R. Gilmore, C. Letellier & N. Romanazzi,
Global topology from an embedding,
Journal of Physics A, 40, 13291-13297, 2007. Online
Abstract: An embedding of chaotic data into a suitable phase space creates a diffeomorphism of the original attractor with the reconstructed attractor. Although diffeomorphic, the original and reconstructed attractors may not be topologically equivalent. In a previous work we showed how the original and reconstructed attractors can differ when the original is three-dimensional and of genus-one type. In the present work we extend this result to three-dimensional attractors of arbitrary genus. This result describes symmetries exhibited by the Lorenz attractor and its reconstructions.
H. Rabarimanantsoa, L. Achour, C. Letellier, A. Cuvelier & J.-F. Muir
Recurrence plots and Shannon entropy for a dynamical analysis of asynchronisms in mechanical non-invasive ventilation,
Chaos, 17, 013115, 2007. Online
Abstract: Recurrence plots were introduced to quantify the recurrence properties of chaotic dynamics. Hereafter, the recurrence quantification analysis was introduced to transform graphical interpretations into statistical analysis. In this spirit, a new definition for the Shannon entropy was recently introduced in order to have a measure correlated with the largest Lyapunov exponent. Recurrence plots and this Shannon entropy are thus used for the analysis of the dynamics underlying patient assisted with a mechanical non invasive ventilation. The quality of the assistance strongly depends on the quality of the interactions between the patient and his ventilator which are crucial for tolerance and acceptability. Recurrence plots provide a global view of these interactions and the Shannon entropy is shown to be a measure of the rate of asynchronisms as well as the breathing rhythm.
C. Letellier, R. Gilmore & T. Jones
Peeling bifurcation of toroidal chaotic attractors,
Physical Review E, 76, 066204, 2007. Online
Abstract: Chaotic attractors with toroidal topology (van der Pol attractor) have counterparts with symmetry that exhibit unfamiliar phenomena. We investigate double covers of toroidal attractors, discuss changes in their morphology under correlated peeling bifurcations, describe their topological structures and the changes undergone as a symmetry axis crosses the original attractor, and indicate how the symbol name of a trajectory in the original lifts to one in the cover. Covering orbits are described using a powerful synthesis of kneading theory with refinements of the circle map. These methods are applied to a simple version of the van der Pol oscillator.
L. Achour, C. Letellier, A. Cuvelier, E. Vérin & J.-F. Muir
Asynchrony and cyclic variability in pressure support noninvasive ventilation,
Computer in Biology and Medicine, 37, 1308-1320, 2007. Online
Abstract: Noninvasive mechanical ventilation is an effective procedure to manage patients with acute or chronic respiratory failure. Most ventilators act as flow generators that assist spontaneous respiratory cycles by delivering inspiratory and expiratory pressures. This allows the patient to improve alveolar ventilation and subsequent pulmonary gas exchanges. The interaction between the patient and its ventilator are therefore crucial for tolerance and acceptability and part of this interaction is the facility to trigger the ventilator at the beginning of the inspiration. This is directly related to patients’ discomfort which is not quantified today. Phase portraits reconstructed from the airflow and first-return maps builton the total breath duration were used to investigate the quality of the patient-ventilator interaction. Phase synchronization can be identified from phase portrait and the breath-to-breath variability is well characterized by return maps. This paper is a first step in the direction of automatically estimating the comfort from measurements and not from a necessarily subjective answer given by the patient. These tools could be helpful for the physicians to set the ventilator parameters.
I. Moroz, C. Letellier & R. Gilmore
When are projections also embeddings ?
Physical Review E, 75, 046201, 2007. Online
Abstract: We study an autonomous four-dimensional dynamical system used to model certain geophysical processes. This system generates a chaotic attractor that is strongly contracting, with four Lyapunov exponents that satisfy , so the Lyapunov dimension is in the range of coupling parameter values studied. As a result, it should be possible to find three-dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior. We study mappings into to determine which can be used as embeddings to reconstruct the dynamics. We find dramatically different behavior in the two simplest mappings: projections from to . In one case the one-parameter family of attractors studied remains topologically unchanged for all coupling parameter values. In the other case, during an intermediate range of parameter values the projection undergoes self-intersections, while the embedded attractors at the two ends of this range are topologically mirror images of each other.
C. Letellier & R. Gilmore,
Symmetry groups for 3D dynamical systems
Journal of Physics A, 40 (21), 5597-5620, 2007. Online
Abstract: We present a systematic way to construct dynamical systems with a specific symmetry group . Each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level. Image attractors can be lifted to many inequivalent covering attractors. These are distinguished by an index that has related topological, algebraic, and group theoretical interpretations. These methods are used to describe dynamical systems with symmetry groups , , and .
C. Letellier, G. Amaral & L. A. Aguirre,
Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models
Chaos, 17, 023104, 2007. Online
Abstract: This paper proposes a procedure by which it is possible to synthesize Rössler and Lorenz dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as co-dimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.
J. Maquet, C. Letellier & L. A. Aguirre,
Global models from the Canadian Lynx cycles as a first evidence for chaos in real ecosystems,
Journal of Mathematical Biology, 55 (1), 21-39, 2007. Online
Abstract: Real food chains are very rarely investigated since long data sequences is required. Typically, if we consider that an ecosystem evolves with a period corresponding to the time for maturation, possessing few dozen of cycles would require to count species over few centuries. One well known example of a long data set is the number of Canadian lynx furs caught by the Hudson Bay company between 1821 and 1935 as reported by Elton and Nicholson in 1942. In spite of the relative quality of the data set (10 undersampled cycles), two low-dimensional global models that settle to chaotic attractors were obtained. They are compared with an ad hoc 3D model which was proposed as a possible model for this data set. The two global models, which were estimated with no prior knowledge about the dynamics, can be considered as direct evidences of chaos in real ecosystems.
C. Letellier, M. Bennoud & G. Martel,
Intermittency and period-doubling cascade on tori in a bimode laser model
Chaos, Solitons & Fractals, 33, 782-794, 2007. Online
Abstract: Since the three types of intermittency have been theoretically described, many experimental observations of such regimes have been reported. Chaotic behaviors occuring after torus breakdowns and quasi-periodic regimes are also very often observed. It is not so surprising that intermittencies on tori were never reported as soon as it is understood that these common characteristic of intermittencies should be investigated in a Poincaré section of a Poincaré section, that is, in a set which is not possible to define. A specific approach is therefore required to identify them as shown in the paper with two examples of type-I intermittency on tori solution to two different systems.
C. Letellier, E. M. Mendes & R. E. Mickens
Nonstandard discretization scheme applied to the conservative Hénon-Heiles system
International Journal of Bifurcations and Chaos, 17 (3), 891-902, 2007. Online
Abstract: The discretization of ordinary differential equations is investigated for the case of the conservative Hénon–Heiles system. Starting from a discrete Hamiltonian function, which is invariant under time reversal, discrete equations of motion are analytically obtained using three different discretization schemes recently proposed and investigated in the literature. In the case where the discretization scheme successfully provide discrete systems in which the trace of the Jacobian matrix corresponding to the property required by a conservative system is preserved, it is shown that they are not necessarily invariant to time reversal. Such models are however quite robust when the time step is increased. For the schemes where the trace of Jacobian matrix does not match the condition required by conservative systems, it is shown that energy conservation is not achieved and the original dynamics is lost. Steps toward the solution to this problem are given.