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Granular matter can behave ‒ depending on the circumstances ‒ like a solid, a fluid or a gas, yet often with an unexpected twist owing to the special nature of this complex, multi-particle medium. Here we present two case studies:

1.Vibrated sand: a counter-intuitive gas. If granular matter is shaken vigorously, the particles fly about, forming a kind of gas. Unexpectedly, however, they do not spread out uniformly over the available space as a standard gas would do, but they tend to cluster together. This spontaneous breakdown of equipartition shows up in a particularly clear-cut fashion when the space is divided in two compartments, as in the so-called “Maxwell’s Demon” experiment. We describe this experiment in terms of Dynamical Systems theory and show how the clustering transition manifests itself as a pitchfork bifurcation.

2. Flowing sand: roll waves as stick-slip oscillations. Our second case study focuses on granular matter flowing down a chute, and more specifically, on the roll wave patterns frequently encountered in this type of flow. These nonlinear waves consist of long rising flanks followed by abrupt falls and can adequately be described by the generalized Saint-Venant equations for shallow granular flows. More surprisingly, as we will demonstrate, they can also be seen as the Fluid Dynamical analogue of the famous stick-slip phenomenon found in many mechanical systems.

Understanding the intricate complexity and dynamics of the human brain is fundamental to elucidating the mechanisms underlying both normal cognitive function and the complex pathophysiology of neurological and psychiatric disorders. Electroencephalography (EEG) and functional Magnetic Resonance Imaging (fMRI) stand as the primary non-invasive neuroimaging modalities, offering complementary perspectives on brain activity. EEG provides a direct measure of neural electrical activity with high temporal resolution, capturing rapid changes in brain states, while fMRI indirectly assesses neural activity through hemodynamic responses, yielding high spatial resolution and the ability to visualize deep brain structures. This report synthesizes recent breakthroughs in both EEG and fMRI techniques, alongside the biomarkers identified using these methods, for studying brain complexity and dynamics in healthy individuals and those with various neurological and psychiatric conditions. The advancements in high-density EEG systems and sophisticated signal processing, the emergence of mobile EEG for naturalistic data acquisition, and the progress in EEG-based Brain-Computer Interfaces are discussed. Furthermore, the report highlights the impact of advanced analytical methods such as dynamic functional connectivity and the transformative role of Machine Learning (ML) with paradigms from my work on cognitve disorders and human – autonomous car interactions. By examining the established and emerging EEG and fMRI biomarkers across a spectrum of brain states and disorders, this report underscores the crucial role of these neuroimaging techniques in advancing our understanding of brain complexity and dynamics and their potential for clinical translation in diagnosis and treatment monitoring.

A deep understanding of the mechanisms behind chaos generation in Bohmian trajectories is essential for addressing a fundamental problem in Bohmian Quantum Mechanics (BQM): the dynamical origin of Born’s Rule (BR). BQM is a well-known interpretation of Quantum Mechanics in which quantum particles follow deterministic trajectories governed by the so-called Bohmian equations of motion. BQM provides deep insights into quantum phenomena from both theoretical and experimental perspectives. It is a highly nonlocal quantum theory, where quantum entanglement (QE) plays a central role in the evolution of Bohmian trajectories. While BR is postulated as an axiom in standard quantum theory, in BQM one can, in principle, begin with an initial particle distribution that does not satisfy BR. This raises a critical question: is BR dynamically accessible from arbitrary initial conditions, and if so, what mechanisms govern this process?…

We introduce the notion of integrability of partial difference equations in two independent variables and how it is related to a consistency relation of maps of a specific type. As a prototypical example, we present the  quations that describe non-relativistic elastic collision of two particles in one dimension. Extending these  quations to an arbitrary associative algebra, relativistic elastic collision equations turn out to be a particular case. Furthermore, we show that these equations can be reinterpreted as difference systems defined on the Z^2 graph. Finally, if time permits, we will show how this reinterpretation relates the linear and the non-linear approach of discrete analytic functions.

We investigate the key factors that shape the dynamic evolution of Day-Ahead spot prices of seven European interconnected electricity markets (Austria, Hungary, Slovenia, Romania, Bulgaria, Greece and Italy), with emphasis on their price surges and discrepancies during the period 2022-2024, that challenge the reliability and efficiency of the European target model. The high differences in the prices of the two groups, has generated political reactions from the countries that ‘suffer’ from these price discrepancies, expressed with different ways (e.g. a noticed reaction is the letter of the Greek Prime Minister sent to European Commission President). To  ‘reveal’ the whole path of surging prices (from north to south), we employ combination of Machine Learning (ML) approaches in learning the causal structure of this phenomenon. Local, causal structures learning (LCSL) and Markov Blanket (MB) learning are combined to ‘lift the blanket’ that covers the ‘true structure’ of the path of causalities, responsible for the price disparity. Markov Blanket Learning is useful for identifying key fundamental variables but should be combined with causal structure learning to uncover true causes of price surges. Finally, we compute the correlation curves of rolling volatility of spot prices as well as of cross-border transfer availabilities (CBTA) identified as crucial factors by MB and LCSL, of all markets, to study their volatility spillover (a tool to detect the entire path of volatility propagation from the upstream to downstream SEE countries)…

In the present lecture we briefly review several methods of temporal and non-linear time series analysis, mainly based on phase space reconstruction such as recurrence plots Quadrant Scan, as well as complex network transformed time series based on the visibility algorithm. We discuss the main characteristics of the methods and the insight they can provide of the underlying physical, engineering systems as well as in other systems sch as financial time series, with special focus on system identification and transition detection, event detection, and spatial variation. We present and discuss applications from magnetohydrodynamics and turbulent flows as well as river systems and car flow incident detection financial data.

In the analysis of multivariate time series, the first objective is the estimation of the connectivity structure of the observed variables (or subsystems), where connectivity is also referred to as inter-dependence, coupling, information flow or Granger causality. Depending on the type of analysis one wants to pursues, also indicated by the size of the data, one selects a connectivity measure to estimate the driving-response connections among the observed variables. For example, if the multi-variate time series is very short, one would rather use a linear measure of bivariate (Granger) causality, or even the linear cross-correlation. On the other extreme of a very long multivariate time series, one would prefer to use a nonlinear and even multivariate measure of causality, where multivariate measure is considered a measure that for the estimation of a driving-response relationship of two of the observed variables, the other observed variables are also considered. When the measure is computed for all directed pairs of observed variables, a complex network is formed, called also connectivity or causality network, where the nodes are the observed variables, and the connections are the estimated inter-dependences. For a network with binary connections the interdependences are discretized to zero (not significant) and one (significant) by applying a criterion for the significance, e.g., arbitrary threshold or statistical testing.

Determining the chaotic or regular nature of orbits of dynamical systems is a fundamental problem of nonlinear dynamics, having applications to various scientific fields. The most employed method for distinguishing between regular and chaotic behavior is the evaluation of the maximum Lyapunov exponent (MLE), because if the MLE>0 the orbit is chaotic. The main problem of using this chaos indicator is that its numerical evaluation may take a long -and not known a priori- amount of time to provide a reliable estimation of the MLE’s actual value. In this talk we will focus our attention on two very efficient methods of chaos detection: the Smaller (SALI) and the Generalized (GALI) Alignment Index techniques. We will first recall the definitions of the SALI and the GALI and will briefly discuss the behavior of these indices for conservative Hamiltonian systems and area-preserving symplectic maps. Then, we will explain how one can use these methods to investigate the dynamics of time-dependent dynamical systems, and we will discuss the applicability of these indicators to dissipative systems. Furthermore, we will present some recently introduced methods to estimate the chaoticity of orbits in conservative dynamical systems from computations of Lagrangian descriptors on short time scales.

Οι σπειροειδείς βραχίονες στους μεγάλους κανονικούς σπειροειδείς γαλαξίες είναι κύματα πυκνότητας που δεν αποτελούνατι πάντα από τα ίδια αστέρια. Θα εξηγήσουμε την θεωρία του κύματος πυκνότητας για την περίπτωση αναλυτικών δυναμικών που προσομειώνουν μεγάλους κανονικούς σπειρεοειδείς γαλαξίες. (grand design galaxies). Θα εξερευνήσουμε τον χώρο των φάσεων σε ένα γαλαξιακό μοντέλο που προσομειώνει τις σπείρες του δικού μας Γαλαξία με σκοπό τον αριθμητικό εντοπισμό των ευσταθών
περιοδικών τροχιών που υποστηρίζουν αυτά τα σπειροειδή κύματα πυκνότητας. Θα δείξουμε ότι οι τροχιές αυτές είναι ελλειπτικές με κύριο άξονα που αλλάζει προσανατολισμό καθώς απομακρυνόμαστε από το κέντρο του γαλαξία και η υπέρθεσή τους σε όλες τις ακτίνες δημιουργεί το σπειροειδές κύμα. Επίσης, θα εξηγήσουμε τη θεωρία εύρεσης προσεγγιστκών αναλυτικών λύσεων των περιοδικών αυτών τροχιών κατασκευάζοντας την Χαμιλτονιανή σε κανονική μορφή (normal form construction) με την βοήθεια των σειρών Lie.

Disk galaxies are complex systems where stars, gas, and dust evolve within dark matter halos through gravitational interactions. Among their most prominent features are bars and spiral arms, whose formation and structure are shaped by nonlinear dynamical processes. Observational data, theoretical orbital studies using analytic potentials, and fully self-consistent N-body simulations collectively indicate that the spiral arms are mainly two-dimensional structures. On the other hand, bars comprise two structural elements: an elongated outer “slim” region and a more compact, vertically extended central “thick” component. To gain insight into how these structures form and persist, we examine the nature of stellar orbits within galactic disks. The orbital behavior in both two- and three-dimensional Hamiltonian systems that model rotating barred potentials provides the foundation for interpreting the observed morphologies of barred-spiral galaxies. Due to the strong departure from axisymmetry introduced by bars and spirals, nonlinear dynamical processes are essential in understanding their evolution. Stellar orbits provide the key to understanding these features. Two-dimensional (2D) orbital models help explain the dynamics of spiral arms and outer bars, where both regular and chaotic orbits may play structural roles. However, to explore the vertically extended, central “thick” parts of bars—often observed as boxy or peanut-shaped (b/p) bulges in edge-on galaxies—three-dimensional (3D) models are essential. These bulges are supported by specific 3D families of periodic orbits that bifurcate from the central planar x1 family at vertical resonances. Although many of these families exhibit complex instability, orbits in their vicinity can still contribute to building the vertical structure, especially through sticky behavior in phase space. This talk will present the key orbital mechanisms supporting bars and spirals, with a focus on how certain orbit families shape b/p bulges and the characteristic X-shaped structures embedded within them.