Springer Proceedings in Physics

Volume 173 More information about this series at http://www.springer.com/series/361 Mustapha Tlidi • Marcel G. Clerc Editors

Nonlinear Dynamics: Materials, Theory and Experiments Selected Lectures, 3rd Dynamics Days South America, Valparaiso 3–7 November 2014

123 Editors Mustapha Tlidi Marcel G. Clerc Physique des systèmes dynamiques Facultad de Ciencias Físicas y Matemáticas Université Libre de Bruxelles Universidad de Chile Brussels Santiago Belgium Chile

ISSN 0930-8989 ISSN 1867-4941 (electronic) Springer Proceedings in Physics ISBN 978-3-319-24869-1 ISBN 978-3-319-24871-4 (eBook) DOI 10.1007/978-3-319-24871-4

Library of Congress Control Number: 2015951395

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Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Preface

Nonlinear dynamics in out equilibrium system has been appealing a great deal of attention for several decades due to their tremendous influence in many physical, chemical, biological, economical, ecological, and social processes. Many efforts in the past have focused on the understanding of complex behavior of low-dimensional system, characterization of primary, spatial, or temporal bifurca- tions, which generate dissipative structures that can be either periodic or localized in space or/and in time. In two-dimensional setting, regular patterns like stripes, hexagons, squares, and super lattices, and the understanding of the macroscopic particle type solutions. Prigogine and Lefever to describe the spontaneous appearance of periodic structures as a result of irreversible processes maintained by exchanges of matter and/or energy with a non-equilibrium environment have introduced the term “dissipative structures.” Dissipative structures generally evolve on macroscopic scales and can only be maintained by continuous application of a non-equilibrium constraint. The spontaneous emergence of dissipative structures arises from a principle of self-organization that can be either in space or in time. A classic example of spatial self-organization is provided in the context of chemical reaction diffusion system referred to as Turing instability. This instability is char- acterized by a symmetry breaking that leads spontaneously to the formation of periodic structures with an intrinsic wavelength. In this case the wavelength is determined solely by the dynamical parameters such as diffusion coefficient and the inverse characteristic time associated with chemical kinetics. However, because of theoretical and numerical advances, and experimental observations are still offering many fundamental questions, which present the challenge of the nonlinear community. Such as characterization and emergence of complex spatiotemporal behavior, self-organization of life matter and nanoscale structure, and geometric characterization of localized structures. To make progress in this problematic, it is necessary to develop new nonlinear concepts, methods, and instruments. This book contains the selected lectures given at the third Dynamics Days South America 2014, which took place at Valparaiso November 3–7, 2014. Since its beginning, a key goal of Dynamics Days South America conference has been to

v vi Preface promote the participation and cross-discipline interaction of young scientists together with more established experts from a wide variety of fields. The lectures were devoted to various aspects of pattern formation, coarsening dynamics, driven systems, front propagation, stochastic process, nonlinear Anderson localization, interface dynamics, defects dynamics, phase transition and nonlinear dynamics in magnetic, fluid, optical, mechanical, and social systems. We want to thank, here, our sponsors and supporters whose interest and help were essential for the success of the meeting. However, the physical realization of the workshop would have not been possible without the decisive collaboration of the Universidad de Chile, Pontificia Universidad Católica de Valparaiso and Center for mathematical modeling.

Brussels, Belgium Mustapha Tlidi Santiago, Chile Marcel G. Clerc Contents

Part I Optics Visualisation of the Intensity and Phase Dynamics of Semiconductor Lasers via Electric Field Reconstructions ...... 3 David Goulding, Thomas Butler, Bryan Kelleher, Svetlana Slepneva, Stephen P. Hegarty and Guillaume Huyet Coarsening Dynamics of Umbilical Defects in Inhomogeneous Medium ...... 31 Raouf Barboza, Umberto Bortolozzo, Marcel G. Clerc, Stefania Residori and Valeska Zambra Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves ...... 45 Sergej Flach Nonlinear Dynamics of Vertical-Cavity Surface-Emitting Lasers: Deterministic Chaos and Random Number Generation ...... 59 Martin Virte, Marc Sciamanna, Hugo Thienpont and Krassimir Panajotov Experimental Observation of Front Propagation in Lugiato-Lefever Equation in a Negative Diffractive Regime and Inhomogeneous Kerr Cavity ...... 71 V. Odent, M. Tlidi, M.G. Clerc and E. Louvergneaux Splitting, Hatching and Transformation of the Repetition Rate in a Mode-Locked Laser ...... 87 A.V. Kovalev and E.A. Viktorov Spontaneous Symmetry Breaking in Nonlinear Systems: An Overview and a Simple Model ...... 97 Boris A. Malomed

vii viii Contents

Experimental Spatiotemporal Chaotic Textures in a Liquid Crystal Light Valve with Optical Feedback ...... 113 Marcel G. Clerc, Gregorio González-Cortés and Mario Wilson Chiral Modes in 2D PT-Symmetric Nanostructures ...... 125 M. Botey, R. Herrero, M. Turduev, I. Giden, H. Kurt and K. Staliunas Stabilization of Broad Area Semiconductor Amplifiers by Spatially Modulated Potentials...... 139 S. Kumar, W.W. Ahmed, R. Herrero, M. Botey, M. Radziunas and K. Staliunas Weakly Nonlinear Analysis and Localized Structures in Nonlinear Cavities with Metamaterials...... 153 N. Slimani, A. Makhoute and M. Tlidi Secondary Instabilities and Chaos in Optical Ring Cavities...... 167 Z. Liu, F. Leo, S. Coulibaly and M. Taki

Part II Mechanics, Fluids and Magnetics Hidden High Period Accelerator Modes in a Bouncer Model...... 179 Tiago Kroetz, André L.P. Livorati, Edson D. Leonel and Iberê L. Caldas How Dissipative Highly Non-linear Solitary Waves Interact with Boundaries in a 1D Granular Medium ...... 193 Lautaro Vergara Inviscid Transient Growth on Horizontal Shear Layers with Strong Vertical Stratification ...... 199 Cristóbal Arratia, Sabine Ortiz and Jean-Marc Chomaz Numerical Solution of a Novel Biofilm Growth Model...... 207 Patricio Cumsille, Juan A. Asenjo and Carlos Conca Dynamics of a One-Dimensional Kink in an Air-Fluidized Shallow Granular Layer ...... 223 J.E. Macías, M.G. Clerc and C. Falcón Measurements of Surface Deformation in Highly-Reflecting Liquid-Metals ...... 233 Pablo Gutiérrez, Vincent Padilla and Sébastien Aumaître Parametric Phenomena in Magnetic Nanostripes ...... 247 Alejandro O. León Hyper-Chaotic and Chaotic Synchronisation of Two Interacting Dipoles ...... 261 D. Urzagasti, D. Becerra-Alonso, L.M. Pérez, H.L. Mancini and D. Laroze Contents ix

Part III Robust Phenomena Finger Dynamics in Pattern Forming Systems ...... 275 Ignacio Bordeu, Marcel G. Clerc, René Lefever and Mustapha Tlidi Random Walk Model for Kink-Antikink Annihilation in a Fluctuating Environment ...... 293 Daniel Escaff Time-Periodic Forcing of Spatially Localized Structures ...... 303 Punit Gandhi, Cédric Beaume and Edgar Knobloch Formation and Interaction of Two-Kink Solitons ...... 317 Mónica A. García-Ñustes Around the Ising Model ...... 329 Fernando Mora, Felipe Urbina, Vasco Cortez and Sergio Rica Nonequilibrium Trade-Investment Model for the Ranked Distribution of Gross Domestic Products Per-capita ...... 347 R. Bustos-Guajardo and Cristian F. Moukarzel

Index ...... 359 Contributors

W.W. Ahmed Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica de Catalunya (UPC), Terrassa, Barcelona, Spain Cristóbal Arratia Departamento de Física FCFM, Universidad de Chile, Santiago, Chile; LadHyX, CNRS-École Polytechnique, Palaiseau, France Juan A. Asenjo Centre for Biochemical Engineering and Biotechnology, Centre for Biotechnology and Bioengineering, University of Chile, Santiago, Chile Sébastien Aumaître Service de Physique de l’Etat Condensé, DSM, CEA-Saclay, Gif-sur-Yvette, France; Laboratoire de Physique, ENS de Lyon, Lyon, France Raouf Barboza Departamento de Física, Facultad de Ciencias Físicas Y Matemáticas, Universidad de Chile, Santiago, Chile Cédric Beaume Department of Aeronautics, Imperial College London, London, UK D. Becerra-Alonso Universidad Loyola Andalucía, Córdoba, Spain Ignacio Bordeu Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile Umberto Bortolozzo Institut Nonlinéaire de Nice, Université de Nice-Sophia Antipolis, CNRS, Valbonne, France M. Botey Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica de Catalunya (UPC), Terrassa, Barcelona, Spain R. Bustos-Guajardo Departamento de Física Aplicada, CINVESTAV del IPN, Mérida, YucatánMéxico Thomas Butler Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology and Tyndall National Institute, Cork, Ireland Iberê L. Caldas Instituto de Física, Universidade de São Paulo, São Paulo, Brazil

xi xii Contributors

Jean-Marc Chomaz LadHyX, CNRS-École Polytechnique, Palaiseau, France Marcel G. Clerc Facultad de Ciencias Físicas y Matemáticas, Departamento de Física, Universidad de Chile, Santiago, Chile Carlos Conca Department of Mathematical Engineering (DIM), Centre for Mathematical Modelling (CMM) (UMI CNRS 2807), and Centre for Biotechnology and Bioengineering (CeBiB), University of Chile, Santiago, Chile Vasco Cortez Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Peñalolén, Santiago, Chile S. Coulibaly PhLAM, Université de Lille 1, Bât. P5-bis; UMR CNRS/USTL 8523, Villeneuve D’ascq, France Patricio Cumsille Group of Applied Mathematics, Department of Basic Sciences, University of Bío-Bío, Chillán, Chile; Centre for Biotechnology and Bioengineering, University of Chile, Santiago, Chile Daniel Escaff Facultad de Ingeniería y Ciencias Aplicadas, Complex Systems Group, Universidad de Los Andes, Las Condes, Santiago, Chile C. Falcón Facultad de Ciencias Físicas y Matemáticas, Departamento de Física, Universidad de Chile, Santiago, Chile Sergej Flach Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, Korea; New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry and Physics, Massey University, Auckland, New Zealand Punit Gandhi Department of Physics, University of California, Berkeley, CA, USA Mónica A. García-Ñustes Instituto de Física, Pontificia Universidad Católica de Valparaíso, Valparaíso, Casilla, Chile I. Giden Department of Electrical and Electronics Engineering, TOBB University of Economics and Technology, Ankara, Turkey Gregorio González-Cortés Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile David Goulding Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology and Tyndall National Institute, Cork, Ireland Pablo Gutiérrez Facultad de Ciencias Físicas y Matemáticas, Departamento de Física, Universidad de Chile, Santiago, Chile; Service de Physique de l’Etat Condensé, DSM, CEA-Saclay, Gif-sur-Yvette, France Stephen P. Hegarty Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology and Tyndall National Institute, Cork, Ireland Contributors xiii

R. Herrero Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica de Catalunya (UPC), Terrassa, Barcelona, Spain Guillaume Huyet Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology, Cork, Ireland; National Research University of Information Technologies, Mechanics and Optics, Saint Petersburg, Russia and Tyndall National Institute, Cork, Ireland Bryan Kelleher Department of Physics, University College Cork and Tyndall National Institute, Cork, Ireland Edgar Knobloch Department of Physics, University of California, Berkeley, CA, USA A.V. Kovalev ITMO University, St. Petersburg, Russia Tiago Kroetz Universidade Tecnológica Federal do Paraná, Pato Branco, Paraná, Brazil S. Kumar Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica de Catalunya (UPC), Terrassa, Barcelona, Spain H. Kurt Department of Electrical and Electronics Engineering, TOBB University of Economics and Technology, Ankara, Turkey D. Laroze Instituto de Alta de Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile; SUPA School of Physics and Astronomy, University of Glasgow, Glasgow, UK René Lefever Faculté des Sciences, Université Libre de Bruxelles (ULB), Bruxelles, Belgium F. Leo Photonics Research Group, Department of Information Technology, Ghent University-IMEC, Ghent, Belgium Edson D. Leonel Departamento de Física, Universidade Estadual Paulista, São Paulo, Brazil Alejandro O. León Facultad de Ciencias Físicas y Matemáticas, Departamento de Física, Universidad de Chile, Santiago, Chile Z. Liu PhLAM, Université de Lille 1, Bât. P5-bis; UMR CNRS/USTL 8523, Villeneuve D’ascq, France André L.P. Livorati Departamento de Física, Universidade Estadual Paulista, São Paulo, Brazil E. Louvergneaux Laboratoire de Physique des Lasers, Atomes Et Molécules, CNRS UMR8523, Université de Lille 1, Villeneuve d’Ascq, France J.E. Macías Facultad de Ciencias Físicas y Matemáticas, Departamento de Física, Universidad de Chile, Santiago, Chile xiv Contributors

A. Makhoute Physique du Rayonnemet et des Interactions Laser-Matiere, Faculté des Sciences, Département de Physique, Université Moulay Ismail, Meknès, Morocco Boris A. Malomed Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel H.L. Mancini Departamento de Física y Matemática Aplicada, Universidad de Navarra, Pamplona, Spain Fernando Mora Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Peñalolén, Santiago, Chile Cristian F. Moukarzel Departamento de Física Aplicada, CINVESTAV del IPN, Mérida, YucatánMéxico V. Odent Laboratoire de Physique des Lasers, Atomes Et Molécules, CNRS UMR8523, Université Lille 1, Villeneuve d’Ascq, France Sabine Ortiz LadHyX, CNRS-École Polytechnique, Palaiseau, France; UME/DFA, ENSTA, Palaiseau, France Vincent Padilla Service de Physique de l’Etat Condensé, DSM, CEA-Saclay, Gif-sur-Yvette, France Krassimir Panajotov Brussels Photonics Team (B-Phot), Department of Applied Physics and Photonics (TONA), Vrije Universiteit Brussel, Brussel, Belgium; Institute of Solid State Physics, Sofia, Bulgaria L.M. Pérez Departamento de Física y Matemática Aplicada, Universidad de Navarra, Pamplona, Spain M. Radziunas Weierstrass Institute for Applied Analysis and Stochastics, Leibniz Institute in Forschungsverbund Berlin e.V., Berlin, Germany Stefania Residori Institut Nonlineaire de Nice, Université de Nice-Sophia Antipolis, CNRS, Valbonne, France Sergio Rica Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Peñalolén, Santiago, Chile Marc Sciamanna OPTEL Research Group, LMOPS EA-4423, CentraleSupelec - Universite de Lorraine, Belin, France Svetlana Slepneva Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology and Tyndall National Institute, Cork, Ireland N. Slimani Physique du Rayonnement et des Interactions Laser-Matière, Faculté des Sciences, Université Moulay Ismail, Meknès, Morocco Contributors xv

K. Staliunas Departament de Fisica i Enginyeria Nuclear, Universitat Politècnica de Catalunya (UPC), Terrassa, Barcelona, Spain; Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain M. Taki PhLAM, Université de Lille 1, Bât. P5-bis; UMR CNRS/USTL 8523, Villeneuve D’ascq, France Hugo Thienpont Brussels Photonics Team (B-Phot), Department of Applied Physics and Photonics (TONA), Vrije Universiteit Brussel, Brussel, Belgium Mustapha Tlidi Faculté des Sciences, Université Libre de Bruxelles (ULB), Bruxelles, Belgium M. Turduev Department of Electrical and Electronics Engineering, TED University, Ankara, Turkey Felipe Urbina Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Peñalolén, Santiago, Chile D. Urzagasti Instituto de Investigaciones Físicas, UMSA, La Paz, Bolivia Lautaro Vergara Departamento de Física, Universidad de Santiago de Chile, USACH, Santiago 2, Chile E.A. Viktorov ITMO University, St Petersburg, Russia Martin Virte Brussels Photonics Team (B-Phot), Department of Applied Physics and Photonics (TONA), Vrije Universiteit Brussel, Brussel, Belgium Mario Wilson DFI, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile Valeska Zambra Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile