupled (nearest neighbor coupling) system, the spatially synchronized (chaotic synchronization) as well as periodic solution may be obtained by the introduction of three long range couplings at the cost of three nearest neighbor couplings.Comment: 5 pages (revtex), 7 figures (eps, included

Various Approaches for Predicting Land Cover in Mountain Areas

Using former maps, geographers intend to study the evolution of the land cover in order to have a prospective approach on the future landscape; predictions of the future land cover, by the use of older maps and environmental variables, are usually done through the GIS (Geographic Information System). We propose here to confront this classical geographical approach with statistical approaches: a linear parametric model (polychotomous regression modeling) and a nonparametric one (multilayer perceptron). These methodologies have been tested on two real areas on which the land cover is known at various dates; this allows us to emphasize the benefit of these two statistical approaches compared to GIS and to discuss the way GIS could be improved by the use of statistical models.Comment: 14 pages; Classifications: Information Theory; Probability Theory & Applications; Statistical Computing; Statistical Theory & Method

Iterated maps for clarinet-like systems

The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.)