Framework Of Dynamical Systems Research Articles

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Nonlinear dynamical system approximation and adaptive control based on hybrid‐feed‐forward recurrent neural network: Simulation and stability analysis

AbstractWe proposed an online identification and adaptive control framework for the nonlinear dynamical systems using a novel hybrid‐feed‐forward recurrent neural network (HFRNN) model. The HFRNN is a combination of a feed‐forward neural network (FFNN) and a local recurrent neural network (LRNN). We aim to leverage the simplicity of FFNN and the effectiveness of RNN to capture changing dynamics accurately and design an indirect adaptive control scheme. To derive the weights update equations, we have applied the gradient‐descent‐based Back‐Propagation (BP) technique, and the stability of the proposed learning strategy is proven using the Lyapunov stability principles. We also compared the proposed method's results with those of the Jordan network‐based controller (JNC) and the local recurrent network‐based controller (LRNC) in the simulation examples. The results demonstrate that our approach performs satisfactorily, even in the presence of disturbance signals.

Contract composition for dynamical control systems: Definition and verification using linear programming

Designing large-scale control systems to satisfy complex specifications is hard in practice, as most formal methods are limited to systems of modest size. Contract theory has been proposed as a modular alternative, in which specifications are defined by assumptions on the input to a component and guarantees on its output. However, current contract-based methods for control systems either prescribe guarantees on the state of the system, going against the spirit of contract theory, or are not supported by efficient computational tools. In this paper, we present a contract-based modular framework for discrete-time dynamical control systems. We extend the definition of contracts by allowing the assumption on the input at a time k to depend on outputs up to time k−1, which is essential when considering feedback systems. We also define contract composition for arbitrary interconnection topologies, and prove that this notion supports modular design, analysis and verification. This is done using graph theory methods, and specifically using the notions of topological ordering and backward-reachable nodes. Lastly, we present an algorithm for verifying vertical contracts, which are claims of the form “the conjunction of given component-level contracts implies given contract on the integrated system”. These algorithms are based on linear programming, and scale linearly with the number of components in the interconnected network. A numerical example is provided to demonstrate the scalability of the presented approach, as well as the modularity achieved by using it.

HiDeS: a higher-order-derivative-supervised neural ordinary differential equation for multi-robot systems and opinion dynamics.

This paper addresses the limitations of current neural ordinary differential equations (NODEs) in modeling and predicting complex dynamics by introducing a novel framework called higher-order-derivative-supervised (HiDeS) NODE. This method extends traditional NODE frameworks by incorporating higher-order derivatives and their interactions into the modeling process, thereby enabling the capture of intricate system behaviors. In addition, the HiDeS NODE employs both the state vector and its higher-order derivatives as supervised signals, which is different from conventional NODEs that utilize only the state vector as a supervised signal. This approach is designed to enhance the predicting capability of NODEs. Through extensive experiments in the complex fields of multi-robot systems and opinion dynamics, the HiDeS NODE demonstrates improved modeling and predicting capabilities over existing models. This research not only proposes an expressive and predictive framework for dynamic systems but also marks the first application of NODEs to the fields of multi-robot systems and opinion dynamics, suggesting broad potential for future interdisciplinary work. The code is available at https://github.com/MengLi-Thea/HiDeS-A-Higher-Order-Derivative-Supervised-Neural-Ordinary-Differential-Equation.

Attitudes towards STEM education: nonlinear effects of teachers’ readiness and the crucial role of affective conditions

IntroductionTeacher attitudes (Att) toward STEM—Science, Technology, Engineering, and Mathematics—education is decisive for its successful integration into contemporary curricula. On the other hand, teachers’ readiness for STEM influences their attitudes and controls their behavior in everyday practice.MethodsIn this study, the four dimensions of readiness for STEM, i.e., affective conditions (Affe), cognitive preparedness (Cogn), self-efficacy (SEff), and STEM commitment (Com), measured via the TRi-STEM scale, were tested as predictors of attitudes using non-linear models. Data were taken from teachers (N = 494) who completed the TRi-STEM questionnaire and the attitudes towards STEM scale for measuring attitudes. Catastrophe theory was applied, and three cusp models, superior to the linear and logistic counterparts, were proposed predicting attitudes (Att) as a function of combinations of Cogn, SEff, Com, and Affe.ResultsThe three models are as follows: Cusp 1 with (Cogn – Affe) as asymmetry and (Cogn + Affe) as bifurcation factors; Cusp 2 with (Com – Affe) as asymmetry and (Com + Affe) as bifurcation factors, and Cusp 3 with (SEff – Affe) as asymmetry and (SEff + Affe) as bifurcation factors. The findings showed that affective conditions involved in a dynamic interplay with other independent variables could lead to sudden and abrupt changes in Att.DiscussionThe empirical evidence for non-linear effects in teacher attitudes (Att) toward STEM informs theory development and practice by supporting the complexity and dynamical system framework as a more realistic premise to describe and interpret potentially occurring phenomena of teachers’ behavior in the context of STEM education.

A multi-modality and multi-dyad approach to measuring flexibility in psychotherapy

Introduction Flexibility, the ability of an individual to adapt to environmental changes in ways that facilitate goal attainment, has been proposed as a potential mechanism underlying psychopathology and psychotherapy. In psychotherapy, most findings are based on self-report measures that have important limitations. We propose a multimodal, multi-dyad approach based on a nonlinear dynamical systems framework to capture the complexity of this concept. Method A new research paradigm was designed to explore the validity of the proposed conceptual model. The paradigm includes a psychotherapy-like social interaction, during which body movement and facial expressiveness data were collected. We analyzed the data using Hankel Alternative View of Koopmann analysis to reconstruct attractors of the observed behaviors and compare them. Results The patterns of behavior in the two cases differ, and differences in the reconstructed attractors correspond with differences in self-report measures and behavior in the interactions. Conclusions The case studies show that information provided by a single modality is not enough to provide the full picture, and multiple modalities are needed. These observations can serve as an initial support for our claims that a multi-modal and multi-dyad approach to flexibility can address some of the issues of measurement in the field.

C3F: Constant Collaboration and Communication Framework for Graph-Representation Dynamic Multi-Robotic Systems

C3F: Constant Collaboration and Communication Framework for Graph-Representation Dynamic Multi-Robotic Systems

Dynamic Cantor Derivative Logic

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics based on the topological closure operator, $d$-logics have not previously been studied in the framework of dynamical systems, which are pairs $(X,f)$ consisting of a topological space $X$ equipped with a continuous function $f\colon X\to X$. We introduce the logics $\bf{wK4C}$, $\bf{K4C}$ and $\bf{GLC}$ and show that they all have the finite Kripke model property and are sound and complete with respect to the $d$-semantics in this dynamical setting. In particular, we prove that $\bf{wK4C}$ is the $d$-logic of all dynamic topological systems, $\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic topological systems, and $\bf{GLC}$ is the $d$-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where $f$ is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems $\bf{wK4H}$, $\bf{K4H}$ and $\bf{GLH}$. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological $d$-logics. Furthermore, our result for $\bf{GLC}$ constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

A novel incremental attribute reduction approach for incomplete decision systems

With the increasing diversification and complexity of information, it is vital to mine effective knowledge from information systems. In order to extract information rapidly, we investigate attribute reduction within the framework of dynamic incomplete decision systems. Firstly, we introduce positive knowledge granularity concept which is a novel measurement on information granularity in information systems, and further give the calculation method of core attributes based on positive knowledge granularity. Then, two incremental attribute reduction algorithms are presented for incomplete decision systems with multiple objects added and deleted on the basis of positive knowledge granularity. Furthermore, we adopt some numerical examples to illustrate the effectiveness and rationality of the proposed algorithms. In addition, time complexity of the two algorithms are conducted to demonstrate their advantages. Finally, we extract five datasets from UCI database and successfully run the algorithms to obtain corresponding reduction results.

Characterizing affect dynamics with a damped linear oscillator model: Theoretical considerations and recommendations for individual-level applications.

People show stable differences in the way their affect fluctuates over time. Within the general framework of dynamical systems, the damped linear oscillator (DLO) model has been proposed as a useful approach to study affect dynamics. The DLO model can be applied to repeated measures provided by a single individual, and the resulting parameters can capture relevant features of the person's affect dynamics. Focusing on negative affect, we provide an accessible interpretation of the DLO model parameters in terms of emotional lability, resilience, and vulnerability. We conducted a Monte Carlo study to test the DLO model performance under different empirically relevant conditions in terms of individual characteristics and sampling scheme. We used state-space models in continuous time. The results show that, under certain conditions, the DLO model is able to accurately and efficiently recover the parameters underlying the affective dynamics of a single individual. We discuss the results and the theoretical and practical implications of using this model, illustrate how to use it for studying psychological phenomena at the individual level, and provide specific recommendations on how to collect data for this purpose. We also provide a tutorial website and computer code in R to implement this approach. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Computationally efficient neural hybrid automaton framework for learning complex dynamics

This paper proposes a computationally efficient and effective data-driven modeling framework for dynamical systems. The proposed modeling framework employs a collection of shallow neural networks known as Extreme Learning Machines (ELMs) to model local system behaviors along with data-driven inferred transitions among local models to establish a neural hybrid automaton model. First, the sampled system inputs are mapped to the corresponding feature spaces to obtain data-driven partitions, which subsequently define the transitions and invariants of the neural hybrid automaton model through a novel data-driven mode clustering process. Then, a collection of ELMs are trained to approximate the local dynamics. The learning processes integrate a segmented data merging procedure for location identification and a local dynamics modeling process. The proposed neural hybrid automaton models can capture behaviors of complex dynamical systems with high modeling precision but significantly lower computational complexities in computationally expensive tasks such as training and verification, which are traditionally considered to be computationally expensive tasks for neural network models. A computationally efficient set-valued reachability analysis method which is commonly used in safety verification is then developed based on interval analysis and a novel Split and Combine process. Finally, applications to modeling the limit cycle and human handwritten motions are presented to show the effectiveness and efficiency of our approach.

A Test of the Behavioral Model of Tic Disorders Using a Dynamical Systems Framework

Tic disorders are a class of neurodevelopmental disorders characterized by involuntary motor and/or vocal tics. It has been hypothesized that tics function to reduce aversive premonitory urges (i.e., negative reinforcement) and that suppression-based behavioral interventions such as habit reversal training (HRT) and exposure and response prevention (ERP) disrupt this process and facilitate urge reduction through habituation. However, previous findings regarding the negative reinforcement hypothesis and the effect of suppression on the urge-tic relationship have been inconsistent. The present study applied a dynamical systems framework and within-subject time-series autoregressive models to examine the temporal dynamics of urges and tics and assess whether their relationship changes over time. Eleven adults with tic disorders provided continuous urge ratings during separate conditions in which they were instructed to tic freely or to suppress tics. During the free-to-tic conditions, there was considerable heterogeneity across participants in whether and how the urge-tic relationship followed a pattern consistent with the automatic negative reinforcement hypothesis. Further, little evidence for within-session habituation was seen; tic suppression did not result in a reduction in premonitory urges for most participants. Analysis of broader urge change metrics did show significant disruption to the urge pattern during suppression, which has implications for the current biobehavioral model of tics.

Analogous response of temperate terrestrial exoplanets and Earth’s climate dynamics to greenhouse gas supplement

Humanity is close to characterizing the atmospheres of rocky exoplanets due to the advent of JWST. These astronomical observations motivate us to understand exoplanetary atmospheres to constrain habitability. We study the influence greenhouse gas supplement has on the atmosphere of TRAPPIST-1e, an Earth-like exoplanet, and Earth itself by analyzing ExoCAM and CMIP6 model simulations. We find an analogous relationship between CO2 supplement and amplified warming at non-irradiated regions (night side and polar)—such spatial heterogeneity results in significant global circulation changes. A dynamical systems framework provides additional insight into the vertical dynamics of the atmospheres. Indeed, we demonstrate that adding CO2 increases temporal stability near the surface and decreases stability at low pressures. Although Earth and TRAPPIST-1e take entirely different climate states, they share the relative response between climate dynamics and greenhouse gas supplements.

A data-driven framework for learning hybrid dynamical systems.

The existing data-driven identification methods for hybrid dynamical systems such as sparse optimization are usually limited to parameter identification for coefficients of pre-defined candidate functions or composition of prescribed function forms, which depend on the prior knowledge of the dynamical models. In this work, we propose a novel data-driven framework to discover the hybrid dynamical systems from time series data, without any prior knowledge required of the systems. More specifically, we devise a dual-loop algorithm to peel off the data subject to each subsystem of the hybrid dynamical system. Then, we approximate the subsystems by iteratively training several residual networks and estimate the transition rules by training a fully connected neural network. Several prototypical examples are presented to demonstrate the effectiveness and accuracy of our method for hybrid models with various dimensions and structures. This method appears to be an effective tool for learning the evolutionary governing laws of hybrid dynamical systems from available data sets with wide applications.

The sources of extreme precipitation predictability; the case of the ‘Wet’ Red Sea Trough

Extreme precipitation events inflict detrimental socio-economic impacts in the Eastern Mediterranean. These are mainly associated with Mediterranean cyclones or the ‘Wet’ Red Sea Trough (WRST). The region's weather forecasters consider the second challenging to forecast, even just a few days in advance. Here, we study the dynamic and thermodynamic factors influencing the intrinsic predictability of WRST events. With this aim, we combine insights from traditional atmospheric analysis techniques, Lagrangian air-parcel backward trajectories, and dynamical systems theory. The latter describes atmospheric states via their local dimension (d) and inverse persistence (θ), which inform us of the intrinsic predictability of the atmosphere in phase space. We compare WRST events of low (upper decile of d and θ) with high (lower decile of d and θ) predictability. We argue that low-predictability events display a significantly different atmospheric pattern. Moreover, the low-predictability events show significantly higher daily precipitation rates, more extensive spatial spread, and greater precipitation variability among events than more predictable ones. On average, low predictability events are initiated by two distinct moisture sources with different water vapor content. We conclude that the dynamical systems framework may become a valuable tool to improve the forecast of extreme precipitation events associated with the WRST by providing a priori information on their intrinsic predictability. We foresee successfully implementing such a framework for other extreme weather events and regions.

An optimal control approach to particle filtering

We present a novel particle filtering framework for the continuous-time dynamical systems with continuous-time measurements. Our approach is based on the duality between estimation and optimal control, which allows for reformulating the estimation problem over a fixed time window into an optimal control problem. The resulting optimal control problem has a cost function that depends on the measurements, and the closed-loop dynamics under optimal control coincides with the posterior distribution over the trajectories for the corresponding estimation problem. By recursively solving these optimal control problems approximately as new measurements become available, we obtain an optimal control based particle filtering algorithm. Our algorithm uses path integrals to compute the weights of the particles and is thus termed the path integrals particle filter (PIPF). A distinguishing feature of the proposed method is that it uses the measurements over a finite-length time window instead of a single measurement for the estimation at each time step, resembling the batch methods of filtering, and improving fault tolerance. The efficacy of our algorithm is illustrated with several numerical examples.

Adaptive Nonlinear Incremental Flight Control for Systems With Unknown Control Effectiveness

This article exposes that although some sensor-based nonlinear fault-tolerant control frameworks including incremental nonlinear dynamic inversion control can passively resist a wide range of actuator faults and structural damage without requiring an accurate model of the dynamic system, their stability heavily relies on a sufficient condition, which is unfortunately violated if the control direction is unknown. Consequently, it is proved in this article that no matter, which perturbation compensation technique (adaptive, disturbance observer, sliding-mode) is implemented, none of the existing nonlinear incremental control methods can guarantee closed-loop stability. Therefore, this article proposes a Nussbaum function-based adaptive incremental control framework for nonlinear dynamic systems with partially known (control direction is unknown) or even completely unknown control effectiveness. Its effectiveness is proved in the Lyapunov sense and is also verified via numerical simulations of an aircraft attitude tracking problem in the presence of sensing errors, parametric model uncertainties, structural damage, actuator faults, as well as inversed and unknown control effectiveness.

A Bayesian switching linear dynamical system for estimating seizure chronotypes

Epilepsy is a disorder characterized by paroxysmal transitions between multistable states. Dynamical systems have been useful for modeling the paroxysmal nature of seizures. At the same time, intracranial electroencephalography (EEG) recordings have recently discovered that an electrographic measure of epileptogenicity, interictal epileptiform activity, exhibits cycling patterns ranging from ultradian to multidien rhythmicity, with seizures phase-locked to specific phases of these latent cycles. However, many mechanistic questions about seizure cycles remain unanswered. Here, we provide a principled approach to recast the modeling of seizure chronotypes within a statistical dynamical systems framework by developing a Bayesian switching linear dynamical system (SLDS) with variable selection to estimate latent seizure cycles. We propose a Markov chain Monte Carlo algorithm that employs particle Gibbs with ancestral sampling to estimate latent cycles in epilepsy and apply unsupervised learning on spectral features of latent cycles to uncover clusters in cycling tendency. We analyze the largest database of patient-reported seizures in the world to comprehensively characterize multidien cycling patterns among 1,012 people with epilepsy, spanning from infancy to older adulthood. Our work advances knowledge of cycling in epilepsy by investigating how multidien seizure cycles vary in people with epilepsy, while demonstrating an application of an SLDS to frame seizure cycling within a nonlinear dynamical systems framework. It also lays the groundwork for future studies to pursue data-driven hypothesis generation regarding the mechanistic drivers of seizure cycles.

Accounting for Individual Trajectories in L2 Learning: A One‑Year Pilot Case Study with Adult Professionals

This paper seeks to explore the links between individual trajectories followed over time by adult professionals and the way these trajectories shape their L2 learning routes. Embracing the theoretical framework of Individual Differences and complex dynamic systems, we sought to study these links via an appraisal of the contribution of non‑classroom integrated telecollaboration as an opportunity for L2 learning which embraces the inherent variability of individual learning pathways. A case study approach was adopted given the size of our sample (10 adult learners from Cnam Paris; 10 students from Cardiff University), the qualitative nature of the collected data, and the reliability of individual longitudinal case studies in determining factors structuring L2 learning. A three-phase protocol (introductory tutorial, telecollaboration sessions, final feedback) was designed for this one‑year project. Data gathered from the semi-structured interviews of the two most successful French participants were analyzed and cross-checked with data from reflective logbooks and feedback reports focusing on factors shaping L2 development over time and perceptions of learning benefits. Our findings further confirm the intrinsic relationship between individual trajectories and language learning, thus calling for the necessity of learner-centeredness in the design process of (time‑limited) institutional L2 courses particularly for adult professionals.

Active extremum Kriging-based multi-level linkage reliability analysis and its application in aeroengine mechanism systems

To improve the computational efficiency and accuracy of dynamic multi-component system reliability analysis involving complex characteristics like dynamic traits, high-nonlinearity, and failure correlation, an active extremum Kriging-based multi-level linkage method (AEK-MLL) is proposed by incorporating the benefits of extremum selection technique, active learning Kriging model, and the multi-level linkage strategy. In AEK-MLL modeling, the extremum selection technique first converts the dynamic candidate sample domain into a steady-state candidate sample domain, and the active learning technique searches for the best training samples, to build the active extremum Kriging model of component-level limit state functions (LSFs); moreover, the multi-level linkage strategy is adopted to take failure correlation into account, to establish a reliability framework for complex dynamic multi-component systems. The proposed method is first validated by a dynamic numerical case and three system numerical cases, and then applied to the dynamic multi-component reliability analysis of a typical aero-engine mechanism system. The numerical cases and engineering case show that the AEK-MLL holds the high-accuracy and high-efficiency in dealing with the complex dynamic multi-component system reliability analysis.