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.