Phase space analysis in complex systems - From quantum dynamics to turbulent flows

Recently, the term explainable AI became known as an approach to produce models from artificial intelligence which allow interpretation. Since a long time, there are models of symbolic regression in use that are perfectly explainable and mathematically tractable: in this contribution we demonstrate how to use symbolic regression methods to infer the optimal control of a dynamical system given one or several optimization criteria, or cost functions. In previous publications, network control was achieved by automatized machine learning control using genetic programming. Here, we focus on the subsequent analysis of the analytical expressions which result from the machine learning. In particular, we use AUTO to analyze the stability properties of the controlled oscillator system which served as our model. As a result, we show that there is a considerable advantage of explainable models over less accessible neural networks.

The interaction between shear driven turbulence and thermal stratification is a key process in a wide array of geophysically relevant flows that spans many orders of magnitude in spatiotemporal scales. Recently it has been suggested that quick numerical model prediction and synthesis of in situ data are needed to gain a better understanding of the interaction between shear production and scalar flux at varying scales (Thompson et al, 2017, Oceanography). Here, we use the resolvent framework (McKeon & Sharma, 2010, JFM) to investigate the effects an active scalar has on incompressible wall-bounded turbulence. We obtain the state of the flow system by applying the linear resolvent operator to the nonlinear terms in the governing Navier-Stokes equations with the Boussinesq approximation, thereby extending the formulation to include the scalar advection equation with the scalar component acting in the wall-normal direction in the momentum equations (Dawson et al., 2018, AIAA). The stable stratification strength in the system is augmented using the friction Richardson number $J_\tau$ and is shown to have a strong effect on the resultant velocity response mode shape and phase. As $J_\tau$ is increased we observe a decrease in mode width and distance from the wall, these results match observations in experiments and simulations. The change in mode phase difference across the critical layer with increasing stratification strength has implications on the nonlinear forcing between modes and on the energy transfer between mode components at varying scales. By utilising the resolvent framework, this study allows us to move towards understanding the effect of an active scalar on the near-wall regeneration cycle, consequently helping to model and extract important spatiotemporal processes and features of oceanic and atmospheric flows.

The motion of a quantum particle induced in a periodic potential by a propagating plane wave can generate rich dynamics. Such systems include, but are not limited to, acoustically driven miniband superlattices (SLs) or driven cold atoms in an optical periodic potential, which exhibit a very anisotropic band structure that results into pronounced nonlinearities. Recently, we theoretically discussed the related acoustoelectric effects in the semiconductor superlattices by means of a non-linearized kinetic model. Remarkably, the average drift velocity of electrons demonstrates strong nonmonotonic dependence on the amplitude of the acoustic wave [1]. Furthermore, the propagating deformation potential can induce quasi-periodic Bloch oscillations of miniband electrons [2]. The appearance of the the above transport effects are associated with bifurcations (dynamic instabilities) developing with an increase of the wave amplitude. These global bifurcations dictate specific topological transformation of the system dynamics in its phase space. Here we present a clear physical interpretation of these topological rearrangements which, in fact, relates to an emission of SL phonons by supersonic electrons, and their back action on the electrons. In particular, these phonons demonstrate features of Smith-Purcell radiation (radiation which is emitted when a superluminal particle propagates through a spatially resonant structure [3]). The underlining radiation mechanisms involve electronic transitions in the potential well formed by the acoustic deformation potential and can be understood as direct generalization of the Ginzburg-Frank-Tamm superluminal effects [3]. Finally, we demonstrate that the accompanying deterministic instabilities are able to provide promising applications in acoustoelectronics, including THz gain of electromagnetic radiation. REFERENCES [1] A. Apostolakis, M.K. Awodele, K.N. Alekseev, F.V. Kusmartsev, A.G. Balanov, Phys. Rev. E 95 062203 (2017). [2] M. Greenaway, A. Balanov, D. Fowler, A. Kent, T. Fromhold, Phys. Rev. B 81 235313 (2010). [3] VL Ginzburg, Physics-Uspekhi 39 10 (1996). The author acknowledges support by the Czech Science Foundation (GAČR) through grant No. 19-03765.

Several recent papers presented exact time-periodic solutions in shear flow simulations at moderate Reynolds numbers. Although some of these studies demonstrated similarities between turbulence and the unstable periodic orbits, whether one can utilize these orbits for turbulence modeling remained unclear. We argue that this can be achieved by measuring the frequency of turbulence's visits to the periodic orbits. To this end, we adapt methods from computational topology and develop a metric that quantifies shape similarity between the projections of turbulent trajectories and periodic orbits. We demonstrate the utility of our method by applying it to one-dimensional Kuramoto--Sivashinsky equation and three-dimensional Navier--Stokes equations under sinusoidal forcing.

Quantum chaotic interacting N-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼ log N. Here, we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing ħ=τ, again given by τ ∼ log N. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasiperiodic recurrences indicating reversibility.

The Ott-Antonsen theory [1] is a powerful tool for describing the low-dimensional collective dynamics of populations of coupled oscillators or (more generally) phase elements. Recently, the ‘circular cumulant’ approach [2,3] was introduced and applied for description of the macroscopic dynamics of populations, which do not obey the OA theory [4,5]. In [2,4], a weak additive intrinsic noise was reported to slightly affect the OA solution and make it attracting also for identical oscillators. The latter invigorate the employment of the original OA theory for populations of identical oscillators, as noise is unavoidably present in real-life systems. In this work, we apply the circular cumulant approach for thermodynamically large populations of phase elements, where the OA properties are violated by a multiplicative intrinsic noise. The infinite cumulant equation chain is derived for the case of a sinusoidal sensitivity of the phase to noise. Two-cumulant model reductions, which serve as a generalization of the Ott-Antonsen ansatz, are reported. The accuracy of these model reductions and the macroscopic (low-dimensional) collective dynamics of the system are explored for the case of a Kuramoto-type global coupling. The Ott-Antonsen ansatz and the Gaussian approximation are found to be NOT uniformly accurate for non-high frequencies. Thus, we demonstrate that the Ott-Antonsen ansatz can be inaccurate for populations of identical elements with vanishing multiplicative noise. [1] E. Ott, T.M. Antonsen, Chaos 18, 037113 (2008). [2] I.V. Tyulkina et al., PRL 120, 264101 (2018). [3] D.S. Goldobin, A.V. Dolmatova, PRRes 1, 033139 (2019). [4] D.S. Goldobin et al., Chaos 28, 101101 (2018). [5] I. Ratas, K. Pyragas, PRE, 100, 052211 (2019). [6] The work was financially supported by the RSF (grant no. 19-42-04120).

Complex systems composed of a large number of degrees of freedom canoften be described on the macroscopic level by only a few interacting coherentstructures. A paradigmatic example for this behavior is the vortex sheddingbehind a cylinder, which was analysed in depth by several modal decomposi-tion techniques. In general many turbulent flows seem to be composed of a fewcoherent structures and it might be possible to find a low dimensional repre-sentation of the flow. In most cases these models cannot be derived a priori bygoverning equations and therefore data-driven methods have to be established.While most reduced order modelling is based on separate estimation of modesand models, we will present a novel framework to extract both at the same time.This new method is based on ideas from [1] and [2]. In contrast to POD basedmethods we work purely data-driven and with nonorthogonal modes.The nonlinear mode decomposition is applied to the canonical cylinder wake.We derive a two and a three-dimensional model capable of correctly modellingthe underlying physics.References[1] F. Kwasniok. The reduction of complex dynamical systems using principalinteraction patterns.Physica D: Nonlinear Phenomena, 92(1,2):28 – 60,1996.[2] C. Uhl, R. Friedrich, and H. Haken. Reconstruction of spatio-temporalsignals of complex systems.Zeitschrift f ̈ur Physik B Condensed Matter,92(2):211–219, 1993.

The transition to turbulence in wall-bounded shear flows such as pipe or channel flow does not occur through a linear instability of the laminar profile. Instead, it is connected with transiently occurring localised turbulent regions that either relaminarise or proliferate. Predicting these relaminarisation events and understanding precursors thereof are of interest for practical applications such as flow control or drag reduction. However, relaminarisation of localised turbulence is a memoryless process with an associated exponential distribution of times spent in the turbulent state, which renders the prediction thereof inherently difficult. The increasing performance of deep learning algorithms shows that these approaches are well suited for high-dimensional classification problems. Here, we use such a deep learning machine learning model to predict relaminarisation events a certain amount of time steps ahead in the future. As a first step and for proof of principle, we analyse turbulent trajectories obtained from numerical simulations of a low-dimensional shear flow model [1] that describes the transition to turbulence and the fundamental dynamical processes of turbulence close to onset. Apart from data reduction, the advantage of this model over direct numerical simulations in this context is the physical interpretability in terms of the identification of relaminarisation precursors. Formulating the task as a classification problem, we compute the prediction performance of a deep neural network. Snapshots of short turbulent trajectories are used as input for the neural network, for which we find that the prediction performance increases for shorter trajectories. The trained classifier is evaluated in terms of its false positives and negatives. To assess the performance of the classifier in the fluid dynamical context, it is applied in real-time to a running simulation in order to obtain live predictions of relaminarisation events. [1] Moehlis, Jeff, Holger Faisst, and Bruno Eckhardt. "A low-dimensional model for turbulent shear flows." New Journal of Physics 6.1 (2004): 56.

Strongly inclined layer convection at Pr=1.07 gives rise to spatially localised intermittent bursts of intense convection. We show how invariant solutions capturing the pattern of these bursts emerge in snakes and ladders bifurcations.

Turbulent flows over spanwise heterogeneous rough surfaces are known to generate secondary motions of Prandtl’s second kind. Even though the strength of these secondary motions amount only a few percent of the mean streamwise momentum they significantly alter the properties of the primary flow. For instance, in a turbulent channel flow with streamwise aligned ridges the secondary motion are observable in the mean velocity field with an upward motion above the ridges and downward motion in the valley. A recent study with spanwise varying roughness has shown, that depending on the mean roughness height a reversal of the flow direction of the mean secondary motion can occur [1]. Besides these observations, the exact physical mechanism which determines the flow direction of the secondary motion is still an ongoing debate. Additionaly, the relation of the secondary motion seen in the time- and spatial-averaged field to instantaneous flow phenomenas is not yet fully understood as well. Our investigation is based on direct numerical simulation of turbulent channel flow with fully resolved spanwise heterogeneous roughness at the walls. By varying the roughness type, for example rectangular ridges or fully resolved roughness patches, and its associated geometrical parameters the simulation data allow us to identify the most relevant parameters for the formation of secondary motions. In order to identify the flow structures with the highest energy content the proper orthogonal decomposition (POD) was applied for secondary motion over streamwise aligned ridges [2]. It was shown that roughly 20 modes are required to capture half of the turbulent kinetic energy, however the first six POD modes are already sufficient to capture the large-scale secondary motion. This reduced order model is able to capture a large amount of the Reynolds stresses and was confirmed by comparison with experimental data. Future investigations based on the dynamic mode decomposition (DMD) are intended to reveal the contribution of instantaneous flow structures and dynamical processes to the formation of secondary motions. Keywords: Secondary Motions, Proper Orthogonal Decomposition, Turbulent Channels, Heterogeneous Roughness References [1] Stroh, A., Schäfer, K., Frohnapfel, B., Forooghi, F., Rearrangement of secondary flow over spanwise heterogeneous roughness J. Fluid. Mech., 885, 2020. [2] Vanderwel, C., Stroh, A., Kriegseis, J., Frohnapfel, B., Ganapathisubramani, B., The instantaneous structure of secondary flows in turbulent boundary layers J. Fluid. Mech., 862:845–870, 2019. K. Schäfer, A. Stroh, J. Kriegseis and B. Frohnapfel, Institute of Fluid Mechanics, Karlsruhe Institute of Technology, Germany. email:kay.schaefer@kit.edu P. Forooghi, Department of Engineering - Aarhus University, Denmark.

Motile cilia on ciliated epithelia in mammalian airways, brain ventricles and oviduct can display coordinated beating in the form of metachronal waves, presumably due to mutual hydrodynamic coupling. Metachronal coordination is important for efficient fluid transport. How the shape of the cilia beat determines the direction and wavelength of metachronal waves is not fully understood, nor is robustness with respect to noise. We developed a multi-scale modeling approach, where a cilia carpet is modeled as an array of noise phase oscillators, similar to a Kuramoto model with local coupling. Importantly, pair-wise hydrodynamic interactions between cilia are accurately computed from hydrodynamic simulations of the Stokes equation, using experimentally measured cilia beat patterns. We numerically determine the set of all possible synchronized states, as well as their linear stability. Remarkably, while we find multiple metastable metachronal wave states, analysis of global dynamics reveals that only few of them have sizable basins of attraction. In the presence of noise, corresponding to active fluctuations of cilia beating, we observe stochastic transitions between different synchronized states. While strong noise reduces synchronization, weak noise biases the dynamics towards a single synchronized state of metachronal coordination. References: - G. Klindt, et al. PRL 117.25, 258101 (2016) - R. Ma, et al. PRL 113.4, 048101 (2014) - B.M. Friedrich, EPJ ST 225.11-12, 2353-2368 (2016)

On the example of various dynamical systems, including autonomous and non-autonomous flows, discrete maps, it is shown that the occurrence of hyperchaos is associated with the development of the Shilnikov chaotic attractor containing a saddle-focus with a two-dimensional unstable manifold. The appearance of a saddle-focus with a two-dimensional unstable manifold occurs as a result of a sequence of Neimark-Sacker bifurcations. The presence of secondary Neimark-Sacker bifurcations can lead to the emergence of a hierarchy of chaotic and hyperchaotic attractors.

The ‘circular cumulant’ approach, recently introduced in [I.V. Tyulkina et al., PRL 120, 264101 (2018)], allows one to deal with low-dimensional collective dynamics of populations of phase elements (directional elements or limit-cycle oscillators), where the applicability conditions of the Ott-Antonsen ansatz are slightly violated. The small inertia in phase elements is one of the most important cases of such violation. In this work, the internal noise and inertia are introduced into the classical “Abrams’ system” [D.M. Abrams et al., PRL 101, 084103 (2008)] with an analytically solvable example of chimera states. A closed system of equations is derived for the dynamics of the order parameters of subpopulations. The effect of small inertia on the system and its interaction with intrinsic noise are studied on the basis of both the low-dimensional equation system and the Fourier expansion of the probability density. For no or small inertia, noise makes the periodic chimeras attracting; however, strong enough inertia overcomes the effect of noise and attracting chimeras turn into quasiperiodic ones. These bifurcations in the system dynamics and the increase of its embedding dimension can be tracked with the low-dimensional model derived with the circular cumulant approach. The work was financially supported by the RSF (grant no. 19-42-04120).

We apply the weak formalism on the Boussinesq equations to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy fluxes in very well-resolved numerical simulations. The local Bolgiano–Oboukhov (BO) length is investigated and it is found that its value may change by an order of magnitude through the domain, in agreement with previous results. We then investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman–Howarth–Monin and Yaglom equations. We have not found the classical BO picture, but evidence of a mixture of BO and Kolmogorov scalings. In particular, all the energy fluxes are compatible with a BO local H{\"o}lder exponent for the temperature and a Kolmogorov 41 for the velocity. This behaviour may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of BO local scales. The scale-by-scale analysis allows us also to compare the theoretical BO length computed from its definition with that empirically extracted through scalings obtained from weak analysis. Scalings are observed, but over a limited range. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties."

In the Vicsek model, identical particles move at constant speed, locally aligning their velocities (or phases) in the presence of noise. It is now well-known that collective motion, i.e. global phase synchronisation, arises for weak-enough noise, even in 2D. The statistically homogeneous ordered phase, well described by the Toner and Tu theory, is separated from the disordered gas phase by a regime characterized by high-density high-order bands traveling in a residual gas. We have investigated the robustness of the ordered phases of the Vicsek model to the introduction of chirality disorder. Each particle is now endowed with its own chirality (frequency), drawn from some distribution, much like in the Kuramoto model. We show, using both particle-based simulations and hydrodynamic theories, that the Toner-Tu phase is unstable to the presence of any level of chirality disorder, and that, on the other hand, the traveling bands can sustain a finite amount of disorder.

We study the impact of different boundary conditions of the velocity and temperature fields on the large-scale structure formation in turbulent Rayleigh-Bénard convection. The investigations are based on direct numerical simulations of Rayleigh-Bénard convection in a large aspect ratio domain. In detail, we combine no-slip / stress-free boundary conditions of the velocity field with constant temperature / constant flux boundary conditions of the temperature field. The constant flux boundary condition case reveals the formation of a single large convection cell, once the simulations are run sufficiently long (~ 1e4 convective free-fall time units).