Complex Dynamical Systems Research Articles

Mapping the self in self‐regulation using complex dynamic systems approach

Complex dynamic systems offer a rich platform for understanding the individual or the person‐specific mechanisms. Yet, in learning analytics research and education at large, a complex dynamic system has rarely been framed, developed, or used to understand the individual student where the learning process takes place. Individual (or person‐specific) methods can accurately and precisely model the individual person, create person‐specific models, and devise unique parameters for each individual. Our study used the latest advances in complex systems dynamics to study the differences between group‐based and individual self‐regulated learning (SRL) dynamics. The findings show that SRL is a complex, dynamic system where different sub‐processes influence each other resulting in the emergence of non‐trivial patterns that vary across individuals and time scales, and as such far from the uniform picture commonly theorized. We found that the average SRL process does not reflect the individual SRL processes of different people. Therefore, interventions derived from the group‐based SRL insights are unlikely to be effective in personalization. We posit that, if personalized interventions are needed, modelling the person with person‐specific methods should be the guiding principle. Our study offered a reliable solution to model the person‐specific self‐regulation processes which can serve as a ground for understanding and improving individual learning and open the door for precision education. Practitioner notesWhat is already known about this topic Self‐regulation is a catalyst for effective learning and achievement. Our understanding of SRL personalization comes from insights based on aggregate group‐based data. What this paper adds Every student has their own unique SRL process that varies from the average in non‐trivial ways. We offer a credible method for mapping the individualized SRL process. SRL dynamics vary across time scales. That is, the trait dynamics are different from the state dynamics, and they should be supported differently. Implications for practice and/or policy Personalization can best be achieved if based on unique person‐specific idiographic methods. Supporting learning and SRL in particular can be more efficient when we understand the differences across time scales and persons and apply insights accordingly. The general SRL average should not be expected to work for everyone.

Design of Car Control Algorithm Based on Deep Reinforcement Learning

Reinforcement learning algorithm is widely used in the field of agent control because of its high robustness and strong adaptability. The problem of Mountain-Car-v0 is a classic control problem, which attracts the attention of many researchers. Traditional control methods, for example, PID control, are widely used in industrial and robot control fields, but they show obvious limitations in the face of such highly nonlinear and unknown environments. Therefore, this paper proposes a car control algorithm based on deep reinforcement learning, aiming at overcoming the shortcomings of traditional methods and realizing efficient control of complex dynamic systems. By combining the automatic extraction of high-dimensional features of deep learning with the dynamic decision-making mechanism of reinforcement learning, an adaptive control algorithm is designed, which can continuously learn and optimize in the interaction with the environment without establishing an accurate system model in advance. The experimental results show that compared with traditional control methods and other intelligent control strategies, the proposed algorithm shows higher efficiency and better adaptability in the task of Mountain-car-V0, and obviously improves the success rate of the Mountain-Car-v0.

AI-Aristotle: A physics-informed framework for systems biology gray-box identification.

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework-named AI-Aristotle-combines the eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility, and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. To test the performance of AI-Aristotle, we use sparse synthetic data perturbed by uniformly distributed noise. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.

Time delay recurrent neural networks via PID control law to complex dynamic systems

This paper presents an extension to the theory on the analysis of stability of complex systems with time delay of fractional order; the previous study is based on the theory of functions of Lyapunov–Krasovskii and the fractional-order PID control law. The analytical results are illustrated by the simulation of fractional nonlinear systems interconnected with time delay which are forced to follow complex trajectories as chaotic systems.

Frequency emergency control strategy in power systems considering the participation of energy storage clusters

Recently, the power systems with a high penetration of renewables and power electronics have come into being. In these power systems, complex system dynamics, emergency faults, and insufficient frequency regulation reserve pose threats to system frequency stability. Based on the clustering development of energy storage, to ensure the system frequency stability when emergency faults occur, this paper proposes a decentralized frequency emergency control (FEC) strategy considering the participation of energy storage clusters (ESCs). First, the overall framework of the optimal-droop-based FEC strategy is introduced, which achieves the coordination between FEC and conventional frequency regulation strategies. Second, to appropriately allocate the unbalanced power among the generators and ESCs, a general design method for optimal droop coefficients is proposed, which is applicable to various control objectives. The optimality of the droop coefficients is rigorously proven. This case study is carried out on an electromagnetic transient simulation platform, and the simulation results verify the effectiveness and optimality of the proposed FEC strategy.

Decoding low-frequency climate variations: A case study on ENSO and ocean surface warming

Current perspectives on lower frequency variations and secular warming have predominantly been shaped by traditional anomalies that assume an annual cycle (AC) with a time-invariant amplitude. However, this anomaly framework falls short in capturing the complexity of multiple periodic modes with intricate waveforms and time dependent amplitude—the traits, in general, shared by externally forced responses of complex dynamical systems. By allowing interannual amplitude modulation of ACs, we show that the monotonic amplitude increases of the first AC of Sea Surface Temperature (SST) are manifested as the basin-wide secular warming of the ocean surface. Notably, the first SST-AC exhibits significant interannual variances and the largest linear warming rates in the Pacific Warm Pool. While the linear warming pattern mirrors that of a long-term mean SST, it depicts an entirely different warming pattern on the surface of the tropical Pacific Ocean compared to those reported so far. Moreover, all interannual warm (El Niño) and cold (La Niña) events in tropical Pacific regions are abnormal interannual modulations in the third and fourth ACs of SST, respectively. Specifically, a strong El Niño event occurs when a positive amplitude modulation leads to the overlap of two consecutive positive phases of the third AC of SST. Conversely, the absence of such overlaps during negative amplitude modulations significantly contributes to the positive skewness of SST anomalies. No systematic decadal changes in the zonal propagation characteristics of SST in the eastern Pacific (EP) and central Pacific (CP) regions were detected. These findings underscore that the secular warming and low-frequency events in EP and CP are intrinsic to three distinct ACs.

A Compositional Semantics of Boolean-Logic Driven Markov Processes

Boolean-logic driven Markov processes (BDMPs) is a prominent dynamic extension of static fault trees to model repairable and complex dynamic systems. While BDMPs are intensively used in an industrial context for dependability analysis of energy systems, its formal semantics has not been systematically treated. To date, BDMPs are defined as a library of the domain-specific dependability-modelling language Figaro, a library that is neither open source nor publicly available. A rigorous semantic underpinning of BDMPs is indispensable for (1) developing BDMP analysis tools and (2) comparing its expressive power to other related reliability modelling languages. This paper presents a formal semantics to BDMPs using Markov automata (MA), an extension of continuous-time Markov chains (CTMCs) with action transitions that can be used to compose complex MA from smaller MA. This enables us to provide a compositional semantics. That is, we express the semantics of each individual BDMP element as an MA and obtain the MA for the entire BDMP by combining the MA of its elements. This makes the semantics comprehensible, for those who are familiar with automata theory, and easily extensible with new BDMP elements, e.g., to model security aspects. After the entire BDMP is considered, the actions in its MA that were used to “glue” the MA of BDMP elements, are ignored. This results in a CTMC that is amenable to exact numerical analysis by, e.g., efficient probabilistic model-checking techniques. We report on a prototypical implementation of our semantics and empirically show that our semantics yields dependability metrics that correspond to the interpretation by the Figaro knowledge base of BDMPs.

Dynamic behavior of enzyme kinetics cooperative chemical reactions

This article uses computational mathematics to investigate the dynamics of cooperative occurrences in chemical reactions inside living organisms. We study the dynamics of complex systems using mathematical models based on ordinary differential equations, paying special attention to chemical equilibrium and reaction speed. Explanations of cooperation, non-cooperation, and positive cooperation are presented in our study. We analyze the stabilities of equilibrium points by a systematic analysis that makes use of the Jacobian matrix and the threshold parameter R0. We next extend our investigation to evaluate global stability and the probability of the model. Variations in k3 have a notable effect on substrate concentration probabilities, indicating that it plays an important role in reaction kinetics. Reducing k3 highlights the substrate's critical contribution to the system by extending its presence in the concentration. We find that different results were obtained for cooperative behavior: higher reaction rates at different binding sites are correlated with positive cooperativity, while slower reactions are induced by negative cooperativity. The Adams–Bashforth method is used to show numerical and graphical solutions with the help of MATLAB. Tables and graphs are used to further explain the effects of the parameters. This study underlines how well ordinary differential equations may represent the complicated system dynamics found in chemical reactions. It also provides elusive insights into cooperative occurrences, which develops our understanding of the phenomenon and serves as a foundation for future research.

A Memory-Efficient Neural Ordinary Differential Equation Framework Based on High-Level Adjoint Differentiation

Neural ordinary differential equations (neural ODEs) have emerged as a novel network architecture that bridges dynamical systems and deep learning. However, the gradient obtained with the continuous adjoint method in the vanilla neural ODE is not reverse-accurate. Other approaches suffer either from an excessive memory requirement due to deep computational graphs or from limited choices for the time integration scheme, hampering their application to large-scale complex dynamical systems. To achieve accurate gradients without compromising memory efficiency and flexibility, we present a new neural ODE framework, PNODE, based on high-level discrete adjoint algorithmic differentiation. By leveraging discrete adjoint time integrators and advanced checkpointing strategies tailored for these integrators, PNODE can provide a balance between memory and computational costs, while computing the gradients consistently and accurately. We provide an open-source implementation based on PyTorch and PETSc, one of the most commonly used portable, scalable scientific computing libraries. We demonstrate the performance through extensive numerical experiments on image classification and continuous normalizing flow problems. We show that PNODE achieves the highest memory efficiency when compared with other reverse-accurate methods. On the image classification problems, PNODE is up to two times faster than the vanilla neural ODE and up to 2.3 times faster than the best existing reverse-accurate method. We also show that PNODE enables the use of the implicit time integration methods that are needed for stiff dynamical systems.

Comparison of substructuring methods with interface reduction

For complex dynamical systems, substructuring techniques can be applied to simulate the large system more efficiently by handling it as a collection of smaller components. In order to couple these components, the description of the interface between components is very important. Classic substructuring techniques, such as the Craig-Bampton method, reduce the interior modes of the system, but don’t reduce the interface description. This can cause unnecessary degrees of freedom kept in the final assembled model. While there have been a few substructuring methods published that do include some amount of interface reduction, they are much more rare and even more rarely applied in practice. Therefore, the goal of this work is to survey these techniques and compare them with a new, line-based orthogonal polynomial interface reduction technique. These methods will be applied to a set of example problems in order to determine their validity and convergence compared to the full model.

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the formdxidt=xiϕi(x1,…,xN),whereNrepresents the number of species andxi, the abundance of speciesi. Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding toϕi(x1,…,xN)=ri−xi+(Γx)i,play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here,riis the intrinsic growth of speciesiandΓstands for the interaction matrix:Γijrepresents the effect of speciesjover speciesi. For largeN, estimating matrixΓis often an overwhelming task and an alternative is to drawΓat random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

The application of the attractor model to cognitive poetics, journalism, and foreign language learning.

The article examines the issue of applying of the complex dynamic systems theory elements in the field of teaching foreign languages, analyzing the use of the Internet, cognitive poetics, and journalism. In all these areas, the attractor model is mainly used. An attractor is a potential state of a system to which it is evolving. The semantics of the term is clearly dominated by the meaning of attraction. The article presents the justification of the methodological aspect in the application of the attractor model in various fields of science. An overview of domestic and foreign works in these fields is given, and various types of attractors are considered. Special attention is paid to cognitive poetics, a relatively new field of cognitive linguistics that systematically explores the relationship between the literary texts structure and their perception and conceptualization. In cognitive poetics, attractors influence the level of discourse by creating attention-attracting text objects. New types of textual attractors are proposed – neologisms (nonword), as well as elements of non-trivial lexical phrase that are not collocations. A lexical unit is a textual attractor if it attracts the reader's attention. The proof that a nonword attracts attention is that it introduces dissonance into the text, or that the lexical solution takes longer time. The proof that a nontrivial phrase can be considered a textual attractor is an increase in the reaction time to this phrase. The main contribution of the author to the study of the topic can be called the presentation of two types of new textual attractors, namely nonword and non-trivial phrase. The study of the structure and methods of producing individual author's occasional nonwords and non-trivial phrases that act as attractors in a V. Pelevin’s dystopian novel is carried out.

Comprehensive Parameter Space Mapping of Cell Cycle Dynamics under Network Perturbations.

Studies of quantitative systems and synthetic biology have extensively utilized models to interpret data, make predictions, and guide experimental designs. However, models often simplify complex biological systems and lack experimentally validated parameters, making their reliability in perturbed systems unclear. Here, we developed a droplet-based synthetic cell system to continuously tune parameters at the single-cell level in multiple dimensions with full dynamic ranges, providing an experimental framework for global parameter space scans. We systematically perturbed a cell-cycle oscillator centered on cyclin-dependent kinase (Cdk1), enabling comprehensive mapping of period landscapes in response to network perturbations. The data allowed us to challenge existing models and refine a new model that matches the observed response. Our analysis demonstrated that Cdk1 positive feedback inhibition restricts the cell cycle frequency range, confirming model predictions; furthermore, it revealed new cellular responses to the inhibition of the Cdk1-counteracting phosphatase PP2A: monomodal or bimodal distributions across varying inhibition levels, underscoring the complex nature of cell cycle regulation that can be explained by our model. This comprehensive perturbation platform may be generalizable to exploring other complex dynamic systems.

Coalescence of limit cycles in the presence of noise.

Complex dynamical systems may exhibit multiple steady states, including time-periodic limit cycles, where the final trajectory depends on initial conditions. With tuning of parameters, limit cycles can proliferate or merge at an exceptional point. Here we ask how dynamics in the vicinity of such a bifurcation are influenced by noise. A pitchfork bifurcation can be used to induce bifurcation behavior. We model a limit cycle with the normal form of the Hopf oscillator, couple it to the pitchfork, and investigate the resulting dynamical system in the presence of noise. We show that the generating functional for the averages of the dynamical variables factorizes between the pitchfork and the oscillator. The statistical properties of the pitchfork in the presence of noise in its various regimes are investigated and a scaling theory is developed for the correlation and response functions, including a possible symmetry-breaking field. The analysis is done by perturbative calculations as well as numerical means. Finally, observables illustrating the coupling of a system with a limit cycle to a pitchfork are discussed and the phase-phase correlations are shown to exhibit nondiffusive behavior with universal scaling.

Dynamic Development in L2 Written Complexity of Chinese EFL Learners

Abstract The present study investigated the developmental trajectories of written language of two Chinese EFL learners in terms of complexity over one semester and explored the relationship between syntactic complexity and lexical complexity under the framework of Complex Dynamic Systems Theory (CDST). The data were processed using dynamic analyses (min-max graphs, moving correlations and Change-point analysis) to visualize the developmental process and variability of each subsystem of complexity with 4 measures (mean length of T-unit, mean length of clause, clauses per T-unit and VocD) and interactions between different subsystems. The results showed, first, that different subsystems of written language complexity developed dynamically and nonlinearly with different degrees of variability, and varied in the two participants. Second, interactions existed between syntactic complexity and lexical complexity, which shaped as competitors for Jenny but as connected growers for Helen. The results further deepened the understanding of the development in learners’ written complexity, providing significant implications for EFL writing teaching.

Deep Potential Molecular Dynamics Study of Propane Oxidative Dehydrogenation.

Oxidative dehydrogenation (ODH) of light alkanes is a key process in the oxidative conversion of alkanes to alkenes, oxygenated hydrocarbons, and COx (x = 1,2). Understanding the underlying mechanisms extensively is crucial to keep the ODH under control for target products, e.g., alkenes rather than COx, with minimal energy consumption, e.g., during the alkene production or maximal energy release, e.g., during combustion. In this work, deep potential (DP), a neural network atomic potential developed in recent years, was employed to conduct large-scale accurate reactive dynamic simulations. The model was trained on a sufficient data set obtained at the density functional theory level. The intricate reaction network was elucidated and organized in the form of a hierarchical network to demonstrate the key features of the ODH mechanisms, including the activation of propane and oxygen, the influence of propyl reaction pathways on the propene selectivity, and the role of rapid H2O2 decomposition for sustainable and efficient ODH reactions. The results indicate the more complex reaction mechanism of propane ODH than that of ethane ODH and are expected to provide insights in the ODH catalyst optimization. In addition, this work represents the first application of deep potential in the ODH mechanistic study and demonstrates the ample advantages of DP in the study of mechanism and dynamics of complex systems.

Machine-learning parameter tracking with partial state observation

Complex and nonlinear dynamical systems often involve parameters that change with time, accurate tracking of which is essential to tasks such as state estimation, prediction, and control. Existing machine-learning methods require full state observation of the underlying system and tacitly assume adiabatic changes in the parameter. Formulating an inverse problem and exploiting reservoir computing, we develop a model-free and fully data-driven framework to accurately track time-varying parameters from partial state observation in real time. In particular, with training data from a subset of the dynamical variables of the system for a small number of known parameter values, the framework is able to accurately predict the parameter variations in time. Low- and high-dimensional, Markovian and non-Markovian, and spatiotemporal nonlinear dynamical systems are used to demonstrate the power of the machine-learning based parameter-tracking framework. Pertinent issues affecting the tracking performance are addressed. Published by the American Physical Society 2024

AI Institute in Dynamic Systems: Developing machine learning and AI tools for scientific discovery, engineering design, and data‐driven control

AbstractThe mission of the AI Institute in Dynamic Systems is to develop the next generation of advanced machine learning (ML) and AI tools for controlling complex physical systems by discovering physically interpretable and physics‐constrained data‐driven models through optimal sensor selection and placement. The research effort is anchored by a common task framework (CTF) that evaluates the performance of ML algorithms, architectures, and optimization schemes for the diverse tasks required in engineering applications. The aim is to push beyond the boundaries of modern techniques by closing the loop between data collection, control, and modeling, creating a unique and cross‐disciplinary architecture for learning physically interpretable and physics constrained models of complex dynamic systems from time series data. The CTF further supports sustainable and open‐source challenge datasets, which are foundational for developing interpretable, ethical, and inclusive tools to solve problems fundamental to human safety, society, and the environment.

Detecting periodic time scales of changes in temporal networks

Abstract Temporal networks are commonly used to represent dynamical complex systems like social networks, simultaneous firing of neurons, human mobility or public transportation. Their dynamics may evolve on multiple time scales characterizing for instance periodic activity patterns or structural changes. The detection of these time scales can be challenging from the direct observation of simple dynamical network properties like the activity of nodes or the density of links. Here, we propose two new methods, which rely on already established static representations of temporal networks, namely supra-adjacency and temporal event graphs. We define dissimilarity metrics extracted from these representations and compute their power spectra from their Fourier transforms to effectively identify dominant periodic time scales characterizing the changes of the temporal network. We demonstrate our methods using synthetic and real-world data sets describing various kinds of temporal networks. We find that while in all cases the two methods outperform the reference measures, the supra-adjacency-based method identifies more easily periodic changes in network density, while the temporal event graph-based method is better suited to detect periodic changes in the group structure of the network. Our methodology may provide insights into different phenomena occurring at multiple time scales in systems represented by temporal networks.

Real-Time Time-Dependent Density Functional Theory for Simulating Nonequilibrium Electron Dynamics.

The explicit real-time propagation approach for time-dependent density functional theory (RT-TDDFT) has increasingly become a popular first-principles computational method for modeling various time-dependent electronic properties of complex chemical systems. In this Perspective, we provide a nontechnical discussion of how this first-principles simulation approach has been used to gain novel physical insights into nonequilibrium electron dynamics phenomena in recent years. Following a concise overview of the RT-TDDFT methodology from a practical standpoint, we discuss our recent studies on the electronic stopping of DNA in water and the Floquet topological phase as examples. Our discussion focuses on how RT-TDDFT simulations played a unique role in deriving new scientific understandings. We then discuss existing challenges and some new advances at the frontier of RT-TDDFT method development for studying increasingly complex dynamic phenomena and systems.