Symbolic sequence analysis using approximated partition,
Chaos, Solitons & Fractals, 36, 32-41, 2008. Online
Abstract: Particular attention is paid to characterizing the dynamical behavior using symbolic sequences when there is no topological criterion to define the partition. In that case, a so-called threshold crossings technique is used. It is shown that maximizing the number of realized sequences provides a partition closer to the topological partition than using a partition leading to equiprobable symbols. After numerical evidences on benchmark maps, an experimental time series from a copper electrodissolution is used to check the applicability of this technique.
C. Letellier, H. Rabarimanantsoa, L. Achour, A. Cuvelier & J.-F. Muir,
Recurrence plots for dynamical analysis of non invasive mechanical ventilation,
Philosophical Transactions of the Royal Society of London A, 366, 621-634, 2008. Online
Abstract: Quantifiers were introduced to convert recurrence plots into a statistical analysis of dynamical properties. It is shown that the Shannon entropy, if properly computed, increases as the chaotic regime is developed as expected. Recurrence plots and a new estimator for the Shannon entropy are then used to identify asynchronisms in non invasive mechanical ventilation. It is thus shown that the phase coherence - easily identified using a Shannon entropy - is relevant in the quality of the mechanical ventilation. In particular, some patients with chronic respiratory diseases or healthy subjects can have a high rate of asynchronisms but a regular breathing rhythm.
L. A. Aguirre, S. B. Bastos, M. A. Avles & C. Letellier,
Observability of nonlinear dynamics: normalized indices and a time series approach,
Chaos, 18, 013123, 2008. Online
Abstract: This paper investigates the observability of nonlinear dynamical systems. Two difficulties associated with previous studies are dealt with. First, normalized observability indices are defined. This permits the comparison of different systems, which was not generally possible before. Second, a time series approach is proposed based on omni-directional nonlinear correlation functions to rank a set of time series of a system in terms of observability without requiring the knowledge of the system equations. The two indices proposed in this paper and a former definition of observability were applied to five benchmark systems and an overall agreement of over 92% was found.
C. Letellier, V. Messager & R. Gilmore,
From quasi-periodicity to toroidal chaos: analogy between the Curry-Yorke map and the van der Pol system,
Physical Review E, 77 (4), 046203, 2008. Online
Abstract: The van der Pol attractor exhibits a wide variety of behavior depending on the control parameter values: limit cycles, quasiperiodic motion on a torus, mode locking, period-doubling, banded chaos, boundary crises, torus wrinkling, breakup of a torus, toroidal chaos. The organization of these phenomena with respect to each other is well described by studying a partition of the control parameter plane of the Curry-Yorke map.
H. Rabarimanantsoa, L. Achour, C. Letellier, J.-F. Muir & A. Cuvelier,
Objective evaluation of patient–ventilator interactions during noninvasive ventilation (NIV),
European Respiratory Review, 17, 22-23, 2008. Online
Abstract: The success of NIV depends on patient–ventilator interactions. These interactions are evaluated with the subjective comfort score which is not always reliable. An objective evaluation is thus required. To evaluate these interactions, we use a statistical measure of the variability of a physiological signal, i.e the Shannon entropy. Our purpose is to show whether estimating Shannon entropy from airway pressure (SP) and from the total duration of ventilatory cycles (ST) may evaluate objectively the patient–ventilator interactions during NIV.
Pressure support NIV was applied to 4 COPD patients, 4 OHS patients in stable state and 4 healthy subjects during six successive 10-min periods with various inspiratory pressure. The flow and the airway pressure signals were recorded with sensors located near the mask. Good patient–ventilator interactions were assumed to correspond to patient well synchronized (low ineffective efforts) and with low ventilatory variability. All the subjects were awaked and both Shannon entropies SP and ST were computed for each ventilatory tracing.
L. A. Aguirre, C. Letellier & J. Maquet,
Forecasting the Time Series of Sunspot Numbers,
Solar Physics, 249 (1), 103-120, 2008. Online
Abstract: Forecasting the solar cycle is of great importance for weather prediction and environmental monitoring, and also constitutes a difficult scientific benchmark in nonlinear dynamical modeling. This paper describes the identification of a model and its use in the forecasting the time series comprised of Wolf’s sunspot numbers. A key feature of this procedure is that the original time series is first transformed into a symmetrical space where the dynamics of the solar dynamo are unfolded in a better way, thus improving the model. The nonlinear model obtained is parsimonious and has both deterministic and stochastic parts. Monte Carlo simulation of the whole model produces very consistent results with the deterministic part of the model but allows for the determination of confidence bands. The obtained model was used to predict cycles 24 and 25, although the forecast of the latter is seen as a crude approximation, given the long prediction horizon required. As for the 24th cycle, two estimates were obtained with peaks of 65±16 and of 87±13 units of sunspot numbers. The simulated results suggest that the 24th cycle will be shorter and less active than the preceding one.
D. José de Oliveira, C. Letellier, M. E. D. Gomes & L. A. Aguirre,
The use of synthetic input sequences in time series modelling,
Physics Letters A, 372, 5276-5282, 2008. Online
Abstract: In many situations time series models obtained from noise-like data settle to trivial solutions under iteration. This Letter proposes a way of producing a synthetic (dummy) input, that is included to prevent the model from settling down to a trivial solution, while maintaining features of the original signal. Simulated benchmark models and a real time series of RR intervals from an ECG are used to illustrate the procedure.
C. Letellier, I. Moroz & R. Gilmore,
A comparison of test for embeddings,
Physical Review E, 78, 026203, 2008. Online
Abstract: It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a newly proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions.
D. Amroun, H. Leblond, M. Brunel, C. Letellier & F. Sanchez,
Stabilization of space-time laser instability through the finite transverse extension of pumping,
Journal of Applied Optics A, 10 (9), 095101, 2008. Online
Abstract: We investigate the space–time dynamics of a homogeneously broadened single-mode laser when diffraction is taken into account. It is well known that such a laser displays instability when pumping reaches the second laser threshold. We show that the laser dynamics can be stabilized by pumping in a domain of finite width. The analysis of stationary solutions to the Maxwell–Bloch equations (evanescent waves, travelling waves, localized solutions) allows the stabilization mechanism to be explained.
F. Lusseyran, L. Pastur & C. Letellier,
Dynamical analysis of an intermittency in an open cavity flow,
Physics of Fluids, 20, 114101, 2008. Online
Abstract: When open flows pass an open cavity, it is known that for medium or large Reynolds numbers, self-sustained oscillations generally appear. When more than one mode is excited, some nonlinear competition between modes may occur. In the configuration investigated here, the underlying dynamics are mainly driven by two dominant modes. The interplay between these two modes is investigated using phase portraits, Poincaré sections, and return maps. The toroidal structure of the phase portrait is then investigated using a symbolic dynamics built from an angular return map. Each symbol can be associated with a specific mode and the interplay described in terms of symbolic sequences, leading to exhibit a switching mode process.