# 2018 Applied Dynamics Seminar Archives

### February 1, 2018

#### Reviewing artificial intelligence and the book Life 3.0

University of Maryland | Department of Mathematics

*Abstract: TBA*

### February 8, 2018

#### Experiments with arbitrary networks in time-multiplexed delay systems

University of Maryland | Department of Physics ; IREAP

*Abstract: Complex networks of coupled oscillators have proven to be systems that can display incredibly rich dynamical behaviors. Despite great theoretical advances in our understanding of coupled oscillator networks, it has proven difficult to design experiments that permit the study of dynamics on large networks with arbitrary topology. Here we present a new experimental approach that allows for the investigation of large networks of truly identical nodes with arbitrary topology. Our approach relies upon the space-time interpretation of systems with time delay in order to construct a network of coupled maps using a single nonlinear, time-delayed feedback loop. This system has many advantages: the network nodes are truly identical, the network is easily reconfigurable, and the network dynamics occur at high speeds. We use this system to study cluster synchronization and chimera states in both small and large networks of different topologies.*

### February 15, 2018

#### The spherical Couette system: simple yet complex

Johns Hopkins University | Department of Earth & Planetary Sciences

*Abstract: The spherical Couette system consists of two concentric spheres rotating differentially about a common axis. The space in between the spheres is filled with a conducting fluid. It is a relatively simple system without any thermal or density stratification and and has potential applications to planetary and stellar interiors. In addition, it is also an extremely interesting fluid dynamical system displaying a host of complex instabilities and other fluid dynamics phenomena. In the first part of the talk, I shall introduce the system and explore the generic regimes of different instabilities. This will be followed by some results using hydrodynamic simulations of this system with two pseudo-spectral codes MagIC and XSHELLS while comparing them with experimental data. Focus will be on the origin of special wave instabilities called 'inertial modes' and the transition to turbulence. In the next part, I will present magnetohydrodynamic simulations exploring the effect of an external magnetic field on inertial modes. In the final part of the talk, I will present simulations of self-consistent dynamo action in this system and the parameter dependence of the same.*

### February 22, 2018

#### Classical-to-quantum correspondence and transitions in chaotic dynamics of out-of-time-ordered correlators

University of Maryland | Department of Physics

*Abstract: One of the most intriguing phenomena in the studies of classical chaos is the butterfly effect, which manifests itself in that small changes in initial conditions lead to drastically different trajectories. It is characterized by a Lyapunov exponent that measures divergence of the classical trajectories. The question how/if this prototypical effect of classical chaos theory generalizes to quantum systems (where the notion of a trajectory is undefined) has been of interest for decades, but became more popular recently, when it was realized that there exist intriguing connections to string theory and general relativity in some quantum chaotic models. At the center of this activity is the so-called out-of-time-ordered correlator (OTOC) - a quantity that in the classical limit seems to approximate the classical Lyapunov correlator. In this talk, I will discuss the connection between the standard Wigner-Dyson approach to "quantum chaos" and that based on the OTOC on the example of a chaotic billiard and a disordered interacting electron system (i.e., a metal). I will also consider the standard model of quantum and classical chaos - kicked rotor - and calculate the correlator and Lyapunov exponents. The focus will be on how classical chaos and Lyapunov divergence develop in the OTOC and cross-over to the quantum regime. We will see that the quantum out-of-time-ordered correlator exhibits a clear singularity at the Ehrenfest time, when quantum interference effects sharply kick in: transitioning from a time-independent value to its monotonous decrease with time. In conclusion, I will discuss many-body generalizations of such quantum chaotic models.*

### March 1, 2018

#### Economic inequality from a statistical physics point of view

University of Maryland | Department of Physics

*Abstract: Inequality is an important and seemingly inevitable aspect of the human society. Various manifestations of inequality can be derived from the concept of entropy in statistical physics. In a stylized model of monetary economy, with a constrained money supply implicitly reflecting constrained resources, the probability distribution of money among the agents converges to the exponential Boltzmann-Gibbs law due to entropy maximization. Our empirical data analysis [1] shows that income distributions in the USA, European Union, and other countries exhibit a well-defined two-class structure. The majority of the population (about 97%) belongs to the lower class characterized by the exponential ("thermal") distribution. The upper class (about 3% of the population) is characterized by the Pareto power-law ("superthermal") distribution, and its share of the total income expands and contracts dramatically during booms and busts in financial markets. Interestingly, the same equations can be also applied to heavy-ion collisions [2]. Globally, energy consumption (and CO2 emissions) per capita around the world shows decreasing inequality in the last 30 years and convergence toward the exponential probability distribution, as expected from the maximal entropy principle. In agreement with our prediction [3], a saturation of the global Gini coefficient for energy consumption at 0.5 is observed for the most recent years. All papers are available at http://physics.umd.edu/~yakovenk/econophysics/.*

*[1] Yong Tao et al., "Exponential structure of income inequality: evidence from 67 countries", Journal of Economic Interaction and Coordination (2017) http://doi.org/10.1007/s11403-017-0211-6 http://arxiv.org/abs/1612.01624*

*[2] Xuejiao Yin et al., "A new two-component model for hadron production in heavy-ion collisions", Advances in High Energy Physics (2017) 6708581, http://doi.org/10.1155/2017/6708581*

*[3] S. Lawrence, Q. Liu, and V. M. Yakovenko, "Global inequality in energy consumption from 1980 to 2010", Entropy 15, 5565 (2013), http://dx.doi.org/10.3390/e15125565*

### March 8, 2018

#### APS March Meeting - No seminar

### March 15, 2018

#### Optimal control of networks: energy scaling and open challenges

University of New Mexico | Department of Mechanical Engineering

*Abstract: Recent years have witnessed increased interest from the scientific community regarding the control of complex dynamical networks. Some common types of networks examined throughout the literature are power grids, communication networks, gene regulatory networks, neuronal systems, food webs, and social systems. Optimal control studies strategies to control a system that minimize a cost function, for example the energy that is required by the control action.We show that by controlling the states of a subset of the nodes of a network, rather than the state of every node, the required energy to control a portion of the network can be reduced substantially. The energy requirements exponentially decay with the number of target nodes, suggesting that large networks can be controlled by a relatively small number of inputs, as long as the target set is appropriately sized. An important observation is that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. However, when the network dynamics is linearized, the linearization is only valid in a local region of the state space and hence the question arises whether optimal control can be used. We provide a solution to this problem by determining the region of state space where the trajectory does remain local and so minimum energy control can still be applied to linearized approximations of nonlinear systems. We apply our results to develop an algorithm that determines a piecewise open-loop control signal for nonlinear systems. Applications include controlling power grid dynamics and the regulatory dynamics of the intracellular circadian clock. This work is in collaboration with Isaac Klickstein and Afroza Shirin (UNM).*

### March 22, 2018

#### Spring Break - No seminar

### March 29, 2018

#### Magnetic Reconnection and Particle Energization

University of Maryland | IREAP

*Abstract: In many plasmas in nature magnetic reconnection is the primary process whereby energy stored in the magnetic field is transformed into kinetic and thermal energy. It lies at the heart of such phenomena as disruptions in fusion experiments, auroras, and solar flares. In addition, it is strongly suspected that reconnection plays a significant role in generating energetic particles (i.e., non-thermal power laws) in these and other systems. I will discuss the basic physics of magnetic reconnection as well as some of the numerical tools (e.g., particle-in-cell simulations) used to study its complex interplay of scales. Finally, I will discuss recent advances in quantifying how and under what conditions reconnection accelerates particles to non-thermal energies.*

### April 5, 2018

#### Large-scale neural network modeling: from neuronal microcircuits to whole-brain complex network dynamics

Qin Liu

University of Maryland | Department of Physics

*Abstract: Neural networks mediate human cognitive functions, such as sensory processing, memory, attention, etc. Computational modeling has proved to be a powerful tool to test hypotheses of network mechanisms underlying cognitive functions, and to better understand human neuroimaging data. Here we present a large-scale neural network modeling study of human brain visual/auditory processing and how this process interacts with memory and attention. We show how our modeling and simulation can relate phenomena across different scales from the neuronal level to the whole-brain network level. The model can perform a number of cognitive tasks utilizing different cognitive functions by only changing a task-specification parameter. Based on the performance and simulated imaging results of these tasks, we proposed hypothesis for the neural mechanisms underlying several important cognitive phenomena, which could be tested experimentally in the future.*

### April 12, 2018

#### Macroscopic Behavior of Systems of Many Interacting Orientable units: The Strong Influence of Dimensionality on the Dynamics

University of Maryland | Department of Physics and IREAP

*Abstract: The Kuramoto model, originally motivated by the dynamics of many interacting oscillators, has been used and generalized for a wide range of applications involving the collective behavior of large heterogenous groups of dynamical units whose states are characterized by a scalar angle variable. One such application in which we are interested is the alignment of velocity vectors among members of a swarm. Despite being commonly used for this purpose, the Kuramoto model can only describe swarms in 2 dimensions, and hence the results obtained do not apply to the often relevant situation of swarms in 3 dimensions. Partly based on this motivation, we study the Kuramoto model generalized to D dimensions, focusing on the 3-dimensional case. We show that in 3 dimensions, as well as for all odd dimensionality, the generalized Kuramoto model for heterogenous units has dynamics that are remarkably different from the dynamics in 2 dimensions. In particular, for odd D the transition of the time asymptotic equilibrium state to coherence occurs discontinuously as the coupling constant K is increased through zero, as opposed to the D=2 case (and, as we will show, also the case of even D) for which the transition to coherence occurs as K increases through a *postitive* critical value K _{c}. We observe that the odd-dimensional Kuramoto models with a large number of swarm elements is not low dimensional in the sense of Ott & Antonsen (2008). However, application of our generalized form of the Ott-Antonsen ansatz does reduce the complexity of the problem when compared with the full system of equations. We generalize our results beyond the Kuramoto model to a wider class of swarm dynamics in high dimensions, and we show that the Ott-Antonsen ansatz can be appropriately generalized for this class of systems. We expect that our results will hence be useful for solving questions involving coupled systems on a sphere in high dimensions, beyond just the Kuramoto model.*

### April 19, 2018

#### Turbulence closure ideas from plasma physics

University of Maryland | Department of Physics

*Abstract: First-principles turbulence simulations are expensive but very useful in many contexts. One approach to improving one's ability to predict experimental or observational data is to design algorithms for ever more processors, allowing ever higher (and more realistic) resolution. But one can also try to work in the opposite direction, developing closures to reduce the resource demands of computations. In a typical high-resolution turbulence simulation that I undertake, there are O(10**9) spectral amplitudes. Only a tiny percentage of these modes are excited to any appreciable level. I will discuss (and seek advice from the audience!) approaches we are pursuing to develop algorithms that solve a closed (or somewhat reduced) first-principles system with as little resolution as should be required. *

### April 26, 2018

#### High-speed prediction of a chaotic system using reservoir computers

Ohio State University | Department of Physics

*Abstract: A reservoir computer is an approach to machine learning that appears to be ideally suited for classifying time varying signals or as a black-box system for forecasting the behavior of a dynamical system. It consists of a recurrent artificial neural network that serves as a “universal” dynamical system into which data are input, where the connections on the input layer and recurrent links within the network are chosen randomly and held fixed. Only the weights of network output layer are adjusted during the training period, which greatly reduces the training time. I will discuss our recent progress on realizing high-speed prediction of the Mackey-Glass chaotic system (>10^8 predictions per second) using a reservoir computer based on a time-delay autonomous Boolean network realized on a field programmable gate array. I will also touch on our efforts to control a dynamical system with a reservoir computer and some recent results on methods to identify the optimum size of the reservoir computer network for a given task.*

### May 3, 2018

#### Heterogenous chaotic attractors

University of Maryland | Department of Mathematics

*Abstract: There is a saying: "there are two kinds of people in the world—the simple-minded and the muddle-headed." I prefer the former. But much of the investigation of chaotic attractors uses models that are sometimes overly simplistic at least for studying high dimensional chaotic attractors. Our goal is to produce more models that are still simplistic but better reflect typical high dimensional chaotic attractors and permit a better but still simple-minded understanding of chaos in high dimensions. This is a report on joint work with Miguel Sanjuan and Yoshi Saiki.*

### May 10, 2018

#### No seminar

### September 13, 2018

#### No seminar

### September 20, 2018

#### Statistical Description of Hamiltonian Mixed Phase Space Systems

Technion University | Department of Physics

*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*

### September 27, 2018

#### Prediction of complex spatiotemporal evolution through machine learning methods improved with the addition of observers

Department of Physics | University of Crete

*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 “help” provided by a random bath and observers, as defined through reservoir computing (RC) [1]. What about using other “standard” 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 [2]. 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 [3]. 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). [1] 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) [2] 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). [3] 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) *

### October 4, 2018

#### Due to unforeseen circumstances, we have a new speaker for this date

#### Climate model shows large-scale wind and solar farms in the Sahara increase rain and vegetation

University of Maryland | Department of Physics

*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 albedo–precipitation–vegetation 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.*

### October 11, 2018

#### Constructing Chaotic Coordinates for non-integrable dynamical systems

Princeton Plasma Physics Laboratory

*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. *

### October 18, 2018

#### Bifurcations in dynamical control systems for aerospace applications

University of Maryland | Department of Aerospace Engineering

*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. *

### October 25, 2018 - Room change: AV Williams 1147

#### A method for numerical computation starting from a quasiperiodic trajectory

George Mason University | Department of Mathematics

*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.*

### November 1, 2018

#### Neuronal coding in the insect olfactory system

University of Maryland | Department of Biology

*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.*

### November 8, 2018

#### Modeling methodologies for personal protection control strategies in vector-borne disease epidemiology: The role of diversity amplification

Dr. Jeff Demers

University of Maryland | Department of Biology

*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.*

### November 15, 2018

#### Transiently Chaotic behavior in Superconducting Metamaterials

Amitava Banerjee

University of Maryland | Department of Physics

*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).*

### November 22, 2018

#### Thanksgiving Break - No seminar

### November 29, 2018

#### Confining charged particle orbits using hidden symmetry

University of Maryland | IREAP

*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 ‘hidden symmetries’. The first, integrability, means the field lines should lie on nested toroidal surfaces, without regions of islands or chaos. The second, called `quasi-symmetry’, 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.*

### December 6, 2018

#### Invariant measures for the stochastic Navier-Stokes equations for compressible flows and the problem of Turbulence

University of Maryland | Department of Mathematics and IPST

*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 I’ll 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.*

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