Dynamical Methods in Data-based Exploration of Complex Systems

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of "Cole-Hopf-type" transformations for dynamics.

In this investigation we use sophisticated machine learning techniques on a network of temperature and precipitation time series taken from gridded stations throughout Germany for the years 1960-2018. In particular, we consider maximized mutual information as the measure of similarity and expand on recent clustering methods for climate modeling by applying a kernel-based k means algorithm. We find robust regional clusters that are shared by networks defined separately by precipitation and temperature time series. Finally, we use the resulting clusters to create a nonstationary model of regional summer temperature extremes throughout Germany. We find that the probability of observing high extreme summer temperature values (>35^{\circ}C), when compared to the last 30 years, has significantly increased.

We developed a new perception model to simulate multistable perception. The model is based on selective adaptation which periodically destabilises one of the perception states. Additive brain noise makes the switches between coexisting perception states random. Although the effects of selective adaptation are well documented in experimental reports, there is a lack of the underlying mechanism that captures the essence of this phenomenon maintaining a minimum number of state variables and assumptions, and having a clear biological correspondence. Keeping this in mind, our perceptual model implies two competitive perceptions which consistently stabilise and destabilise themselves depending on time scales and noise. In our model, each perceptive state has a slowly varying memory state which is connected so as to provide a negative feedback loop with delay. The number of switches between different perception states is found to be monotonically increasing with noise. The dominance time or the duration of a particular perceptual state maximises at a certain intermediate level of the duty cycle of a biased stimulus which prefers that state. The decreasing dominance results from the selective adaptation which causes self-destabilisation. For small duty cycles, increasing brain noise leads to a decrease in the dominance times due to noise-dependent switches. On the other hand, for large duty cycles, where selective adaptation forces are keeping longer biased stimulations from increasing the dominance, increasing brain noise causes an increase in the dominance. This apparently contradictory result can be explained by the fact that the slowly adapting memory cannot follow fast random variations of the perceptual signal due to much a larger time scale of the memory. This leads to decreasing selective adaptation; the higher duty cycle of the biased stimulus makes the dominance higher. The distribution of dominance times has been widely reported to follow a gamma function in humans. Our magnetoencephalography (MEG) experiments with subjects observing an ambiguous Necker cube flickering image with its two perceptional interpretations yield similar results. The results of the simulations with our model are in a good agreement with our MEG experiments. The simulations reveal that increasing brain noise tends to change the distribution of dominance times from Gaussian to Gamma and then finally to Exponential. The range of noise values in our model corresponding to Gamma distribution should only be used to compare the model predictions and experimental observations. In the same range of brain noise, we observe a monotonic decrease in the dominance time with increasing brain noise, both in the model and experiment. The results of this work provide interesting and valuable insights into the human brain. This simple yet versatile mechanism may turn out to be radical in understanding human perception. Keywords: perception, selective adaptation, brain, noise, MEG.

We know that the reliable logical response can be extracted from a noisy bistable system at an intermediate value of noise strength when two random, two-level, square waveform serve as the inputs. The asymmetry of the potential plays a very important role and dictates the type of logical operation, such as OR or AND, exhibited by the system. Here we have shown that one can build logic gates with symmetric bistable potential if the two states of the double-well are thermalized with two different heat baths. We have found that if a given state is kept at a sufficiently low temperature compared to the other, the system shows one type of logic behavior (say, OR). Interestingly, the system's response turns into the other kind (say, AND) if the temperature of the initial low-temperature well is increased slowly and the quality of the logical response first improves and then becomes weak after passing through a maximum at a particular value of temperature. However, the reliability of the second kind of logical response(AND) is not as good as the first kind (OR) and it depends on the amplitude of the inputs. Still one can construct both types of logic gates with maximum reliability by properly choosing the initial low-temperature well.

In this poster we will present a collection of intuitive, easy-to-use and performant software packages for nonlinear dynamics and chaos, which compose the fully open source GitHub organization "JuliaDynamics". All implementations were made from the ground up, based on the principles of clarity and intuition, taking full advantage of new programming paradigms. Examples of features that we will show include Lyapunov exponents, categorizing chaotic behavior, attractor dimensions, recurrence plots, Poincare sections, feature-full billiard evolution, spatiotemporal timeseries prediction and much more. Importantly, obtaining the result for any of these features requires typing typically 5-10 lines of code (examples will be shown on the poster). The software that we will present are: - DynamicalSystems.jl: Award winning software library for exploring dynamical systems - DynamicalBilliards.jl: Easy-to-use, modular and extendable software for dynamical billiards in two dimensions - TimeseriesPrediction.jl: Prediction of timeseries using methods of nonlinear dynamics and timeseries analysis - InteractiveChaos: Interactive applications for the exploration of chaos and nonlinear dynamics. All these software are written in Julia, a relatively new computer programming language that has an impressive list of features: dynamic, intuitive syntax, multiple dispatch, performance approaching C/FORTRAN and being open source.

Embryonic stem cells are derived from the early blastocyst-stage embryos. Through differentiation programs they can generate every cell type in the body. These differentiation programs are usually driven by extracellular signals. We are interested in how the information carried by these external inputs is encoded by the cell in signaling networks activity. We quantitatively analyze signaling dynamics in individual mouse embryonic stem cells in response to an extracellular stimulus. These time series display some dynamic events that look like pulses of signaling activity, interspersed with intervals of noise. We aim to distinguish noise fluctuations from genuine dynamical activity.To this end we developed a set of local observables employing statistical physics and information theory concepts. Together with interpeak intervals statistics, we explore how the extracellular stimulus concentrations correlate with signaling dynamic signatures.

Deep polar ice cores provide us with long, detailed accounts of Earth's ancient climate system: tens to hundreds of thousands of years’ worth of perspective and clues about temperature, accumulation rates, volcanic activity, and more. But this length and precision come at a high cost. Collection of an ice core is very expensive and extraction of proxy data from it is time consuming---as well as susceptible to both human and machine error. Ensuring the accuracy of these data is as challenging as it is important. Many of these cores are only fully sampled one time and most are unique in the time period and region that they ``observe," making comparisons and statistical tests challenging and outliers difficult to identify. However, recent advances in information theory and ice-core measurement technology have provided us the means to begin tackling this problem. In this talk, we demonstrate that estimates of the Shannon entropy rate of the water-isotope data from the West Antarctica Ice Sheet Divide ice core, calculated using permutation entropy techniques, identify regions of the core that merit further investigation. To date this approach has flagged regions containing missing ice, data-post processing errors, instrumentation irregularities, signatures of geothermal heating and identified several intervals in the data that may be of direct relevance to paleoclimate interpretation---including periods of abrupt climate change.

Phase oscillators evolving by a Moebius transformation of the unit circle according to the Watanabe-Strogatz theory in principle cannot contract in phase space to form more than one cluster. In numerical simulation, however, Kuramoto-Sakaguchi phase oscillators under common multiplicative noise, which alone leads to the synchronization of the oscillators, and under repulsive coupling, which alone leads to the desynchronization of the oscillators, have been shown to form stable multicluster states. Using Fokker-Planck formulation, we show that two-cluster states under such a model are non-attractive. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. Monitoring the induced change in certain integrals of motion we quantify these errors. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function, as well as additional amplitude dynamics, multiclusters can occur naturally. Thus that common noise can induce clustering in general oscillator systems under repulsive coupling remains valid.

In recent years, data-driven discovery of mathematical models has emerged as a promising alternative to more traditional modeling approaches. While promising, existing approaches have several major weaknesses. Most notably, they break down for data with high levels of noise and have to be tuned empirically to produce meaningful results, making them ill-suited for analyzing experimental data. We show how these weaknesses can be addressed using a combination of sparse regression and a weak formulation of the model PDE. The weak formulation has substantial freedom that makes it quite powerful and flexible, but a question arises how this freedom can be used to robustly obtain the most accurate model. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show that the approach can be optimized in the limits of low and high noise. In particular, we derive the scaling that relates the accuracy of the model, the parameters of our method, and the properties of the data such as its spatial and temporal resolution or the level of noise.

Superextreme events in Parametrically and Externally Excited Liénard system S. Leo Kingston Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland. In this poster, we presented the effect of the time-varying parameter in the forced Liénard system. The system exhibits superextreme events which denote that the sudden expansion of the system state variables more than the standard extreme events qualifier threshold. In the absence of time-varying parameter, the system is in the chaotic or periodic state. When we include the time-varying parameter into the system, superextreme events have emerged. The influence of parametric perturbation is studied in the different parameters of the model system. We explored its dynamical origin and emerging mechanism using different characterization techniques. References: 1. S. L. Kingston, K. Thamilmaran, P. Pal, U. Feudel, and S. Dana, Phys. Rev. E 96, 052204 (2017). 2. C. Bonatto and A. Endler, Phys. Rev. E 96, 012216 (2017). 3. S. Albeverio, V. Jentsch, and H. Kantz, Extreme events in nature and society (Springer,2006). 4. X. Han, Q. Bi, P. Ji, and J. Kurths, Phys. Rev. E 92 (2015). 5. A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2, 011015 (2012).

The continuously increasing number of newly discovered worlds outside of our own solar system requires as precise as possible parameter estimations such as planetary masses, orbital characteristics, bulk density, etc. Comprehensive statistical methods and inverse dynamical analyses have been worked out to obtain system parameters from astronomical observations. Nevertheless, the time domain measurements as scalar time series transformed into complex networks serve a powerful tool to investigate dynamical systems via network topology. Many recent works make significant effort to explore the causality relations and coupling directions between connected dynamical systems. In this study a new estimation procedure of planetary masses is presented making use of eclipse time variation in multi-planetary systems. Due to the gravitational coupling the motion of planets differs from pure Keplerian ellipse resulting in variable orbital periods. Measuring this tiny effect for nearly co-planar planets one is able to reconstruct the trajectories sharing the same phase space. Transforming then the obtained state vectors of the entangled dynamical systems into network representation, it can be shown that the coupling directions between the interacting sub-networks are related to planetary masses relative to each other.

Real world dynamic networks are open systems receiving inputs from their environment. There can be many origins for these inputs, including systems malfunction, attacks or model errors. In general, these inputs affect the state of the dynamic system and need to be taken into account for state estimation, data assimilation or prediction. However, in many cases not even the state nodes of the system targeted by these unknown inputs are known. In this poster, we present criteria to decide, whether the state nodes targeted by unknown inputs can be identified from output measurements. In cases were the exact localisation of the inputs is impossible, we present algorithms to localise the region in the dynamic network, where the unknown input applies. Our results provide a principled way for error localisation in complex dynamic networks with potential applications in many areas including network biology, engineering and communications. Authors: Dominik Kahl, Philipp Wendland and Maik Kschischo Department of Mathematics and Technology, University of Applied Sciences Koblenz, Joseph-Rovan-Allee 2, 53424 Remagen, Germany

Experimental study of networks of coupled oscillators attracts the attention of many researchers. Convenient objects for the experimental investigation of collective dynamics in networks are coupled electronic oscillators. Such networks have been experimentally studied for various electronic oscillators in the nodes and various types of coupling. However, the experimental study of network dynamics becomes a difficult problem if it is necessary to examine complex networks with a large number of nodes and a large number of couplings between them. This problem is even more difficult in the case of complex coupling functions and a necessity to change the coupling strength within the experiment. We propose an approach to the experimental study of complex networks of coupled electronic oscillators, which makes it possible to assign an arbitrary architecture and type of couplings between the oscillators. At the first step, we digitize the analog signals of all electronic oscillators using a multichannel analog-to-digital converter. Then, the required type of coupling between the oscillators is realized in a program way. At last, the signals that are responsible for the coupling of oscillators are converted to analog form using a multichannel digital-to-analog converter and fed to the inputs of oscillators. Using this approach, we constructed an experimental setup, which allows us to implement almost any type of coupling (linear coupling, nonlinear coupling, delayed coupling, etc.) between the electronic oscillators. The setup operates in real time and provides a possibility to tune the coupling strength if necessary. We experimentally studied the collective dynamics in different complex networks of coupled electronic oscillators. The processes of synchronization and the formation of chimera states in complex networks are experimentally examined. This study was funded by the grant of the President of Russian Federation, MK-1199.2019.8.

We analyze temperature autocorrelation functions from European station data with the method of detrended fluctuation analysis (DFA). This method is known to give a scaling exponent indicating long range correlations in time for temperature anomalies i.e. deviations from the climatological average on each calendar day. By a more careful look at the fluctuation function we are able to explain the emergent scaling behavior by short time relaxation, the yearly cycle and one additional process. Obtaining such a model of characteristic timescales is possible due to recent progress in theoretical understanding of DFA. It turns out that for many stations the inter-annual variability can be described as an oscillatory mode with a period length of approximately 7-8 years, which is consistent with results of other methods. We discuss the spatial patterns in the parameters of our model and validate the finding of the 7-8 year period by comparing stations with and without this mode.

A flux-controlled memristor model is introduced into an existing 5D hyperchaotic autonomous system. The dynamics of the new system obtained is investigated in terms of equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, phase portraits, time series and attraction basins. The system is hyperchaotic under a line or a plane of equilibria. Other attractive dynamics observed are: hidden extreme multistability, transient chaos, bursting and offset boosting phenomena.

A 6D hyperchaotic system generates hidden attractors with the unusual feature of having plane and line equilibrium under different parameter conditions. Its dynamical behaviors are characterized through sets of chaos indicators. Appropriate values of parameters lead to rich nonlinear dynamics such as limit cycles, quasi-periodicity, chaos, hyperchaos, bursting and hidden extreme multistability.

The effect of delay on a network of indirectly coupled Euler's beams is analyzed in this work. The system consists of a number of hinged-hinged beams indirectly interconnected to piezoelectric patches in a parallel conformation. A stability analysis is provided in order to give the threshold values of the parameter space for the onset of unstable motion. It is found that the stable region is reduced as the time delay increases. Disturbance-induced by time-delay on the synchronization state and the strong amplitude reduction (SAR) state are also presented. It is conventionally known that delay induces instability in coupled systems, but we demonstrated here that this delay can also contribute to stabilize these systems by synchronizing them.

Dilute ensembles of granular matter (granular gases) are nonlinear systems which exhibit fascinating dynamical behavior far from equilibrium, including unusual cooling properties, clustering and violation of energy equipartition. So far, most studies have been theoretical or numerical, and only few experiments, mainly in two dimensions (2D), have been performed. Falcon et al. showed dynamical clustering in a first sounding rocket experiment [1], where no analysis on the grain scale level was possible. Currently, an instrument is being prepared for the International Space Station [2]. The experimental realization of low excitation or cooling regimes of granular gases in particular requires microgravity of high quality, e.g. on suborbital rocket flights or in drop towers [3]. Another important issue is the trackability of particles in large ensembles in 3D. In gases of rod-like particles, the mean free path is substantially reduced as compared to gases of spherical grains of identical volume fraction [4]. One particular problem in the data analysis is the reliable detection and tracking of the rods in 3D, especially at volume fractions beyond the very dilute limit. Up to now, the experimental data have been analyzed mostly manually. We developed a software for automatic 3D tracking of position and orientation of elongated particles in the ensemble, based on two-perspective video data analysis. Two-dimensional localization of particles is performed with help of the Mask R-CNN neural network [5]. Then, the problem of 3D matching of the particles is solved by minimization of the total reprojection error. Finally, the particle trajectories are tracked and ensemble statistics can be extracted. Depending on the required accuracy, the system can be used fully automatically or serve as a base for subsequent manual correction. The approach can be extended to other 3D and 2D particle tracking problems. References [1] É. Falcon et al., Cluster Formation in a Granular Medium Fluidized by Vibrations in Low Gravity, Phys. Rev. Lett., 83 (1999), 440-443 [2] M. Noirhomme et al., Threshold of gas-like to clustering transition in driven granular media in low-gravity environment, EPL, 123 (2018), 14003 [3] K. Harth, T. Trittel, K. May, S. Wegner and R. Stannarius, Three-dimensional (3D) experimental realization and observation of a granular gas in microgravity, Advances in Space Research, 55 (2015), 1901 – 1912 [4] K. Harth, T. Trittel, S. Wegner, and R. Stannarius, Free cooling of a granular gas of rodlike particles in microgravity, Phys. Rev. Lett., 120 (2018), 214301 [5] K. He, G. Gkioxari, P. Dollár, and R. B. Girshick, “Mask R-CNN”, CoRR, https://arxiv.org/abs/1703.06870 (2017)

Deep Reinforcement Learning (DRL) is applied to control a nonlinear, chaotic system governed by the one-dimensional Kuramoto-Sivashinsky (KS) equation. DRL provides a framework for the the determination of optimal control solutions by approximating the value function and the control policy using deep Neural Networks. In this work, we show that model-free DRL controllers, used in combination with localized actuators and sensors, are capable of stabilizing the dynamics of the KS system around its unstable fixed solutions, here considered as target states. We believe that the performance/robustness of the defined policy in controlling the KS system pave the way for the application of RL methods in control of complex systems.

Recent developments in understanding the structure and dynamics of networks have transformed research in many fields, however, the Geosciences have not benefited much from this conceptual framework. In this presentation, we use river deltas as a case study to illustrate the potential of an integrated approach that relies on network theory, remote sensing and modelling tools to understand and predict the structure and dynamics of geomorphic systems. By studying river deltas through the lens of their channel networks, we show how we can make significant strides toward solving the inverse problem of inferring process from form, establish a methodology to compare, contrast and classify river deltas, and understand and predict their dynamic response under different forcing scenarios. We also present new results that extend the single network theory to multilayer networks for studying geomorphic systems which have more than one connectivity structure emerging from the simultaneous action of multiple processes characterized by different time scales of transport.

The auto-covariance function of a stationary time series contains information of all oscillatory and relaxational time scales of the dynamical system the time series is taken from. For non-stationary time series instead of using an ensemble or infinite time average we propose to use an exponential time average with sufficiently long averaging time scale in order to calculate a time-varying auto-covariance function and from that via a Pade approximation the poles in the negative complex half plane corresponding to the relaxation spectrum of the time series. Slow changes in the relaxation spectrum may indicate slow changes in the underlying dynamics and we propose to use this analysis to detect early warning signs for a change of stability in the system equilibrium. We present a set of linear filters in form of simple ODEs which can be used to directly generate the Fourier-Laplace transform of the time series auto-covariance function in real time. This method can also be extended to a wavelet-like frequency analysis. A subsequent reduction of the time-frequency data via Pade approximation and complex root finding gives the finite time averaged relaxation spectrum.

Heteroclinic networks provide a promising candidate attractor to generate reproducible sequential series of metastable states. While from an engineering point of view it is known how to construct heteroclinic networks to achieve certain dynamics, a data based approach for the inference of heteroclinic dynamics is still missing. Here, we present a method by which a template system dynamically learns to mimic an input sequence of metastable states. For this purpose, the template is unidirectionally, linearly coupled to the input in a master-slave fashion, so that it is forced to follow the same sequence. Simultaneously, its eigenvalues are adapted to minimize the difference of template dynamics and input sequence. Hence, after the learning procedure, the trained template constitutes a model with dynamics that are most similar to the training data. Our approach may thus be applied to infer the topology and the connection strength of a heteroclinic network from data in a dynamic fashion.