**Prof. M. Lakshmanan**, Bharathidasan U.

### Integrability and Chaos in Simple and Complex Systems

#### April 25, 2022

**Abstract:** In my talk, I will present a broad overview of some of the fascinating collective dynamical states which arise in the study of integrable and nonintegrable, including chaotic, nonlinear systems of simple and complex types, and illustrate them with specific examples. These include degenerate and nondegenerate solitons, breathers, rogue waves, bullets, and vortices in nonlinear dispersive systems, and desynchronized states, synchronized states, clusters, chimeras, traveling and solitary waves, chimera death states, and so on in complex nonlinear dissipative dynamical systems. Suitable characterizing measures such as strength of inhomogeneity, discontinuity measure, strength of mixed synchronization symmetry, etc. will be introduced to describe the latter states. Some applications in nonlinear optical systems, magnetic spin systems/spintronics, nonlinear electronic circuits, and mechanical systems including Mathews- Lakshmanan oscillator, Liénard type nonlinear oscillator, and Murali-Lakshmanan-Chua circuit will be touched upon.

**Prof. Muthusamy Lakshmanan** specializes in the areas of Nonlinear Dynamics/Theoretical Physics with special reference to solitons, nonlinear evolution equations, and chaos. He has made varied and in-depth contributions to the general theory of solitons, integrable systems, magnetic and optical solitons, and classical chaos including bifurcations, control, synchronization, and secure communications, as well as quantum chaos and spatiotemporal patterns. Known for his research on nonlinear dynamics and for the development of *Murali-Lakshmanan-Chua (MLC) Circuit*, Professor Lakshmanan is a Fellow of all the three Academies of Science of India and is an elected Foreign Member of the Royal Academy of Sciences, Uppsala, Sweden. He is also an elected Fellow of the Academy of Sciences of the Developing Countries (FTWAS 2009).

Photo credits: https://en.wikipedia.org/wiki/Muthusamy_Lakshmanan

**Prof. Katharina Krischer**, TU Munich

Between Synchrony and Turbulence: The Rich Dynamics of Globally Coupled Stuart-Landau Oscillators

#### March 28, 2022

**Abstract**: Ensembles of coupled oscillators are a class of seemingly simple dynamical systems that can take on a rich variety of emergent behaviors and have provided insights in virtually every discipline, from the natural sciences to sociology. In many studies, weak coupling between the oscillators has been assumed. In this case, the dynamics of an individual oscillator can be approximated by the evolution of its phase only, and emergent behavior is studied in the framework of coupled phase oscillators, such as the famous Kuramoto model. In this talk, Prof. Krischer will consider situations where the phase approximation breaks down, and the dynamics of the ensemble is instead captured by coupled Stuart-Landau oscillators. First, Prof. Krischer will focus on 2-cluster states and discuss that all possible 2-cluster states emerge from a codimension-2 bifurcation, the so-called cluster singularity, which acts as organizing center for the saddle-node bifurcations creating the 2-cluster states as well as for transverse bifurcations alternating their stability. When coupled nonlinearly, two-cluster states can further differentiate in cluster-splitting cascades, leading to multi-cluster states of widely different sizes. These states may then collide with mirror images in phase space, destroying some of the clusters and rendering the dynamics chaotic. A cascade of such symmetry-increasing bifurcations eventually produces a completely incoherent state in which each oscillator has its own asynchronous dynamics. The penultimate state of this cascade consists of one large synchronized cluster and otherwise only individual oscillators, i.e., a so-called chimera state. Chimera states have received much attention during the last decade. In our example, it is thus just one state of two cascades of other coexistence states that link the synchronous state to a state of complete incoherence.

**Prof. Katharina Krischer** investigates fundamental aspects of self-organization and pattern formation under non-equilibrium conditions. Her work includes experiments, which mainly focus on nonlinear phenomena at the solid/liquid interface, and theory. In the theoretical work she bridges the gap between system specific models and normal form type approaches. Since 2002 she is a professor of physics at the Technical University of Munich, Germany, and has

coauthored about 130 publications in peer reviewed journals as well as a text book on ‘Physics of Energy Conversion’. She was elected a fellow of the International Society of Electrochemistry and is a member of the German Physical Society (DPG) and the Society of German Chemists (GDCh).

Photo credits: https://www.ph.tum.de/about/diversity/gender/?language=en

**Dr. Louis Pecora**, NRL Washington (with Dr. Thomas L. Carroll)

Statistics of Attractor Embeddings in Reservoir Computing

#### February 28, 2022

**Abstract: **The question why a reservoir computer (RC), driven by only one time series from a drive system, can be trained to recreate all dynamical time series signals from the drive leads to the idea that the RC must be recreating the attractor from the drive signal, i.e. creating an embedding of the drive attractor in the RC dynamics. There have been some mathematical advances that move that argument closer to a general theorem. However, for RCs constructed from actual physical systems like interacting lasers or analog circuits, the RC dynamics may not be known well or known at all. And many of the existing embedding theorems have restrictive assumptions on the dynamics. We first show that the best way to analyze RC behavior is to first treat it properly like a dynamical system, which it is. This will lead to some conflict with existing ideas about RCs, but also a clarification of those ideas. Secondly, in the absence of complete theories on RCs and attractor embeddings, we show several ways to analyze the RC behavior to help understand what underlying processes are in place, especially regarding good embeddings of the drive system in the RC. We show that a statistic we developed for other uses can help test for homeomorphisms between a drive system and the RC by using the time series from both systems. This statistic is called the continuity statistic and it is modeled on the mathematical definition of a continuous

function. We show the interplay of dynamical quantities (e.g. Lyapunov exponents, Kaplan- Yorke dimensions, generalized synchronization, etc.) and embeddings as exposed by the continuity statistic and other statistics based on ideas from nonlinear dynamical systems theory.

**Dr. Lou Pecora** is currently a research physicist at the Naval Research Laboratory, Washington, DC, where he heads the section for Magnetic Materials and Nonlinear Dynamics in the Materials and Sensors branch. He received his B.S. degree in physics from Wilkes College and he then received a PhD from Syracuse University in Solid State Science in 1977. In the same year, he was awarded an NRC postdoctoral fellowship at the Naval Research Laboratory where he worked on applications of positron annihilation techniques in determining electronic states in copper alloys. This led to a permanent position at NRL. In the mid-1980’s Dr. Pecora moved into the field of nonlinear dynamics in solid-state systems. Subsequent work has focused on the applications of chaotic behavior, especially the effects of driving systems with chaotic signals and coupling nonlinear dynamical systems in complex networks. This has resulted in the discovery of synchronization of chaotic systems, control and tracking, and dynamics of many coupled, nonlinear systems. His research interests turned to quantum chaos briefly and then to the collective behavior of oscillators in large complex networks which led to the development of the master stability function and, more recently, the use of techniques of computational group theory to study cluster synchronization. Recently, he has expanded his collective behavior research to driven networks of oscillators called reservoir computers. Dr. Pecora has published over 160 scientific papers and has 5 US patents for the applications of chaos. His original paper on the synchronization of chaotic systems has over 14000 (ISI) citations and is the 11th most cited paper ever in Physical Review Letters. In 1995 he received the Sigma Xi award for Pure Science for the study of synchronization in chaotic systems. He is also a Fellow of the American Physical Society (APS) and of the American Association for the Advancement of Science (AAAS). Recently, he and his colleague, Tom Carroll won a Clarivate Citation Award as Clarivate Laureates “Researchers of Nobel Class” by Clarivate (citation index provider) in 2020.

Photo credits: U.S. Naval Research Laboratory

**Prof. Steven Brunton**, U. Washington

### Machine Learning for Scientific Discovery, with Examples in Fluid Mechanics

#### January 31, 2022

**Abstract:** This work describes how machine learning may be used to develop accurate and efficient nonlinear dynamical systems models for complex natural and engineered systems. Prof. Brunton will explore the sparse identification of nonlinear dynamics (SINDy) algorithm, which identifies a minimal dynamical system model that balances model complexity with accuracy, avoiding overfitting. This approach tends to promote models that are interpretable and generalizable, capturing the essential “physics” of the system. He will also discuss the importance of learning effective coordinate systems in which the dynamics may be expected to be sparse. This sparse modeling approach will be demonstrated on a range of challenging modeling problems in fluid dynamics, and Prof. Brunton will discuss how to incorporate these models into existing model-based control efforts. Because fluid dynamics is central to transportation, health, and defense systems, he will emphasize the importance of machine learning solutions that are interpretable, explainable, generalizable, and that respect known physics.

**Dr. Steven L. Brunton** is a Professor of Mechanical Engineering at the University of Washington. He is also an Adjunct Professor of Applied Mathematics and Computer Science, and a Data Science Fellow at the eScience Institute. Steve received B.S. in Mathematics from Caltech in 2006 and Ph.D. in Mechanical and Aerospace Engineering from Princeton in 2012. His research combines machine learning with dynamical systems to model and control systems in fluid dynamics, biolocomotion, optics, energy systems, and manufacturing. He is a co-author of three textbooks, received the University of Washington College of Engineering junior faculty and teaching awards, the Army and Air Force Young Investigator Program (YIP) awards, and the Presidential Early Career Award for Scientists and Engineers (PECASE).

Photo credits: https://www.me.washington.edu/facultyfinder/steve-brunton

**Prof. Edward Ott**, U. Maryland

### Prediction of Chaotic Dynamical Systems using Machine Learning

#### November 29, 2021

**Prof. Edward Ott** will discuss the use of machine learning for predicting the future evolution of dynamical systems, including systems that are very large, complex, and chaotic. He will explain reservoir computing, the basic machine learning method used in this talk. Following that, he will illustrate prediction on simple systems, hybrid prediction combining machine learning with physical knowledge, and a parallel configuration for treating large spatiotemporally chaotic systems. Illustrations and recent progress on applications to terrestrial weather and climate prediction will be presented.

Professor Ott’s current research is on the basic theory and applications of nonlinear dynamics. Some of his current research projects are in wave chaos, dynamics on large interconnected networks, chaotic dynamics of fluids, and weather prediction. Professor Ott is a fellow of the American Physical Society, the Institute of Electrical and Electronics Engineers, and the Society for Industrial and Applied Mathematics (SIAM). He is the recipient of the APS Julius Edgar Lilienfield Prize for 2014.

Photo credits: https://www.ae-info.org/ae/Member/Ott_Edward

**Prof. Kazuyuki Aihara**, U. Tokyo

### Data Analysis on Critical Transitions in Complex Systems and its Application to Early Precision Medicine

#### October 25, 2021

**Prof. Kazuyuki Aihara** reviewed his group’s recent studies on DNB (Dynamical Network Biomarkers) that provide early warning signals of imminent bifurcation from a healthy state to a disease state through a pre-disease state. He also explained the possible application of DNB for early precision medicine.

Kazuyuki Aihara received a B.E. degree in electrical engineering and Ph.D. degree in electronic engineering from the University of Tokyo (UTokyo), Tokyo, Japan, in 1977 and 1982, respectively. He led the ERATO (Exploratory Research for Advanced Technology) Aihara Complexity Modelling project by JST (Japan Science and Technology Agency) from 2003 to 2008 and the FIRST Innovative Mathematical Modelling project by JSPS (Japan Society for the Promotion of Science) through the FIRST (Funding Program for World-Leading Innovative R&D Science and Technology) program from 2010 to 2014 designed by CSTP (Council for Science and Technology Policy). Currently, he is University Professor and Professor Emeritus of UTokyo, Deputy Director at the International Research Center for Neurointelligence (IRCN) at UTokyo, and Project Manager of the Moonshot project by the cabinet office of the Japanese government on “Comprehensive Mathematical Understanding of the Complex Control System between Organs and Challenge for Ultra-Early Precision Medicine.”

Photo credits: https://ircn.jp/en/mission/people/kazuyuki_aihara

**Prof. Jürgen Kurths**, PIK Germany

### Exploring Predictability of Extreme Climate Events via a Complex Network Approach

#### September 27, 2021 – 04:00 pm (Central European Time)

Earth is a complex system whose dynamics involve innumerous interactions and multiple feedbacks. This makes predictions and risk analysis of very strong, and sometimes extreme events such as floods, landslides, heatwaves, earthquakes, etc., a challenging task. Here, I will introduce a recently developed approach via complex networks to analyze strong climate events. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? Towards this, we propose a method to reconstruct and analyze a complex network from observational and reanalysis data. This approach enables us to uncover relations to global and regional circulation patterns in oceans and atmosphere, which leads to substantially better predictions of high-impact phenomena, in particular of the Indian Summer Monsoon, El Nino events, droughts in the central Amazon, extreme rainfall in the Eastern Central Andes, and the Pacific decadal oscillation. I argue that network-based approaches can significantly complement numerical modeling for better predictions of the extreme weather events and lead to a better understanding of meteorological data.

**Prof. Jürgen Kurths** is Senior Advisor at Research Department for “Complexity Science” at Potsdam Institute for Climate Impact Research (PIK) as well as Professor and Senior Advisor at Humboldt University Berlin.

Photo credits: https://www.pik-potsdam.de/members/kurths/homepage