Abstract: Typical physical systems follow deterministic behavior. This behavior can be sensitive to initial conditions, such that it is very difficult to predict their behavior in the longtime limit. The resulting motion is chaotic and looks stochastic or random. In many cases the motion is described by a Hamiltonian and the energy is conserved. The motion can be also regular, that is predictable. In the work reported here we studied systems where depending on initial conditions the motion is either regular or chaotic. The simplest systems of this type are of two degrees--of--freedom, or periodically kicked systems with one degree--of--freedom. For this type of systems transport in the chaotic regions of phase space is dominated by sticking to complicated structures in the vicinity of the regular region. The probability to stay in the vicinity of the initial point is a power law in time characterized by some exponent. The question of the value of this exponent and its universality is the subject of a long controversy. We have developed a statistical description for this type of systems, where statistics are with respect to parameter or family of systems rather than to initial conditions. Following previous studies, it is based on a scaling of periodic and quasi-periodic orbits in a way which relies heavily on number theory. We have found an indication that the statistics of scaling is parameter independent and might be relevant for a wider universality class including the systems we explored. This statistical description is implemented in a stochastic Markov model proposed by Meiss and Ott in 1986. Even though many approximations are used, it predicts important results quantitatively, showing the power law decay exponent to be approximately 51.57 in agreement with direct simulations done in this work and also other works. Its universality is inferred from the universality of the scaling statistics. The model systems used in this work are paradigms for chaotic dynamics (the H'enon map and the standard map) therefore it might indicate a wider universality class. Quantum manifestation of this phenomenon and its relevance for time correlations, is showing different behavior for increasing effective Planck's constant, namely, the Planck's constant divided by the typical action. By using recent results regarding the universality of wave function transmission across barriers in phase space, we generalize the use of the Markov model to describe the results after some modification.
The work reported was done in collaboration with Or Alus, James Meiss and Mark Srednicki
Abstract: Can we use machine learning (ML) to predict the evolution of complex, chaotic systems? The recent Maryland-based work showed that the answer is conditionally affirmative once we use some additional âhelpâ provided by a random bath and observers, as defined through reservoir computing (RC) . What about using other âstandardâ ML methods in forecasting the future of complex systems? The ETH-MIT group showed that the long-short-term-memory (LSTM) method may work in general spatiotemporal evolution of the Kuramoto type . Our work (Crete-Harvard) focused on the following question: Under what circumstances ML can predict spatiotemporal structures that emerge in complex evolution that involves nonlinearity as well as some form of stochasticity? To address this question we used two extreme phenomena, one being turbulent chimeras while the second involves stochastic branching. The former phenomenon generates partially coherent structures in highly nonlinear oscillators interacting through short or long range coupling while the latter appears in wave propagation in weakly disordered media. Examples of the former include biological networks, SQUIDs (superconducting quantum interference devices), coupled lasers, etc while the latter geophysical waves, electronic motion in a graphene surface and other similar wave propagation configurations.
In our work we applied and compared three ML methods, viz. LSTM, RC as well as the standard Feed-Forward neural networks (FNNs) in the two extreme spatiotemporal phenomena dominated by coherence, i.e. chimeras, and stochasticity, i.e. branching, respectively . In order to increase the predictability of the methods we augmented LSTM (and FNNs) with observers; specifically we assigned one LSTM network to each system node except for "observer" nodes which provide continual "ground truth" measurements as input; we refer to this method as "Observer LSTM" (OLSTM). We found that even a small number of observers greatly improves the data-driven (model-free) long-term forecasting capability of the LSTM networks and provide the framework for a consistent comparison between the RC and LSTM methods. We find that RC requires smaller training datasets than OLSTMs, but the latter requires fewer observers. Both methods are benchmarked against Feed-Forward neural networks (FNNs), also trained to make predictions with observers (OFNNs).
 Z. Lu Z, J. Pathak, B. Hunt, M. Girvan, R. Brockett and E. Ott, Reservoir observers: Model free inference of unmeasured variables in chaotic systems. Chaos 27, 041102 (2017); J. Pathak, B. Hunt,M. Girvan, Z. Lu and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Let. 120, 024102 (2018)
 P. R. Vlachas, W. Byeon, Z. Y. Wan, T. P. Sapsis and P. Koumoutsakos, Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc.R.Soc.A 474, 20170844 (2018).
 G. Neofotistos, M. Mattheakis, G. D. Barmparis, J. Hizanidis, G. P. Tsironis and E. Kaxiras, Machine learning with observers predicts complex spatiotemporal evolution, arXiv 1807.10758 (2018)
Abstract: Wind and solar farms offer a major pathway to clean, renewable energies. However, these farms would significantly change land surface properties, and, if sufficiently large, the farms may lead to unintended climate consequences. In this study, we used a climate model with dynamic vegetation to show that large-scale installations of wind and solar farms covering the Sahara lead to a local temperature increase and more than a twofold precipitation increase, especially in the Sahel, through increased surface friction and reduced albedo. The resulting increase in vegetation further enhances precipitation, creating a positive albedoâprecipitationâvegetation feedback that contributes ~80% of the precipitation increase for wind farms. This local enhancement is scale dependent and is particular to the Sahara, with small impacts in other deserts.
Abstract: Action-angle coordinates can be constructed for so-called integrable Hamiltonian dynamical systems, for which there exists a foliation of phase space by surfaces that are invariant under the dynamical flow. Perturbations generally destroy integrability. However, we know that periodic orbits will survive, as will cantori, as will the "KAM" surfaces that have sufficiently irrational frequency, depending on the perturbation. There will also be irregular "chaotic" trajectories. By "fitting" the coordinates to the invariant structure that are robust to perturbation, action-angle coordinates may be generalized to non-integrable dynamical systems. These coordinates "capture" the invariant dynamics and neatly partition the chaotic regions. These so-called chaotic coordinates are based on a construction of almost-invariant surfaces known as ghost surfaces. The theoretical definition and numerical construction of ghost surfaces and chaotic coordinates will be described and illustrated.
Abstract: This talk will discuss bifurcations in several dynamical control systems that arise in aerospace engineering applications. First, I will present the swimming dynamics and control of a flexible underwater robot based on closed-loop control of an internal reaction wheel. The feedback law stabilizes a limit cycle about the desired heading angle and produces forward swimming motion. Analysis of a global bifurcation in the dynamics under feedback control reveals the set of control gains that yields the desired limit cycle. Second, I will discuss a nonlinear control system consisting of a single vortex in a freestream near an actuated cylinder that represents an airfoil under a conformal mapping. Using heaving and/or surging of the cylinder as input stabilizes the vortex position relative to the cylinder. The closed-loop system utilizes a linear state-feedback control law, which gives rise to several bifurcations by varying the control gains. Lastly, time permitting, I will discuss a state-space model for representing the lift of an airfoil at high angles of attack. A feedback controller stabilizes a limit cycle in the angle of attack that provides greater (average) lift than a static pitch angle. In all three examples, incorporating dynamical systems theory complements the state-space modeling and control design.
Abstract: A trajectory is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus for which F has the form of a rigid rotation on the torus with rotation vector rho. There is an extensive literature on determining the rotation vector associate with F, as well finding Fourier components to establish these conjugacies. I will present two new methods with very good convergence rates: the Weighted Birkhoff Method and the Embedding Continuation Method. They are based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. I will illustrate these for one- and two-dimensional examples ideas by computing rotation vectors or numbers, computing Fourier components for conjugacies, and distinguishing chaos versus quasiperiodic behavior.
Abstract: The world is full of volatile chemical cues that animals must decipher to detect the presence of prey, predators, and even potential mates. The olfactory system is burdened with the task of interpreting a near infinite amount of odors given a limited repository of chemoreceptors. Studies emphasizing invertebrates have provided tremendous insight into the basic mechanisms of olfaction, and the highly analogous organization of invertebrate and mammal olfactory systems suggests that such studies can shed light upon how our own sense of smell functions. In this seminar, I will discuss data from Drosophila melanogaster and locusts revealing how olfactory information is transformed at subsequent stages of processing. Finally, I will discuss data from my own laboratory showing how neuromodulatory neurons that alter the sensory processing interact with the olfactory system.
Abstract: Personal Protection measures, such as bed nets and personal repellents, are important tools for the suppression of vector-borne diseases like malaria and Zika, and the ability of health agencies to distribute protection and encourage its use plays an important role in the efficacy of community-wide disease management strategies. Recent modeling studies have shown that a counterintuitive diversity-driven amplification in community-wide disease levels can result from a population's partial adoption of personal protection measures, potentially to the detriment of disease management efforts. This finding, however, may overestimate the negative impact of partial personal protection as a result of implicit restrictive model assumptions regarding host compliance, access to, and longevity of protection measures. We establish a new modeling methodology for incorporating community-wide personal protection distribution programs in vector-borne disease systems which flexibly accounts for compliance, access, longevity, and control strategies by way of a flow between protected and unprotected populations. Our methodology yields large reductions in the severity and occurrence of amplification effects as compared to existing models.
Abstract: In this seminar, we attempt to connect two of the major research vistas in nonlinear dynamics, namely, chimera states and chaos. We consider a simplified mathematical model of a one-dimensional lattice of coupled superconducting quantum interference devices (SQUIDs) driven by an external magnetic field [1,2]. We numerically simulate chimeras and other collective states in the magnetic flux oscillations through the SQUIDs and show that they are born through chaotic dynamics on finite time scales. We demonstrate the signatures of transient chaos in flux oscillations with fluctuating amplitudes, exponential escape time distribution, and fractal Wada basins of attraction for chimera states [1,3]. This study complements the identification of chimeras as transiently chaotic states themselves [4,5], and may be useful for prediction, characterization and control of such states.
References: 1. A. Banerjee and D. Sikder, Phys. Rev. E 98, 032220 (2018). 2. M. Trepanier, D. Zhang, O. Mukhanov, V. P. Koshelets, P. Jung, S. Butz, E. Ott, T. M. Antonsen, A. V. Ustinov, and S. M. Anlage, Phys. Rev. E 95, 050201(R) (2017). 3. Y.-C. Lai and T. Tel, Transient Chaos (Springer, New York, 2011). 4. M. Wolfrum and O. E. Omelchenko, Phys. Rev. E 84, 015201(R) (2011). 5. M. Wolfrum, O. E. Omelchenko, S. Yanchuk, and Y. L. Maistrenko, Chaos 21, 013112 (2011)."
Abstract: Toroidal magnetic fields can confine charged particles, which can be exploited for basic physics studies or potentially for fusion energy. The magnetic field should lack axisymmetry (continuous rotational symmetry), or else a large electric current is needed inside the confinement region. However, the magnetic field should possess two properties that could be termed âhidden symmetriesâ. The first, integrability, means the field lines should lie on nested toroidal surfaces, without regions of islands or chaos. The second, called `quasi-symmetryâ, generalizes the conservation of canonical angular momentum in the presence of strong magnetic fields. This second property arises because the Lagrangian for particle motion in strong magnetic fields can be expressed in terms of the strength of the field, independent of its direction. Magnetic fields with these properties can be found using optimization or using a new constructive procedure.
Abstract: Statistically stationary solutions to randomly forced systems have been of fundamental importance from both theoretical and practical points of view.
From one hand the existence of invariant measures provides information on the long time dynamics of randomly forced systems and from the other, under certain ergodicity assumptions, it provides a link between experimental observations and theoretical predictions.
In this talk Iâll present results on the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system, describing the motion of a compressible viscous fluid.
The existence of an invariant measure for the Markov process generated by strong solutions will be discussed.
Abstract: The propagation of ultra-short (~100 fs), intense (~10â100 TW/cm2) laser pulses in the atmosphere is rich in nonlinear physics and may have a broad range of applications. Experiments using terawatt pulses with durations less than a picosecond demonstrate the formation and long-distance propagation of plasma and optical filaments, white light generation, and the emission of secondary radiation far from the laser frequency. Controlling the propagation of these laser pulses over long atmospheric paths is scientifically and technologically challenging. In this talk, we discuss the various physical mechanisms governing the atmospheric propagation of ultrashort laser pulses and report on several theoretical, computational, and experimental studies carried out by the Naval Research Laboratory (NRL). These studies include recent experiments demonstrating extended channeling through very strong atmospheric turbulence enabled by nonlinear self-focusing of laser pulses in air. In addition, we discuss theoretical considerations for increasing the laser power that can propagated through the atmosphere.
Abstract: Time-delayed optoelectronic oscillators (OEOs) are at the center of a very large body of scientific literature. The complex behavior of these nonlinear oscillators has been thoroughly explored both theoretically and experimentally, leading to a better understanding of their dynamical properties. Beyond fundamental research, these systems have also inspired a wide and diverse set of applications, such as optical chaos communications, pseudo-random number generation, optoelectronic reservoir computing, ultra-pure microwave synthesis, optical pulse-train generation, and sensing. In this communication, we will provide a comprehensive overview of this field, outline the latest achievements, and discuss the main challenges ahead.
Abstract: Positive Lyapunov exponents are one of the key characteristics of chaos in classical dynamical systems. Here we discuss the notion of Lyapunov exponents in quantum many-body systems focusing on a recent definition of a whole spectrum of quantum Lyapunov exponents (https://arxiv.org/abs/1809.01671). The talk will not assume prior knowledge of the subject, although some knowledge of quantum mechanics will be helpful.
Abstract: The purpose of this work is to perform a mathematically rigorous study of Lagrangian chaos and passive scalar turbulence in incompressible fluid mechanics. We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, Sobolev-in-space stochastic forcing in a periodic box. We prove that if the forcing satisfies suitable non-degeneracy conditions, then these flows are chaotic in the sense that the top Lyapunov exponent is strictly positive. Our main results are for the 2D Navier-Stokes equations and the hyper-viscous regularized 3D Navier-Stokes equations (at arbitrary Reynolds number and hyper-viscosity parameters). For the passive scalar problem, we study statistically stationary solutions to the advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's 1949 law of passive scalar turbulence in over a suitable inertial range -- the constant flux law analogous to the Kolmogorov 4/5 law. To our knowledge, this work is the first to provide a complete mathematical proof of any such scaling law from fundamental equations of fluid mechanics. The work combines ideas from random dynamical systems (the Multiplicative Ergodic Theorem and an infinite dimensional variation of Furstenberg's Criterion) with elementary approximate control arguments and infinite-dimensional hypoellipticity via Malliavin calculus. Joint work with Alex Blumenthal and Sam Punshon-Smith.
Abstract: We present experiments on the effects of laminar flows on the spreading of the excitable Belousov-Zhabotinsky chemical reaction and on the motion of swimming bacteria. The results of these experiments have applications for a wide range of systems including microfluidic chemical reactors, cellular-scale processes in biological systems, and blooms of phytoplankton in the oceans. To predict the behavior of reaction fronts, we adapt tools used to describe chaotic fluid mixing in laminar flows. In particular, we propose "burning invariant manifolds" (BIMs) that act as one-way barriers that locally block the motion of reaction fronts. These barriers are measured experimentally in a range of vortex-dominated 2- and 3-dimensional fluid flows. A similar theoretical approach predicts "swimming invariant manifolds" (SwIMs) that are one-way barriers the impede the motion of microbes in a flow. We are conducting experiments to test the existence of SwIMs for both wild-type and smooth swimming Bacillus subtilis in hyperbolic and vortex-dominated fluid flows.
Abstract: The dynamo effect is a class of macroscopic phenomena responsible for generation and maintaining magnetic fields in astrophysical bodies. It hinges on hydrodynamic three-dimensional motion of conducting gases and plasmas that achieve high hydrodynamic and/or magnetic Reynolds numbers due to large length scales involved. The existing laboratory experiments modeling dynamos are challenging and involve large apparatuses containing conducting fluids subject to fast helical flows. Here we propose that electronic solid-state materials -- in particular, hydrodynamic metals -- may serve as an alternative platform to observe some aspects of the dynamo effect. In this talk, I will discuss two candidate systems -- Well semimetals and critical fluctuating superconductors, where electronic turbulence and dynamo effect appear within experimental reach.
 V. Galitski, M. Kargarian, and S. Syzranov, "Dynamo Effect and Turbulence in Hydrodynamic Weyl Metals," Phys. Rev. Lett. 121, 176603 (2018)
 Y. Liao and V. Galitski, "Two-Fluid Hydrodynamics and Viscosity Suppression in Fluctuating Superconductors,"
Abstract: In his famous undergraduate physics lectures, Richard Feynman remarked about the problem of fluid turbulence: "Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences.â This statement was already false when Feynman made it. Unbeknownst to him, Lars Onsager decades earlier had made an exact mathematical analysis of the high Reynolds-number limit of incompressible fluid turbulence, using a method that would now be described as a non-perturbative renormalization group analysis and discovering the first âconservation-law anomalyâ in theoretical physics. Onsagerâs results were only cryptically announced in 1949 and he never published any of his detailed calculations. Onsagerâs analysis was finally rescued from oblivion and reproduced by the speaker in 1994. The ideas have subsequently been intensively developed in the mathematical PDE community, where deep connections emerged with John Nashâs work on isometric embeddings. Furthermore, the method has more recently been successfully applied to new physics problems, compressible fluid turbulence and relativistic fluid turbulence, yielding many new testable predictions. This talk will briefly review Onsagerâs exact analysis of the original incompressible turbulence problem and subsequent developments. Then a new application to kinetic plasma turbulence will be described, with novel predictions for turbulence in nearly colllisionless plasmas such as the solar wind and the terrestrial magnetosheath.
Abstract: Josephson junctions with topological materials as weak links pervade the research of Majorana bound states. Yet these junctions exhibit many complex phenomena, some which are accessible due to new material and fabrication technology. A comprehensive picture is important both for elucidation of the physics of Majorana bound states and fundamental research into Josephson junctions. In this talk I will detail the physics of Josephson junctions and the chaotic behavior we observe in high-quality, graphene-based Josephson junctions under application of RF radiation. We quantify a instability measured in the AC Josephson regime which is analyzed in terms of crisis-induced intermittency. Further, these observations cast doubt over arguments that AC Josephson effect in the low RF drive amplitude region would offer the opportunity to observe 4-Ï current phase relation in topological Josephson junctions.
Abstract: The human brain accounts for just 2% of the body's mass but metabolizes 25% of its calories, producing significant metabolic waste. However, waste buildup links to neurodegenerative diseases like Alzheimer's and Parkinson's. The brain removes waste via the recently-discovered glymphatic system, a combination of spaces and channels through which cerebrospinal fluid flows to sweep away toxins like amyloid-beta. With an interdisciplinary group of neuroscientists and physical scientists, I study the physical processes of the glymphatic system: Where does fluid flow, and how fast? What drives flow? Does flow shear cause waste accumulation? What characteristics of the system enable essential functions? How can we improve waste removal? Can we use glymphatic flow to deliver drugs? The team combines physics tools like particle tracking and newly-invented front tracking with biological tools like two-photon imaging through cranial windows in order to address these questions with in vivo flow measurements. I will talk about recent results showing that glymphatic flow proceeds along vessels with near-optimal shapes, pulses with the heart, is driven by artery walls, and can be manipulated by changing the wall motion.
Abstract: I will describe the design of a smart controller for dynamical systems based on reservoir computers. I use this approach to control fixed points, unstable period orbits, and arbitrary orbits for the Lorenz chaotic system. We have also applied the controller to other systems, such as a mathematical model of a drone (quadcopter). The controller to easily adopt to a partial degradation of the system allowing the drone to maintain stable flight. I will discuss our progress on using this approach to controlling the dynamics of a chaotic electronic circuit.
Abstract: In this talk, we will describe two branches of research related to neural attractor dynamics: the first focuses on *identifying* attractors, in order to understand what a trained network has learned; and the second focuses on *harnessing* attractors, in order to teach a network useful computations. More specifically, the first part of this talk presents "directional fibers," which are mathematical objects that can be used to systematically enumerate fixed points in many dynamical systems. Directional fibers are curves in high-dimensional state space that contain the fixed points of a system and can be numerically traversed. We will define directional fibers, derive their important theoretical properties, and describe empirical results of applying directional fibers to locate fixed points in recurrent neural networks. For example, directional fibers revealed that a network trained on the Lorenz system will have fixed points in correspondence with the Lorenz system fixed points, even though the Lorenz fixed points were not included in the training data. The second part of this talk presents a "Neural Virtual Machine" (NVM), which is a purely neural system that can emulate a Turing-complete computer architecture. The NVM uses local learning rules and itinerant neural attractor dynamics to represent, learn, and execute symbolic computer programs written in an assembly-like language. We will present the dynamical equations of the NVM, explain how it can be used to carry out algorithmic tasks, and present results of computer experiments that quantify its performance and scaling requirements. In particular, we demonstrate that the number of neurons required is only linear in the size of the programs being emulated.
Abstract: In dynamical systems, basins of attraction are defined as the set of initial conditions leading to a particular asymptotic behavior. Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. A special case of fractal boundaries appears when a single boundary separates three or more different basins of attraction. Then we say that the set of basins has the Wada property and initial conditions near that boundary become particularly unpredictable. Although it could seem an odd situation, many physical systems showing this topological property appear in the literature. In this talk, I will review some basic aspects on Wada basins, and then I will describe some new recently developed methods to ascertain the Wada property in dynamical systems. These new methods present important advantages with respect to the previously known method, provide new perspectives on the Wada property and broaden the situations where it can be verified. Also, I will show how the novel concept of basin entropy helps us quantify the unpredictability associated to different basins of attraction and its relation with Wada basins.
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