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2019 Applied Dynamics Seminar Archives

January 31, 2019

No seminar

February 7, 2019

Propagation of Ultrashort, Intense Laser Pulses Through the Atmosphere

Joseph Peñano

Naval Research Laboratory

Abstract: The propagation of ultra-short (~100 fs), intense (~10–100 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.

February 14, 2019

Time-delayed optoelectronic oscillators: theory and applications

Yanne Chembo

University of Maryland | Department of Electrical and Computer Engineering/IREAP

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.

February 21, 2019

Quantum Lyapunov Spectrum

Brian Swingle

University of Maryland | Department of Physics

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.

February 28, 2019

Lagrangian chaos and passive scalar turbulence

Jacob Bedrossian

University of Maryland | Department of Mathematics

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.

March 7, 2019

APS March Meeting - No seminar

March 14, 2019

Reaction fronts and swimming organisms in laminar flows: manifolds and barriers

Tom Solomon

Bucknell University | Department of Physics

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.

March 21, 2019

Spring Break - No seminar

March 28, 2019

Turbulence and dynamo effect in electronic materials

Victor Galitski

University of Maryland | Department of Joint Quantum Institute

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.

[1] V. Galitski, M. Kargarian, and S. Syzranov, "Dynamo Effect and Turbulence in Hydrodynamic Weyl Metals," Phys. Rev. Lett. 121, 176603 (2018)

[2] Y. Liao and V. Galitski, "Two-Fluid Hydrodynamics and Viscosity Suppression in Fluctuating Superconductors,"

https://arxiv.org/abs/1903.08666

April 4, 2019

'We have no good fundamental theory (of turbulence) at all': Was Feynman right?

Gregory Eyink

Johns Hopkins University | Department of Applied Mathematics

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.” 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 “conservation-law anomaly” in theoretical physics. Onsager’s results were only cryptically announced in 1949 and he never published any of his detailed calculations. Onsager’s 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 Nash’s 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 Onsager’s 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.

April 11, 2019

Chaotic Dynamics of Driven Graphene Josephson Junctions

James Williams

University of Maryland | Department of Physics

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-π current phase relation in topological Josephson junctions.

April 18, 2019

The physical processes of brain waste removal

Douglas Kelley

University of Rochester | Department of Mechanical Engineering

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.

April 25, 2019

Controlling dynamical systems using a deep reservoir computer

Daniel Gauthier

Ohio State University | Department of Physics

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.

May 2, 2019

Cluster Synchronization in Multilayered Networks

Louis Pecora

Naval Research Laboratory

Abstract: The behavior of dynamical systems (nodes or oscillators) that are coupled together in complex networks are greatly affected by the structure of those networks. Various patterns of synchronization among subsets of nodes are possible when the network has symmetries and input or balanced partitions. Symmetries and partitions will be introduced in this talk along with the dynamical patterns they can support. The basic machinery from group theory will be used to analyze the patterns, especially for their dynamical stability. The main portion of the talk will be about extending these ideas to networks with multiple oscillator types called multilayered networks. This extension while looking straightforward has some interesting structures that govern the behavior of synchronous clusters in the network as a whole as well as the symmetries of the network.

May 9, 2019

Identifying and Harnessing Attractor Dynamics in Neural Systems with Applications to Chaotic Time-Series Prediction and Turing-Complete Program Learning

Garrett Katz

Syracuse University | Department of Computer Science

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.

May 16, 2019

New methods for fractal and Wada basins

Álvar Daza

King Juan Carlos University | Applied Physics

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.

September 5, 2019

Population collapse in elite-dominated societies: A differential equation model without differential equations

James Yorke and Naghmeh Akhavan

University of Maryland | Department of Mathematics

Abstract: We discuss models of interactions with the environment by human populations, both between poor and rich people, "Commoners'' and "Elites''. The Elites control the society's wealth and consume it at a higher rate than Commoners, whose work produces the wealth. We say a model is “Elite-dominated” when the Elites' per capita population change rate is always at least as large as the Commoners'. We can show the model always exhibits population crashes for all choices of parameter values for which it is Elite-dominated. But any such model with explicit equations raises questions of how the resulting behaviors depend on the details of the models. How important are the particular design features codified in the differential equations? We discard the differential equations, replacing them with qualitative conditions that the original model satisfies, and we prove these conditions imply population collapse must occur. In particular, one condition is that the model is Elite-dominated. Our approach of introducing qualitative mathematical hypotheses can better show the underlying features of the model that lead to collapse. We also ask how societies can avoid collapse.

September 12, 2019

Strong neuron-to-body coupling implies weak neuron-to-neuron coupling in motor cortex

Woodrow Shew

University of Arkansas | Department of Physics

Abstract: Cortical neurons can be strongly or weakly coupled to the network in which they are embedded, firing in sync with the majority or firing independently. Both these scenarios have potential computational advantages in motor cortex. Commands to the body might be more robustly conveyed by a strongly coupled population, whereas a motor code with greater information capacity could be implemented by neurons that fire more independently. Which of these scenarios prevails? Here we measure neuron-to-body coupling and neuron-to-population coupling for neurons in motor cortex of freely moving rats. We find that neurons with high and low population coupling coexist, and that population coupling was tunable by manipulating inhibitory signaling. Importantly, neurons with different population coupling tend to serve different functional roles. Those with strong population coupling are not involved with body movement. In contrast, neurons with high neuron-to-body coupling are weakly coupled to other neurons in the cortical population.

September 19, 2019

No seminar. We invite you to attend the Paint Branch Distinguished Lecture at 4pm, 1101 A. James Clark Hall.

Paint Branch Distinguished Lecture: From Nonlinear Optics to High-Intensity Laser Physics

Donna Strickland

University of Waterloo | Department of Physics and Astronomy

Abstract: The laser increased the intensity of light that can be generated by orders of magnitude and thus brought about nonlinear optical interactions with matter. Chirped pulse amplification, also known as CPA, changed the intensity level by a few more orders of magnitude and helped usher in a new type of laser-matter interaction that is referred to as high-intensity laser physics. In this talk, I will discuss the differences between nonlinear optics and high-intensity laser physics. The development of CPA and why short, intense laser pulses can cut transparent material will also be included. I will also discuss future applications.

September 26, 2019

Tree-like approximations and critical network cascades

Sarthak Chandra

University of Maryland | Department of Physics

Abstract: Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called "locally tree-like approximation" (which assumes that one can ignore small loops in a network). This is, however, usually an `uncontrolled' approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In our work, we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of network cascades (i.e., a network dynamical process describing the spread of activity through a network). For this widely applicable problem, we discuss the conditions under which tree-like approximations give good results, and also explain the reasons for deviation from this approximation. More specifically, we show that these deviations are negligible for networks with a large number of network links, justifying why tree-based theories appear to work well for most real-world networks.

October 3, 2019

Can we use future observations to improve current forecasts without cheating?

Slides available here.

Eugenia Kalnay

University of Maryland | Department of Civil and Environmental Engineering, Department of Mechanical Engineering, Department of Atmospheric and Oceanic Science

Abstract: Co-authored with Tse-Chun Chen and Daisuke Hotta. The National Weather Service computes operational weather forecasts using a process called “data assimilation”: A 6 hour forecast is computed starting from the current “analysis”. The 6 hour forecast is then optimally combined with the observations collected 6 hours later to create the new analysis which serves as initial conditions for the next forecast. This process, known as “analysis cycle”, is repeated every 6 hours. Miyakoda (personal communication, ~1980) pointed out that using any future information to improve current forecasts should be considered “cheating” because it cannot be done in operational forecasting. Chen (2018, PhD thesis), Chen and Kalnay (2019a) MWR, and Chen and Kalnay (2019b, under review), developed an application of Ensemble Forecast Sensitivity to Observations (EFSO, Kalnay et al., 2012, Tellus) combined with Proactive Quality Control (PQC, Hotta et al., 2017). It uses future data (e.g., observations obtained 6 hours after the present analysis) to identify and delete current detrimental observations (in the present analysis). We found that making a late correction of every current analysis after the new observations have been received, accumulates improvements with time. The accumulated improvement is found to be much larger than the last correction that cannot be used in order to avoid cheating, so that forecasts are significantly improved “without cheating”.

October 10, 2019

Quantum impulse control

Christopher Jarzynski

University of Maryland | Department of Chemistry & Biochemistry and Department of Physics

Abstract: The quantum adiabatic theorem governs the evolution of a wavefunction under a slowly time-varying Hamiltonian. I will consider the opposite limit of a Hamiltonian that is varied impulsively: a strong perturbation U(x,t) is applied over a time interval of infinitesimal duration ε → 0. When the strength of the perturbation scales like 1/ ε 2, there emerges an interesting dynamical behavior characterized by an abrupt displacement of the wave function in coordinate space. I will solve for the evolution of the wavefunction in this situation. Remarkably, the solution involves a purely classical construction, yet describes the quantum evolution exactly, rather than approximately. I will use these results to show how appropriately tailored impulses can be used to control the behavior of a quantum wavefunction.

October 17, 2019

Using machine learning to assess short term causal dependence and infer network links

Amitava Banerjee

University of Maryland | Department of Physics and IREAP

Abstract: The general problem of determining causal dependences in an unknown time evolving system from observations is of great interest in many fields. Examples include inferring neuronal connections from spiking data, deducing causal dependences between genes from expression data, discovering long spatial range influences in climate variations, etc. Previous work has tackled such problems by consideration of correlations, prediction impact, or information transfer metrics. Here we propose a new method that leverages the ability of machine learning to generalize from examples, combined with concepts from dynamical systems theory. We test our proposed technique on numerical examples obtaining results that suggest excellent performance for a large range of situations. An important, somewhat surprising, conclusion is that, although our rationale is based on noiseless deterministic systems, dynamical noise can greatly enhance our technique's effectiveness.

October 24, 2019

Observations of Laminar Chaos

Rajarshi Roy

University of Maryland | Department of Physics and IREAP and IPST

Abstract: TBD

October 31, 2019

A putative mechanism for implicit learning in biological and artificial neural systems

Zhixin Lu

University of Pennsylvania | School of Engineering and Applied Science

Abstract: The human brain is capable of diverse feats of intelligence. A particularly salient example is the ability to implicitly learn dynamics from experiencing the physical world. Analogously, artificial neural systems such as reservoir computing (RC) networks have shown great success in learning the long-term behavior of various complex dynamical systems from data, without knowing the explicit governing equation. Regardless of the marked differences between biological and artificial neural systems, one fundamental similarity is that they are essentially dynamical systems that are fine-tuned towards the imitation of other dynamical systems. To shed some light on how such a learning function may emerge from biological systems, we draw inspiration from observations of the human brain to propose a first-principles framework explicating its putative mechanisms. Within this framework, one biological or artificial dynamical system, regardless of its specific composition, implicitly and adaptively learns other dynamical attractors (chaotic or non-chaotic) by embedding the dynamical attractors into its own phase space through the invertible generalized synchronization, and imitates those attractors by sustaining the embedded attractors through fine-tuned feedback loops. To demonstrate this general framework, we construct several distinct neural network models that adaptively learn and intimate multiple attractors. With these, we observe and explain the emergence of five distinct phenomena reminiscent of cognitive functions: (i) imitation of a dynamical system purely from learning the time series, (ii) learning of multiple dynamics by a single system, (iii) switching among the imitations of multiple dynamical systems, either spontaneously or driven by external cues, (iv) filling-in missing variable from incomplete observations of a learned dynamical system, and (v) deciphering superimposed input from different dynamical systems.

November 7, 2019

Nonlinear and quantum phenomena in whispering-gallery mode crystalline resonators

Yanne Chembo

University of Maryland | Department of Electrical and Computer Engineering and IREAP

Abstract: Whispering-gallery mode (WGM) resonators are disks, toroids or spheres with micro- or millimetric radius and (sub-)nanometer surface roughness. They have the capability to trap laser light by total internal reflection for a duration higher than a microsecond. In these ultra-high Q resonators, the small volume of confinement, high photon density and long photon lifetime ensures a very strong light-matter interaction, which may excite the WGMs through various nonlinear effects, namely Kerr, Raman, or Brillouin. Quantum phenomena such as twin-photon generation, entanglement, and squeezing can also occur in these optical cavities. In this talk, we discuss some of the main challenges related to the understanding of nonlinear and quantum phenomena in WGM resonators, and present as well as some of the principal applications in aerospace and communication engineering.

November 14, 2019

Data-Assisted Forecasting of Chaotic Dynamical Systems using Partial State Measurements

Jaideep Pathak

University of Maryland | Department of Physics

Abstract: We consider the problem of data-driven forecasting of chaotic dynamical systems when the available data is from a sparse spatial sampling, i.e., the full state of the dynamical system cannot be observed directly. Recently, there have been several promising data-driven approaches to forecasting of chaotic dynamical systems using machine learning. Particularly promising among these are hybrid approaches that combine machine learning with a knowledge-based model, where a machine learning technique is used to correct the imperfections in the knowledge-based model. Such a hybrid approach is promising when a knowledge-based model is available but is imperfect due to incomplete understanding of the physical processes in the underlying dynamical system. However, previously proposed data-driven forecasting approaches assume knowledge of the full state of the dynamical system. We seek to relax this assumption by using a data assimilation technique along with Machine Learning in a novel technique that improves forecasts. We demonstrate that using partial measurements of the state of the dynamical system we can train a machine learning model to correct model error in an imperfect knowledge-based model.

November 21, 2019

Cybersecurity Applications of Chaos

Andrew Pomerace

Potomac Research LLC

Abstract: In this talk, I will describe experiments on a chaotic electronic circuit that can be used as a high speed true random number generator. This circuit can be modified to act as a Physically Unclonable Function, which are novel cybersecurity devices used for device authentication, tamper-proofing, and key generation.

November 28, 2019

Thanksgiving Break - No seminar

December 5, 2019

Scaling and universality in bistable optical cavities

Said Rodriguez

AMOLF | Interacting Photons (Group Leader)

Abstract: Driven nonlinear dynamical systems can reside in two steady states at a single driving condition. This feature, known as bistability, is associated with emergent phenomena in phase transitions, scaling, and universal behavior. In descriptions of bistable systems, it is typically assumed that the nonlinear force responsible for bistability acts instantaneously on the system. In addition, the role of quantum fluctuations on bistability was until recently largely assumed to be irrelevant to experiments. In this talk, I will present two experiments where these two assumptions were challenged. Both of these experiments were based on nonlinear optical cavities driven by light, but similar physics is expected in other systems. The experiments we performed consisted of scanning a driving parameter (e.g. laser intensity or frequency) across an optical bistability at various speeds, and analyzing the resultant dynamic optical hysteresis. Intriguingly, both quantum fluctuations and non-instantaneous interactions lead to a universal power law decay of the hysteresis area as a function of the scanning speed. However, whereas quantum fluctuations lead to universal scaling behavior in the limit of slow scans, non-instantaneous interactions lead to a universal scaling behavior in the limit of fast scans. I will conclude with perspectives for realizing lattices of bistable optical cavities, and the opportunities that these open for performing analog computation and for studying stochastic nonlinear dynamics with light.

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