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Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime.

We analyse the dynamics of a network of semiconductor lasers coupled via their mean intensity through a non-linear optoelectronic feedback loop. We establish experimentally the excitable character of a single node, which stems from the slow-fast nature of the system, adequately described by a set of rate equations with three well separated time scales. Beyond the excitable regime, the system undergoes relaxation oscillations where the nodes display canard dynamics. We show numerically that, without noise, the coupled system follows an intricate canard trajectory, with the nodes switching on one by one. While incorporating noise leads to a better correspondence between numerical simulations and experimental data, it also has an unexpected ordering effect on the canard orbit, causing the nodes to switch on closer together in time. We find that the dispersion of the trajectories of the network nodes in phase space is minimized for a non-zero noise strength, and call this phenomenon canard resonance.

Canard-induced phenomena have been extensively studied in the last three decades, both from the mathematical and from the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for Mixed-Mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, in particular in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analysing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show on an example how to construct a (linear) global return and obtain robust PWL MMOs., to appear in SIAM Review (2016)

Die Analysis qualitativer Eigenschaften von nichtlinearen partiellen Differentialgleichungen (PDEs) durch Methoden aus der Theorie dynamischer Systeme ist ein aktives Forschungsgebiet. Ein wichtiges Thema dabei ist die Analyse der Existenz, Stabilit��t und Verzweigung von speziellen L��sungen, die wesentliche Merkmale der zu untersuchenden PDE enthalten. Insbesondere f��r PDEs mit singul��ren St��rungen oder Singularit��ten hat sich dabei eine Kombination aus Methoden aus der Theorie dynamischer Systeme, Methoden der singul��ren St��rungstheorie und numerischen Berechnungen als sehr effektiv erwiesen. Dieses Projekt befasst sich mit zwei Problemen dieser Art, die neuartige Multiskalenmerkmale aufweisen. Im ersten Problem untersuchen wir die Euler-Lagrange-Gleichung der Regularisierung eines nichtkonvexen Variationsproblems, das als einfaches mathematisches Modell f��r Mikrostrukturen in Formged��chtnislegierungen auftritt. F��r dieses singul��r gest��rte Hamiltonsche System beweisen wir die Existenz einer Klasse von periodischen L��sungen und untersuchen ihre Abh��ngigkeit von den wesentlichen Parametern durch asymptotische Methoden und numerische Fortsetzung. Das Ziel ist ein besseres Verst��ndnis der Struktur von minimierenden L��sungen und ihres ungew��hnlichen Skalierungsverhaltens. Mittels numerischer Pfadverfolgung werden zus��tzlich neue Typen von L��sungen gefunden. Das zweite Problem betrifft die Asymptotik und Verzweigung von station��ren L��sungen eines Modells f��r mikroelektromechanische Systeme (MEMS). Dieses Modell wurde k��rzlich als Regularisierung eines einfacheren Modells vorgeschlagen, von dem bekannt ist, dass es in endlicher Zeit Singularit��ten entwickelt. F��r dieses Problem wird das numerisch berechnete Verzweigungsdiagramm erkl��rt, indem die Interaktion des Regularisierungsterms mit der f��r das Touchdown-Ph��nomen verantwortlichen Singularit��t im Detail untersucht wird. Dabei wird die Blow-up-Methode verwendet, um die Dynamik in der N��he der Singularit��t zu analysieren und rigorose Ergebnisse zu erhalten, welche die bereits existierende formale Asymptotik und Numerik beweisen und erg��nzen. Ein zentraler aspekt dabei ist die Untersuchung einer speziellen Sattel-Knoten Verzweigung, deren numerische Untersuchung aufgrund ihres singul��ren Charakters sehr schwierig ist. Diese neuartige Erweiterung der Blow-up-Methode zur Analyse von Randwertproblemen und singul��ren Grenzenwerten in Verzweigungsproblemen hat das Potential auch in anderen Zusammenh��ngen von Nutzen zu sein.

International audience; Minimum time control of slow-fast systems is considered. In the case of only one fast angle, averaging techniques are available for such systems. The approach introduced in Dargent (2014) and Bombrun et al. (2013) is recalled, then extended to time-dependent systems by means of a suitable filtering operator. The process relies upon approximating the dynamics by means of sliding windows. The size of these windows is an additional parameter that provides intermediate approximations between averaging over the whole fast angle period and the original dynamics. The motivation is that averaging over an entire period may not provide a good enough approximation to initialize a convergent numerical resolution of the original system; considering a continuous set of intermediate approximations (filtering over windows of size varying from the period to zero) may ensure that convergence. The method is illustrated on problems coming from space mechanics.

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds.

Probability and its Applications; 57 illus., Hardcover; ISBN: 1-84628-038-9; International audience; Stochastic differential equations play an increasingly important role in modeling the dynamics of a large variety of systems in the natural sciences, and in technological applications. This book is aimed at advanced undergraduate and graduate students, and researchers in mathematics, physics, the natural sciences, and engineering. It presents a new constructive approach to the quantitative description of solutions to systems of stochastic differential equations evolving on well-separated timescales. The method, which combines techniques from stochastic analysis and singular perturbation theory, allows the domains of concentration for typical sample paths to be determined, and provides precise estimates on the transition probabilities between these domains. In addition to the detailed presentation of the set-up and mathematical results, applications to problems in physics, biology, and climatology are discussed. The emphasis lies on noise-induced phenomena such as stochastic resonance, hysteresis, excitability, and the reduction of bifurcation delay.

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow–fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow–fast spatial dynamical system.