Special Session 112: Nonlinear Dynamics: Methods, Models, and Applications

Generalized synchronization and its detection via recurrent neural networks

Jose M Amigo Universidad Miguel Hernandez Spain Co-Author(s):

Abstract: Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder "follows" the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of this communication, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this communication, we will discuss only the noiseless scenario, i.e., when noise is not present in the driver. To reveal generalized synchronization, we check the existence of synchronization maps using recurrent neural networks and predictability. The results demonstrate the capability of our method.

Deep learning for dynamical systems analyses

Roberto Barrio University of Zaragoza Spain Co-Author(s):    Roberto Barrio, Flavio Fenton, Alvaro Lozano, Ana Mayora-Cebollero, Carmen Mayora-Cebollero

Abstract: Dynamical systems are usually analysed using standard techniques such as Lyapunov exponents. However, most of these classical methods are computationally expensive and often not feasible for studying real-world data. In this communication, we propose to use Deep Learning to overcome such limitations. We apply Deep Learning to detect chaotic regions in the parameter space of classical dynamical systems and to analyse chaotic dynamics in biological time series like experimental frog heart data. Furthermore, we use Deep Learning to detect more complex dynamical regimes by approximating Lyapunov exponents from single-variable time series. These analyses show the potential of Deep Learning in the study of dynamical systems.

A Second-Order Unconditionally Stable Scheme for Long-Time Dynamics of Nonlinear Models

Jack L Coleman Missouri University of Science and Technology USA Co-Author(s):    Jack Coleman, Daozhi Han, Xiaoming Wang

Abstract: We propose a highly efficient second-order numerical scheme for approximating the long-time dynamics of a class of finite-dimensional nonlinear models arising in geophysical fluid dynamics. The method is unconditionally stable and requires solving only a fixed symmetric positive definite linear system at each time step. We prove that the numerical solutions remain uniformly bounded for all time and that the scheme accurately captures the long-time behavior of the underlying system. In particular, the global attractors and invariant measures of the scheme converge to those of the original model as the time step approaches zero. Numerical experiments on the Lorenz-96 system demonstrate that the scheme efficiently approximates long-time statistical properties and captures the invariant measures of the system.

A stage-structured seasonal model for the spread of flavescence dor\`ee in vineyards

Attila Denes Bolyai Institute, University of Szeged Hungary Co-Author(s):    Abel Garab, Nigussie Abeye Shiferaw

Abstract: Flavescence dor\`ee is one of the most severe phytoplasma diseases affecting grapevines, posing an increasing threat to vineyards worldwide. The bacterial agent causing the disease is spread by the American grapevine leafhopper. We present a compartmental model for the spread of flavescence dor\`ee incorporating stage structure in the vectors. To account for seasonal weather variations and the behaviour of the vectors, we consider periodic transmission, birth and death rates. We calculate the basic reproduction number as the spectral radius of a linear operator and show that it serves as a threshold parameter for disease persistence. Finally, we provide numerical simulations to assess the effect of varying key model parameters.

Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals

Omer Eryilmaz University of Birmingham England Co-Author(s):    Omer Bahadir Eryilmaz, Cihan Katar, Max A Little

Abstract: Recurrent signals generate trajectories that repeatedly return near previously visited states in state space. Analysing such data requires a principled notion of similarity that determines how local neighbourhoods are defined and scaled. This choice is also critical for topology-preserving denoising, where the aim is to reduce noise without distorting the underlying trajectory structure. We introduce a flow-aware ellipsoidal filtration for persistent homology based on a spatio-temporal covariance construction. The method estimates local flow geometry by combining temporal and spatial neighbours, and assigns an ellipsoid to each point, with orientation and axis lengths determined by local variances. In contrast to isotropic constructions such as the Vietoris--Rips filtration, this approach adapts to the directional structure of the data. When a dominant $H_1$ feature represents the main recurrent loop, its persistence interval provides a natural, data-driven scale selection rule. Experiments on synthetic and real signals show improved topology-preserving denoising and more accurate first-return-time estimation compared to isotropic filtrations.

Discrete Lyapunov functional for cyclic systems of differential equations with time-variable or state-dependent delay

Abel Garab University of Szeged Hungary Co-Author(s):    Istv\`{a}n Bal\`{a}zs

Abstract: We consider nonautonomous cyclic systems of delay differential equations (DDEs) with variable delay. Under suitable feedback assumptions, we define an integer valued Lyapunov functional related to the number of sign changes of the coordinate functions of solutions. We prove that this functional possesses properties analogous to those established by Mallet-Paret and Sell for the constant delay case and by Krisztin and Arino for the scalar case. This may serve as an efficient tool in the study of the global dynamics of DDEs with variable delays. For example, the results can be applied to cyclic systems of DDEs with state-dependent delays to obtain a Morse-decomposition of the global attractor.

Topological and Dynamical Approaches to Cardiorespiratory Data Analysis

Grzegorz Graff Gdansk University of Technology Poland Co-Author(s):    Beata Graff, Pawel Pilarczyk, Maja Elstad, Krzysztof Narkiewicz

Abstract: Physiological signals such as heart rate, blood pressure, and respiration reflect the dynamics of a complex and tightly coupled regulatory system. Their analysis therefore calls for mathematical methods capable of capturing nonlinear interactions, multiscale structure, and temporal variability. Among such approaches, ordinal pattern analysis and topological methods, including persistent homology, provide promising tools for investigating the organization of physiological data.The aim of this work is to study how these methods can be applied to the analysis of interactions within the cardiorespiratory system, with particular emphasis on the role of respiration in shaping cardiovascular dynamics. By comparing selected signal characteristics across physiological states and between healthy and pathological conditions, we seek to identify features that may indicate changes in regulatory mechanisms

Double-Partitions Homology for Coupling Complexity Analysis of Multivariate Time Series

Taichi Haruna Tokyo Woman's Christian University Japan Co-Author(s):    Taichi Haruna

Abstract: Ordinal persistent homology (OPH) and its generalizations were proposed as a framework for investigating coupling complexity in multivariate time series. Total persistence (TP), as defined in OPH, is a measure of coupling complexity in the following sense: it is zero for completely synchronized time series, takes a specific small value with high probability for asynchronous time series satisfying a certain independence condition, and can be large somewhere between the two extreme cases. TP has been successfully applied to analyze coupling complexity in several high-dimensional nonlinear systems. In this presentation, we propose an alternative, simpler homological method to study coupling complexity in multivariate time series. In this method, we consider two sufficiently long time intervals separated in time and partition the set of time series using the ordinal patterns for each time interval. We construct a simplicial complex from the two partitions by assigning a simplex to each part of the partitions. We show that the first Betti number of the simplicial complex can serve as a coupling complexity measure better than TP: it takes zero not only in the synchronous case, but also in the asynchronous case with high probability.

Sparse modeling of high dimensional time series and the role of lag causality

Dimitris Kugiumtzis Aristotle University of Thessaloniki Greece Co-Author(s):    Petros Petridis

Abstract: Machine learning and deep learning models are widely utilized for modeling and forecasting multivariate time series but they have no closed form and do not give insight onto the true dynamics. This study explores methods that elucidate the structural form of the equations representing the components of the multivariate time series and conducts a comparative analysis of their forecasting errors. Specifically, the study evaluates the performance of regression models bearing analytic form on multivariate time series of stochastic and nonlinear deterministic systems. These models are: 1) polynomial regression models, 2) the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm and 3) symbolic regression models. In addition, we consider sparse versions of the above models by restricting the input lag variables to the most informative to the response, as found by the Partial Mutual Information from Mixed Embedding (PMIME) algorithm. The models are compared with respect to multi-step ahead prediction on multivariate time series of stochastic and deterministic systems of varying dimension. The results of the simulations highlight the importance of dimensionality reduction through lag variable selection in the autoregressive models of any of the three examined types.

A Unified Ordinal-Pattern Framework for Testing Temporal Symmetries in Time Series

Giorgio Micali University of Twente Netherlands Co-Author(s):    Annika Betken, Manuel Ruiz Mar\`{i}n

Abstract: We propose a unified framework for testing temporal symmetries in time series based on the distribution of ordinal patterns. Existing ordinal-pattern approaches typically focus on specific symmetry properties, such as time reversibility, by testing equalities between selected pattern probabilities, often pairing each pattern with its reverse. These approaches implicitly induce a partition of the permutation space, restricting the scope of the analysis to predefined symmetry structures. Our framework generalizes this idea by allowing symmetry tests to be constructed from arbitrary partitions of the ordinal-pattern space, providing a flexible and systematic way to design tests for a wide range of temporal properties. Depending on the chosen partition, the proposed methodology recovers classical ordinal-pattern tests as special cases while enabling the exploration of previously unaddressed symmetry structures. We derive asymptotic results for the proposed test statistics under broad classes of stationary processes, ensuring theoretical validity. Extensive experiments on synthetic and real-world time series demonstrate that the proposed tests are highly sensitive to structural temporal asymmetries, while remaining fully data-driven, computationally efficient, and easy to implement.

Classes of multivariate ordinal patterns linked to motion

Alexander Schnurr Siegen University Germany Co-Author(s):    Svenja Fischer, Marco Oesting

Abstract: The classification of movement in space is one of the key tasks in environmental science. Various geospatial data such as rainfall or other weather data, data on animal movement or landslide data require a quantitative analysis of the probable movement in space to obtain information on potential risks, ecological developments or changes in future. Usually, machine-learning tools are applied for this task. Yet, machine-learning approaches also have some drawbacks, e.g. the often required large training sets and the fact that the algorithms are often hard to interpret. We propose a classification approach for spatial data based on ordinal patterns. Ordinal patterns have the advantage that they are easily applicable, even to small data sets, are robust in the presence of certain changes in the time series and deliver interpretative results. They, therefore, do not only offer an alternative to machine-learning in the case of small data sets but might also be used in pre-processing for a meaningful feature selection. In this talk, we introduce the basic concept of multivariate ordinal patterns, classify them and provide the corresponding limit theorem. The approach is applied to rainfall radar data.

Investigating Dynamical Similarity of Unknown Systems through Time Series and Samples of Data

Justyna Signerska Gdansk University of Technology Poland Co-Author(s):

Abstract: A key tool in comparing and classifying dynamical systems is the notion of topological conjugacy. In this talk, we consider the problem of testing (semi-)topological conjugacy between two trajectories arising from unknown dynamical systems when only finite samples of these trajectories are available. We will discuss a range of tests inspired by methods from topological data analysis and present various numerical examples demonstrating their effectiveness. Moreover, we will show that these methods can be applied to determine the optimal embedding dimension, referring to the well-known Takens` Embedding Theorem, which provides a theoretical framework for time series analysis in a wide range of applications. The talk is based on joint work with Pawel Dlotko (Dioscuri Centre in TDA), Michal Lipinski (IST Austria), and Marta Marszewska (Gdansk University of Technology and Dioscuri Centre in TDA).

Disease transmission on networks - when is a hypergraph not a hypergraph

Michael Small University of Western Australia Australia Co-Author(s):    Eugene Tan, Shannon D. Algar

Abstract: Contact networks are a paradigmatic model for understanding disease transmission in heterogenous populations. Typical methods approach this via an averaging over nodes of the same degree. We (with Moore and Wang) have show that a modified concept of network dimension can achieve a better prediction of behaviour including endemic state in SIS models. Recent work has shifted attention to transmission on hypergraphs. In this setting the higher order connection are introduced to model higher order interaction process and introduce additional.complication to the analysis. We show that the dynamics observed in these higher order models is always equivalent to dynamics on a lower-order (network) model with time varying disease parameters. While the hypergraphs may be better suited to modelling certain situations they do not introduce dynamical behaviour not present in lower order models.