SCHOOL PROGRAMME ADVANCED COMPUTATIONAL AND EXPERIMENTAL TECHNIQUES IN NONLINEAR DYNAMICS MAY 6-10, 2013, CUSCO, PERU

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1 SUNDAY 5 SCHOOL PROGRAMME ADVANCED COMPUTATIONAL AND EXPERIMENTAL TECHNIQUES IN NONLINEAR DYNAMICS MAY 6-10, 2013, CUSCO, PERU ROOM Hotel San Agustin Internacional", Salón de Eventos 11:00-19:00 REGISTRATION 19:00-20:00 WELCOME RECEPTION MONDAY 6 ROOM Hotel San Agustin El Dorado, Salón de Eventos 8:00-8:45 REGISTRATION SPEAKER CARLOS PANDO, 8:45-9:00 WELCOME ADDRESS AND OUTLINE OF THE SCHOOL Instituto de Física, BUAP, Puebla, México. 9:00-9:40 EUSEBIUS DOEDEL Concordia University, Montréal, Canada. 10:00 10:40 EUSEBIUS DOEDEL COFFEE 11:20-12:00 WIESLAW KROLIKOWSKI The Australian National University, Canberra, Australia. 12:20-13:00 WIESLAW KROLIKOWSKI LUNCH 16:00-16:40 ITAMAR PROCACCIA Weizmann Institute of Science, Rehovot, Israel. 17:00-17:40 ITAMAR PROCACCIA - 1 -

2 COFFEE 18:20-19:00 SHORT TALKS (20 MINUTES EACH). 19:20-20:00 SHORT TALKS TUESDAY 07 9:00-9:40 EUSEBIUS DOEDEL 10:00 10:40 EUSEBIUS DOEDEL COFFEE 11:20-12:00 PUNIT PARMANANDA IIT Bombay, Powai, Maharashtra, India. 12:20-13:00 PUNIT PARMANANDA LUNCH 16:00-16:40 ITAMAR PROCACCIA 17:00-17:40 ITAMAR PROCACCIA COFFEE 18:20-19:00 SHORT TALKS (20 MINUTES EACH). 19:20-20:00 SHORT TALKS WEDNESDAY 08 - FREE THURSDAY 09 9:00-9:40 JORGE GALÁN Universidad de Sevilla, Spain. 10:00 10:40 JORGE GALÁN - 2 -

3 COFFEE 11:20-12:00 SERGEY PRANTS Pacific Oceanological Institute RAS, Vladivostok, Russia. 12:20-13:00 SERGEY PRANTS LUNCH 16:00-16:40 RAJARSHI ROY University of Maryland, MD, USA. 17:00-17:40 RAJARSHI ROY COFFEE 18:20-19:00 KENNETH SHOWALTER West Virginia University, Morgantown, USA. 19:20-20:00 SHORT TALKS FRIDAY 10 9:00-9:40 WENDELL T. HILL, III University of Maryland, MD,USA. 10:00 10:40 WENDELL T. HILL, III 11:20-12:00 JORGE GALÁN COFFEE 12:20-13:00 RAJARSHI ROY END OF SCHOOL - 3 -

4 ROOM Paraninfo of UNSAAC (Auditorio Principal) RAJARSHI ROY University of Maryland, MD, USA. 19:00-20:00 Outreach Lecture:

5 EUSEBIUS DOEDEL Faculty of Engineering and Computer Science, Concordia University, Montréal, Canada. AN INTRODUCTION TO NUMERICAL CONTINUATION METHODS WITH APPLICATION TO SOME PROBLEMS FROM PHYSICS In these lectures I will cover the basic concepts of numerical continuation, a tool that is extremely useful and efficient in the study of solutions of nonlinear equations. I will discuss the persistence of solutions (the implicit function theorem), with various illustrative examples. The emphasis is on numerical methods for following such solution families, namely, parameter continuation and pseudo-arclength continuation. Illustrative examples will include a chemical reaction model, a predator-prey model, the Gelfand-Bratu boundary value problem, and families of periodic orbits in the circular restricted 3-body problem (CR3BP). I will also discuss the following of folds and Hopf bifurcations. Some details on the implementation of such algorithms in the software AUTO will be presented. Demos used in the course will be available, including extensive demos for computing families of periodic orbits and their stable and unstable manifolds in the CR3BP

6 JORGE GALÁN VIOQUE Matemática Aplicada II Escuela Técnica Superior Ingeniería, Universidad de Sevilla, Spain COMPUTATIONAL METHODS FOR CONSERVATIVE SYSTEMS In these three lectures we present basic and advanced methods for dynamical systems with conserved quantities and symmetries that appear frequently in applications from classical and quantum mechanics. Lecture 1 Introduction to conservative dynamical systems. Examples. Phase portrait. IVP integration. Poincaré sections. Periodic orbits and Floquet multpliers. Symmetries, reversibilities and conserved quantities. Lecture 2 Boundary value formulation. Continuation of periodic orbits in conservative and reversible systems. Lecture 3 Application to the three body problem in Celestial Mechanics: The figure eight and the Horseshoe solution. References 1. Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems (Understanding Complex Systems) (2007). B Krauskopf, H. Osinga and J. Galan Vioque

7 WENDELL T. HILL, III JQI, IPST and Physics University of Maryland College Park, MD 20742, USA ULTRACOLD GAS DYNAMICS IN CONFINED GEOMETRIES: NOT YOUR GRANDFATHER S DYNAMICS Ultra cold gas ensembles in confined geometries offer unique opportunities to investigate novel aspects of well-studied dynamics in ways never possible in the past. The confluence of laser cooling of atoms and an ability to create trapping potentials with nontrivial shapes has opened the door to the emerging field of atomtronics, a re-articulation of electronics with neutral atoms. From creating analogs of basic electrical elements (e.g., resistors, capacitors and inductors [1]) to refining our understanding of quantum flow dynamics (e.g., persistent flow [2]), atomtroncis has the potential to revolutionize and redefine architectures for information transmission. In this presentation we will explore dynamics of ultra cold gases at two ultra cold extremes: (1) (a nearly perfect ideal, classical gas) and (2) (a Bose-Einstein condensate or quantum gas), where is the thermal de Broglie wavelength,, the atom-atom interaction range and mean atomic separation. Specifically, we will present results that reveal dynamics that are both similar and distinct to those found in superconductors and superfluid He, such as SQUID-like behavior. We will also look a possible ways to measure directly critical flow velocities that were predicted nearly 60 years ago by Feynman [3]. References [1] J. Lee, et al. First Demonstration of an Atomtronic Analog of Basic Electronic Circuit Elements, Sci. Rep. 3, 1034 (2013). [2] A. Ramanathan, et al. Superflow in a toroidal Bose-Einstein condensate: An atom circuit with a tunable weak link, Phys. Rev. Lett. 106, (2011). [3] R. P. Feynman, Progress in Low Temperature Physics, Vol. 1, Amsterdam: North-Holland, (1955)

8 WIESLAW KROLIKOWSKI Laser Physics Centre, Australian National University Canberra ACT 0200, Australia Lecture 1 NONLINEAR WAVES IN NONLOCAL MEDIA In nonlinear media with a nonlocal nonlinear response the nonlinear response in a particular point is not determined solely by the amplitude of the wave in that very point (as it is the case for a local medium), but depends also on the waveamplitude in its neighborhood. It appears that nonlocality is a generic feature of a large range of nonlinear systems, ranging from optics to matter waves. It may result from certain transport processes, such as atom diffusion, heat transfer, drift of electric charges, etc., or long range inter-particle interaction. It turns out that nonlocality may dramatically affect the propagation of waves and their stability. In this lectures I will concentrate on the role of spatial nonlocality on propagation of waves and formation of spatial solitons. In particular I discuss such phenomena as modulational instability of plane waves, stability properties of complex soliton states and interaction of bright and dark solitons. Lecture 2 SPATIAL SOLITONS IN OPTICAL LATTICES IN PHOTOREFRACTIVE CRYSTALS Spatial optical solitons form as a balance between diffraction which tends to spread the beam and nonlinearity. In homogeneous media the sign of diffraction is always fixed. Therefore in order to form a bright (dark) soliton the nonlinear response of the medium must be of self-focusing (self-defocusing) type. On the other hand in media with spatially periodic modulation of refractive index diffraction diffraction may be positive, negative or even zero. As a result the same nonlinear medium may support either bright or dark solitons. In this lecture I will discuss formation of spatial optical solitons, including gap solitons, in periodic nonlinear media in both bulk and planar geometries

9 PUNIT PARMANANDA IIT Bombay, Powai, Maharashtra, India. Lecture 1 SYNCHRONIZATION AND CHAOS: AN OXYMORON? Synchronization and Chaos are frequently observed in nonlinear systems. Since these two phenomena are ubiquitous in nature they inevitably overlap (co-exist). In the present talk, the different domains of chaotic synchronization are characterized for a nonlinear chemical system. Numerical calculations are verified using experimental measurements. Lecture 2 SYNCHRONIZATION IN AN ENSEMBLE OF SPATIALLY MOVING OSCILLATORS We study, numerically, synchronization in an ensemble of oscillators residing in a finite 2- Dimensional space. Both phase oscillators as well as normal nonlinear oscillators are considered. The interaction between the oscillators is governed by their spatial movement. The coupling is unidirectional and is of the intermediate kind, with the spatial vision of each oscillator deciding the number of oscillators it is coupled to. Both pulse mediated coupling (phase oscillators) and continuous coupling (normal oscillators) scenarios are entertained. We analyze how the extent of synchronization, characterized by the order-parameter, depends upon the system parameters such as: spatial vision, spatial motion, and strength of the coupling

10 SERGEY V. PRANTS Pacific Oceanological Institute of the Russian Academy of Sciences, Vladivostok, RUSSIA Lecture 1 CHAOTIC ADVECTION IN FLUIDS: ANALYTICAL, NUMERICAL, EXPERIMENTAL AND PRACTICLE ASPECTS Chaotic advection of passive scalars in fluids along with its topological, dynamical, and fractal properties is considered from the standpoint of the theory of dynamical systems and Hamiltonian chaos. Analytic and numerical results on chaotic advection in simple kinematic and dynamic models of oceanic and amospheric flows are described for topographical eddies and meandering jet currents. Laboratory experiments on chaotic advection as an imitation of geophysical transport and mixing of water masses are described. Some practical applications of chaotic advection are discussed. Lecture 2 DYNAMICAL SYSTEMS THEORY METHODS FOR STYDING MIXING AND TRANSPORT IN THE OCEAN We apply nonlinear dynamical system theory approach to study surface largescale transport and mixing in the ocean. The motion of water parcels is simulated in the velocity fields derived from satellite altimeter measurements of anomalies of the sea height. We apply the Lagrangian methods to reveal geometric structures hidden in ocean flows. Some of them are well known stable and unstable manifolds of hyperbolic trajectories and the other ones are new structures to be called Lagrangian fronts, which are boundaries between water masses with different properties and history. As an application of the methods developed, we identify Lagrangian fronts with favourable fishery conditions in the North Pacific Ocean and simulate radionuclide propagation from the Fukushima-Daiichi nuclear plant accident. It is demonstrated that the methods of that theory in the framework of the Lagrangian approach work well with real velocity fields. We discuss the rerspectives of the Lagrangian approach in oceanography and its possible practical applications

11 ITAMAR PROCACCIA Weizmann Institute of Science, Rehovot, Israel. In my school lectures I will provide background and method to understand what is known about the glass transition, and about the glasses that form as a result of this transition. I will unfold the theory of elasticity and plasticity in amorphous (glassy) solids, and the instabilities that lead to their yield at high values external strains. Long standing questions like the existence of a typical length scale that increases towards the glass transition, the identification of the mechanism of plasticity and yield, and how to predict the temperature and strain rate dependence of the yield stress will be answered. Lecture 1 THE GLASS TRANSITION, WHAT IS KNOWN AND WHAT IS MYSTERIOUS Lecture 2 THE HESSIAN MATRIX: EIGENFUNCTIONS, EIGENVALUES, AND THE INFAMOUS BOSON PEAK Lecture 3 MECHANICAL PROPERTIES OF AMORPHOUS SOLIDS: WHAT IS PLASTICITY? Lecture 4 YIELD, SHEAR LOCALIZATION AND SHEAR BANDS IN AMORPHOUS SOLIDS

12 RAJARSHI ROY University of Maryland, College Park, MD, USA Lecture 1 OPTICAL NETWORKS: DYNAMICS, PATTERNS AND SYNCHRONIZATION I Techniques for the generation of time evolving waveforms that can serve as carriers of information will be introduced. We examine the role of nonlinearity, time delays and feedback on the dynamics of optical systems designed to generate a wide variety of waveforms. How do we model these devices and can we test the models experimentally? Can the concept of synchrony between an experimental system and a numerical model be applied for the determination of system parameters and prediction of the time evolution of a chaotic dynamical system? We will describe quantitative results regarding the prediction of the dynamics of nonlinear dynamical systems. Lecture 2 OPTICAL NETWORKS: DYNAMICS, PATTERNS AND SYNCHRONIZATION II Experimental observations of synchrony between coupled systems and its application to communication and sensing will be described. Synchrony in simple network motifs will be examined and examples of communication with synchronized systems will be used as illustrations. Can synchrony be maintained even when the coupling varies irregularly in time? Techniques for adaptively maintaining synchrony in networks and for tracking changes in coupling strength will be given. Lecture 3 OPTICAL NETWORKS: DYNAMICS, PATTERNS AND SYNCHRONIZATION III The concept of optimal synchronization in real networks will be studied with examples of results from numerical computations and experiments. How do systems converge to synchrony? Synchronization error is affected by parameter mismatches and by noise sources. Robustness of synchrony will be studied and we will show that the effects of network topology are important and can influence the rates of convergence significantly

13 KENNETH SHOWALTER Department of Chemistry West Virginia University Morgantown, WV USA COLLECTIVE BEHAVIOR OF INTERACTING PARTICLE-LIKE WAVES Unstable waves are stabilized by global feedback that affects the overall excitability of the medium, and the motion of these waves is controlled by imposing excitability gradients that are regulated by a secondary feedback loop. The locus of steady-state wave size as a function of excitability defines the boundary for spiral wave behavior in active media. Intricate patterns of wave propagation are exhibited with spatiotemporal feedback. Waves interacting with boundaries and with other waves are observed when interaction terms are incorporated into the control algorithm, such as a Lennard-Jones type potential in which there are attractive and repulsive forces. Processional motion is the most common behavior, where waves align with one another to varying degrees depending on the strength of the potential. Rotational motion is also observed, which may occur for the same parameters as processional motion depending on initial conditions. We also discuss other modes of behavior and an analysis of the wave interaction in terms of the gradient of the potential