R. Gilmore, J.-M. Ginoux, T. Jones, C. Letellier & U. S. Freitas
Connecting curves for dynamical systems,
Journal of Physics A, 43, 255101, 2010. Online

Abstract: We introduce one-dimensional sets to help describe and constrain the integral curves of an n-dimensional dynamical system. These curves provide more information about the system than zero-dimensional sets (fixed points). In fact, these curves pass through the fixed points. Connecting curves are introduced using two different but equivalent definitions, one from dynamical systems theory, the other from differential geometry. We describe how to compute these curves and illustrate their properties by showing the connecting curves for a number of dynamical systems.

A. Cuvelier, L. Achour, H. Rabarimanantsoa, C. Letellier, J.-F. Muir & B. Fauroux
A noninvasive method to identify ineffective triggering in patients with noninvasive pressure support ventilation
Respiration, 80, 198-206, 2010. online
Abstract: Background: Ineffective inspiratory triggering efforts are a major cause of poor patient-ventilator interactions during mechanical ventilation, but their routine identification requires the insertion of an esophageal catheter.
Objectives: We developed a mathematical analysis of ventilatory tracings recorded under noninvasive pressure ventilation in order to identify ineffective triggering efforts and their consequences without recording esophageal pressure.
Methods: We assessed 2,183 cycles from 44 pressure support tracings in 14 children with cystic fibrosis treated by noninvasive home ventilation. Airway pressure, flow and esophageal pressure time series were visually analyzed and manually counted. Airway pressure versus time and flow versus time were then analyzed using a dedicated algorithm written by us. Esophageal pressure was only used for validation.
Results: A mathematical treatment of flow time series allowed us to draw phase portraits that had specific periodic trajectories for triggered ventilatory cycles and ineffective triggering efforts. From flow and pressure tracings, our algorithm correctly identified 100% of triggered cycles and 53/56 (94.6%) of ineffective triggering efforts. Ineffective triggering was associated with a significant reduction in minute ventilation, inspiratory flows and a significant increase in inspiratory efforts.
Conclusions: A noninvasive analysis of flow and airway pressure can reliably identify ineffective triggering efforts during noninvasive pressure support ventilation. This approach may be a valuable tool for evaluating patient-ventilator interactions and their consequences during long-term recordings.

C. Letellier & L. A. Aguirre,
Interplay between synchronization, observability, and dynamics,
Physical Review E, 82, 016204, 2010. Online
Synchronizing nonidentical chaotic oscillators is very often achieved by using various types of couplings. In the practice of synchronization the “right choice” of the coupling variable—y for the Rössler system, x for the Lorenz equations, and so on—is usually stated rather than explained or justified. Using the Rössler and Rucklidge system, in this paper, it is shown that such “optimal” choices are strongly related to observability properties which, in turn, can be quantified. In this paper it will also be shown that synchronizability does not only depend on the observability of the system but it is also a consequence of the dynamical regimes under study. The paper aims at providing important insight into the critical problem of making the “right choice” when it comes to choosing the coupling variable in synchronization schemes.

J.-M. Ginoux, C. Letellier & L. O. Chua
Topological analysis of chaotic solution to a three-element memristive circuit,
International Journal of Bifurcation & Chaos, 20 (11), 3819-3827, 2010. Online
The simplest electronic circuit with a memristor was recently proposed. Chaotic attractors solution to this memristive circuit are topologically characterized and compared to Rössler-like attractors.

C. Letellier & V. Messager
Influences on Otto E. Rössler’s earliest paper on chaos,
International Journal of Bifurcation & Chaos, 20 (11), 3585-3616, 2010. Online
Dedicated to Otto E. Rössler for his 70th birthday
Otto E. Rössler is well-known in “chaos theory” for having published one of the most often used benchmark systems producing chaotic attractors. His contribution is mostly reduced to this simple chaotic system published in 1976. Our aim is to show that a slightly earlier paper contains, in fact, much more and reveals a deep topological understanding of how chaotic attractors are organized in phase space. Moreover it is shown that Otto had three main influences: Andronov, Khaikin and Vitt’s textbook, the 1963 Lorenz paper and Li and Yorke’s theorem “period-three implies chaos”. In this paper, these three contributions are clearly identified as the main influences on Rössler’s earliest paper on chaos. The content of the latter is briefly compared to other works that appeared (or were available as reprints) before its own publication.

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