Inherent global stabilization of unstable local behavior in coupled map lattices

The behavior of two-dimensional coupled map lattices is studied with respect to the global stabilization of unstable local fixed points without external control. It is numerically shown under which circumstances such inherent global stabilization can be achieved for both synchronous and asynchronous updating. Two necessary conditions for inherent global stabilization are derived analytically.Comment: 17 pages, 10 figures, accepted for publication in Int.J.Bif.Chao

Stabilization of causally and non-causally coupled map lattices

Two-dimensional coupled map lattices have global stability properties that depend on the coupling between individual maps and their neighborhood. The action of the neighborhood on individual maps can be implemented in terms of "causal" coupling (to spatially distant past states) or "non-causal" coupling (to spatially distant simultaneous states). In this contribution we show that globally stable behavior of coupled map lattices is facilitated by causal coupling, thus indicating a surprising relationship between stability and causality. The influence of causal versus non-causal coupling for synchronous and asynchronous updating as a function of coupling strength and for different neighborhoods is analyzed in detail.Comment: 15 pages, 5 figures, accepted for publication in Physica

Stability analysis of coupled map lattices at locally unstable fixed points

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for exploring fundamental features of CMLs, such as their stability properties. Since CMLs can be considered as graphs, we apply methods of spectral graph theory to analyze their stability at locally unstable fixed points for different updating rules, different coupling scenarios, and different types of neighborhoods. Numerical studies are found to be in excellent agreement with our theoretical results.Comment: 22 pages, 6 figures, accepted for publication in European Physical Journal

On the critical nature of plastic flow: one and two dimensional models

Steady state plastic flows have been compared to developed turbulence because the two phenomena share the inherent complexity of particle trajectories, the scale free spatial patterns and the power law statistics of fluctuations. The origin of the apparently chaotic and at the same time highly correlated microscopic response in plasticity remains hidden behind conventional engineering models which are based on smooth fitting functions. To regain access to fluctuations, we study in this paper a minimal mesoscopic model whose goal is to elucidate the origin of scale free behavior in plasticity. We limit our description to fcc type crystals and leave out both temperature and rate effects. We provide simple illustrations of the fact that complexity in rate independent athermal plastic flows is due to marginal stability of the underlying elastic system. Our conclusions are based on a reduction of an over-damped visco-elasticity problem for a system with a rugged elastic energy landscape to an integer valued automaton. We start with an overdamped one dimensional model and show that it reproduces the main macroscopic phenomenology of rate independent plastic behavior but falls short of generating self similar structure of fluctuations. We then provide evidence that a two dimensional model is already adequate for describing power law statistics of avalanches and fractal character of dislocation patterning. In addition to capturing experimentally measured critical exponents, the proposed minimal model shows finite size scaling collapse and generates realistic shape functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science for the special issue in honor of Victor Berdichevsky, 201

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

Smart control of light in edge-emitting lasers

Tesi en modalitat de compendi de publicacions, amb diferents seccions retallades per drets dels editorsThe invention of the laser triggered the study of light-matter interactions. In turn, the advent of artificial structured materials on micro- and nanometer scales has become a fruitful playground to tailor the propagation and generation of light, even in exotic or counterintuitive ways, uncovering novel physical phenomena. In this thesis, we precisely propose using recently discovered properties of artificial photonic materials, and new schemes, to control the spatiotemporal dynamics of broad area semiconductor lasers and improve their performance. Semiconductor lasers are replacing other laser sources due to their efficiency, compactness and affordable prices, however suffering from a major drawback. The quality of the emitted beam intrinsically deteriorates when power increases, if the aperture of the laser is very broad as compared to wavelength. The highly multimode and unstable emission limits possible applications of these lasers. Although different mechanisms have been proposed to save this obstacle, obtaining a stable and bright emission remains a longstanding open question. This thesis aims at contributing to this goal without compromising their compact design, and to the new field of non-Hermitian Photonics providing new insights into the control of wave dynamics in artificial complex media. Indeed, the physics of open-dissipative, non-Hermitian systems offers new possibilities to utilize the gain and loss for steering optical processes, and is beyond the recent focus on non-Hermitian Photonics. As initially demonstrated in the frame of Quantum Mechanics, systems with gain and losses may still present real eigenvalues of the Hamiltonian (energy) as the purely conservative ones, yet holding other unexpected physical behaviors, derived from an asymmetric coupling between modes. In particular, this was first observed in systems invariant under parity (P-) and time (T-) symmetry ¿ referred as PT-symmetric ¿. Optical systems with complex permittivity are flexible and achievable classical analogs of such quantum systems to realize and explore these effects. As a first step, we propose to use a chirped modulation of the refractive index (chirped photonic crystal) for intracavity filtering the multimode emission of EELs. To numerically assess the filtering performance, we developed a full (2+1)-dimensional spatio-temporal model, including both transverse and longitudinal dimensions plus time, for the evolution of the electric field and carriers. The good agreement between predictions with actual experimental results demonstrates the proposal while validating the model which is used throughout the thesis, with corresponding modifications. We then analyze the effect of intrinsically imposing in phase refractive index and gain modulations within the semiconductor laser, and use the interplay between real and imaginary parts of the non-Hermitic potential to achieve spatial and temporal stabilization. Taking one step further, we propose to divide the EEL cavity into two mirror-symmetric half-spaces, both holding PT-symmetry but with opposite mode coupling. With this geometry, we expect to obtain a two-fold benefit: on the one hand, achieving a spatial-temporal stabilization of the laser, and on the other, localizing the generated field along the symmetry axis. We numerically demonstrate regimes of simultaneous localization and stabilization leading to an enhanced output and improved beam quality. Finally, while thinner lasers show a more stable new temporal and synchronization instabilities arise in EELS arrays (bars) from the coupling between neighboring lasers, leading again to irregular spatiotemporal behaviors. We show that the proposed mirror symmetric non-Hermitian configuration may be extended to couple individual EELs in the array, by a lateral shift between the pump and index profiles. In all cases, the obtained localized and stable output beam may facilitate a direct coupling of the emitted beam to optical fibers.La invenció del làser va representar el tret de sortida per nombrosos estudis de la interacció entre la llum i la matèria. A banda, el desenvolupament de nous materials fotònics artificials en escales micro i nanomètriques ha esdevingut un camp fructífer per al control de la propagació i generació de la llum, fins i tot de maneres exòtiques o contra intuïtives, revelant nous fenòmens físics. En aquesta tesi proposem, precisament, utilitzar nous materials fotònics artificials i nous esquemes per controlar la dinàmica espai-temporal dels làsers de semiconductor d'apertura ampla, per millorar-ne les propietats. Els làsers de semiconductor estan reemplaçant altres fonts de llum làser gràcies a la seva eficiència, format compacte i preu assequible. Malgrat tot, pateixen un gran inconvenient: el deteriorament del feix emès en augmentar la potència, especialment si l'amplada del làser és molt gran respecte la longitud d'ona. Quan l'emissió esdevé altament multimode i inestable en limita les possibles aplicacions. Encara que s'han proposat diferents mecanismes per superar aquest problema, aconseguir una emissió estable sense comprometre'n el format compacte, és encara una qüestió oberta. Aquesta tesi té com a objectiu contribuir a la millora dels làsers de semiconductor i a l'estudi del control de la dinàmica dels làsers mitjançant el nou camp de la fotònica no hermítica. De fet, la física dels sistemes oberts no Hermítics ofereix noves possibilitats per utilitzar la permitivitat complexa per dominar processos òptics i és la causa del recent interès en la fotònica no hermitiana. Primer, es va demostrar en el marc de la Mecànica Quàntica que els sistemes oberts o no Hermítics, tot i tenir guanys o pèrdues poden presentar autovalors reals del Hamiltonià (valors constants de l'energia) i altres comportaments físics inesperats, derivats d'acoblaments asimètrics entre modes. Efecte observat inicialment en sistemes invariants sota la paritat (P-) i la simetria de temps (T-), anomenats PT-simètrics. Els sistemes òptics amb permitivitat complexa són anàlegs clàssics, flexibles i assequibles d’aquests sistemes quàntics per realitzar i explorar aquests nous efectes. Primer, proposem fer servir una modulació de l’índex de refracció per al filtrat, intacavitat, de l'emissió multimode d'amplificadors i làsers de semiconductor. Per l'anàlisi numèrica desenvolupem un model espai-temporal complet, que inclou dues dimensions espacials, transversal i longitudinal, més l'evolució temporal del camp elèctric i dels portadors. Aquest model s'utilitza al llarg de tota la tesis amb les modificacions corresponents i és contrastat també experimentalment. A continuació, analitzem l'efecte d'imposar modulacions intrínseques de l’índex de refracció i el guany, en fase i, dins del làser de semiconductor. Gràcies a la interacció entre parts reals i imaginàries del potencial no hermític s’aconsegueix una estabilització espacial i temporal. Fent un pas més, dividim la cavitat làser en dos espais PT-simètrics (simetria de mirall) amb un acoblament en sentit oposat. Amb aquesta geometria, esperem obtenir un doble benefici: d'una banda, aconseguir una estabilització espacial-temporal del làser, i per una altra banda, localitzar el camp generat en l'eix de simetria. Es demostra numèricament règims de localització i d'estabilització simultànies, augmentant la potència emesa tot millorant la qualitat dels feix. Finalment, tot i que els làsers més estrets mostren una emissió més estable, els làsers propers s’acoblen quan formen part d'una matriu. Demostrem que l'acoblament asimètric també pot ser utilitzat en barres de làsers de semiconductor per estabilitzar-los temporalment i concentrar-ne l’emissió. L'acoblament asimètric es produeix mitjançant un desplaçament lateral entre el bombeig i l'índex de refracció. En tots els casos, el feix de sortida localitzat i estable obtingut pot facilitar un acoblament directe del feix emès a les fibres òptiques.Postprint (published version

The mathematics behind chimera states

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed