Modeling Of Dynamic Systems Research Articles

Integration of Augmented Reality and Modeling Techniques in Dynamic Systems for Teaching Theorems and Demonstrations in University Mathematics Courses: A Differential Equations-Based Approach and Its Applications.

Teaching theorems and proofs in mathematics at the university level poses a considerable challenge due to their high abstraction and conceptual complexity. Augmented reality (AR) emerges as an innovative tool that transforms learning by facilitating the visualization and understanding of complex mathematical concepts, particularly in the realm of differential equations, which are fundamental for modeling dynamic systems. This technology allows students to interact with three-dimensional representations, enhancing their comprehension and analytical skills by visualizing the effects and solutions of equations in real time. Despite technological advancements in education, traditional teaching methods remain static and abstract, limiting the assimilation of concepts. The research employed a quantitative approach and a documentary methodology to analyze the effectiveness of integrating AR and modeling techniques, providing data that evidences their positive impact on academic performance. A thorough review of relevant literature was conducted, consolidating a theoretical framework that highlights the importance of these emerging technologies. The findings suggest that implementing AR in the classroom not only improves knowledge retention but also enriches the discussion on mathematics education in higher education.

Analysis of Multistability of Financial Risk Chaos Systems and Its Application to Voice Cryptography

In the chaos literature, the application of modeling and control of dynamic systems in chaos theory arising in several fields is investigated. In this article we analyze complex financial chaos systems with countries as interest rates, investment demand, and price indices. The proposed chaotic flow's dynamic behavior is examined using phase portraits, eigenvalues, bifurcation diagrams, and Lyapunov exponent spectra. A significant quantity of research on secure communication systems has been published in recent years as a result of the major advancements in communications equipment and encryption techniques. A new voice encryption algorithm design is given using a financial chaos model. An application for voice encryption is conducted using the suggested algorithm, and the outcomes are described.

Data‐driven variational method for discrepancy modeling: Dynamics with small‐strain nonlinear elasticity and viscoelasticity

AbstractThe effective inclusion of a priori knowledge when embedding known data in physics‐based models of dynamical systems can ensure that the reconstructed model respects physical principles, while simultaneously improving the accuracy of the solution in the previously unseen regions of state space. This paper presents a physics‐constrained data‐driven discrepancy modeling method that variationally embeds known data in the modeling framework. The hierarchical structure of the method yields fine scale variational equations that facilitate the derivation of residuals which are comprised of the first‐principles theory and sensor‐based data from the dynamical system. The embedding of the sensor data via residual terms leads to discrepancy‐informed closure models that yield a method which is driven not only by boundary and initial conditions, but also by measurements that are taken at only a few observation points in the target system. Specifically, the data‐embedding term serves as residual‐based least‐squares loss function, thus retaining variational consistency. Another important relation arises from the interpretation of the stabilization tensor as a kernel function, thereby incorporating a priori knowledge of the problem and adding computational intelligence to the modeling framework. Numerical test cases show that when known data is taken into account, the data driven variational (DDV) method can correctly predict the system response in the presence of several types of discrepancies. Specifically, the damped solution and correct energy time histories are recovered by including known data in the undamped situation. Morlet wavelet analyses reveal that the surrogate problem with embedded data recovers the fundamental frequency band of the target system. The enhanced stability and accuracy of the DDV method is manifested via reconstructed displacement and velocity fields that yield time histories of strain and kinetic energies which match the target systems. The proposed DDV method also serves as a procedure for restoring eigenvalues and eigenvectors of a deficient dynamical system when known data is taken into account, as shown in the numerical test cases presented here.

Inference on spatiotemporal dynamics for coupled biological populations.

Mathematical models in ecology and epidemiology must be consistent with observed data in order to generate reliable knowledge and evidence-based policy. Metapopulation systems, which consist of a network of connected sub-populations, pose technical challenges in statistical inference owing to nonlinear, stochastic interactions. Numerical difficulties encountered in conducting inference can obstruct the core scientific questions concerning the link between the mathematical models and the data. Recently, an algorithm has been proposed that enables computationally tractable likelihood-based inference for high-dimensional partially observed stochastic dynamic models of metapopulation systems. We use this algorithm to build a statistically principled data analysis workflow for metapopulation systems. Via a case study of COVID-19, we show how this workflow addresses the limitations of previous approaches. The COVID-19 pandemic provides a situation where mathematical models and their policy implications are widely visible, and we revisit an influential metapopulation model used to inform basic epidemiological understanding early in the pandemic. Our methods support self-critical data analysis, enabling us to identify and address model weaknesses, leading to a new model with substantially improved statistical fit and parameter identifiability. Our results suggest that the lockdown initiated on 23 January 2020 in China was more effective than previously thought.

Computing the Plankton-Oxygen Dynamics Model Using Deep Neural Networks in the Context of Climate Change

This study proposes a novel approach to computing the plankton-oxygen dynamics model using deep neural networks (DNNs) within the context of climate change. By leveraging advanced computational methods, particularly deep learning algorithms, we aim to enhance our understanding of how plankton populations and oxygen concentrations interact in response to changing environmental conditions. The integration of DNNs offers several advantages, including the ability to capture complex nonlinear relationships and patterns from large datasets, making them well-suited for modeling dynamic systems such as aquatic ecosystems. By training DNNs on observational data and environmental variables, we can develop predictive models that simulate the behavior of plankton-oxygen dynamics under different climate scenarios. This research builds upon existing studies in ecological modeling and deep learning techniques to advance our knowledge of plankton-oxygen dynamics and their implications for ecosystem resilience in the face of climate change. By computationally modeling these dynamics, we can gain valuable insights into the mechanisms driving ecosystem responses to environmental stressors and inform conservation efforts and policy decisions.

Анализ компромисса между точностью и сложностью идентифицированных моделей динамических систем

One of the rapidly developing areas of the modern control theory is the identification of systems associated with the construction of mathematical models of systems in the form of a set of mathematical relationships that adequately reflect the basic properties of the system. Symbolic regression methods are becoming more and more popular in the problems of structural-parametric identification of systems based on available observations and experimental data, which allow building regression models in the form of codes of mathematical expressions in a symbolic form. Among the known numerical evolutionary methods of symbolic regression, the universally acknowledged “favorite” is the method of genetic programming”, the application of which allows us to describe the search for solving a problem as the construction of a regression model by enumerating various arbitrary superpositions of functions from some predetermined set. In this case, the important indicators determining the quality of identification of the mathematical model of the system are the accuracy and complexity of the identified model. Often, the system model obtained as a result of solving identification problem is not accurate enough or excessively complex. As a result, the solution to the identification problem is inextricably linked to ensuring sufficient accuracy and simplicity of the identified model. In this connection, it is natural to adhere to the principle of balanced identification, which indicates the search for a compromise between the accuracy of reproduction and the measure of complexity of the identified model. The purpose of this paper, which develops the concept of balanced identification, is to analyze the trade-off between the accuracy and complexity of models of dynamic systems identified by genetic programming. In this paper we introduce a functional "accuracy-complexity", which allows us to balance the trade-off between these key indicators of the identified models when solving the identification problem. The effectiveness of the proposed functional is demonstrated on the example of computer identification by genetic programming of a Lorentz dynamical system.

A Novel ANN-ARMA Scheme Enhanced by Metaheuristic Algorithms for Dynamical Systems and Time Series Modeling and Identification

A Novel ANN-ARMA Scheme Enhanced by Metaheuristic Algorithms for Dynamical Systems and Time Series Modeling and Identification

Long short-term memory (LSTM) neural networks for predicting dynamic responses and application in piezoelectric energy harvesting

Long Short-Time Memory (LSTM) deep neural networks are capable of learning order dependence in sequence problems and capturing long-term, non-linear temporal dependencies between the input and out of a system. With the long-term vision to model dynamical systems to which analytical or numerical methods are impossible or difficult to apply, this paper presents a study of modeling system dynamics and predicting responses using the LSTM networks, which have demonstrated excellent capability in predicting single-mode responses in a prior study. However, the LSTM network exhibits difficulties in modeling and predicting multi-mode responses accurately. To resolve the multi-mode issue, this paper presents an approach that obtains an equivalent network consisting of a set of sub-networks learned on isolated modes, and demonstrates its effectiveness on a simulated 2-degree-of-freedom mass-spring-damper system of nonlinear Duffing springs. The second part of the paper is focused on the application of the proposed approach in piezoelectric energy harvesting. Experiments are conducted on a harvester subjected to random base-motion excitation and exhibiting nonlinearity in its multi-mode response. Both the direct and mode-separation LSTM modeling approaches are applied to predict the output voltage given a random base-motion excitation. The mode-separation approach outperforms the direct approach significantly, and yields an excellent match between the actual and predicted responses. Specifically, for a test electrical voltage response of RMS value 0.2241 V, the difference between the actual test and predicted responses by using the mode-separation approach has an RMS value of 0.0504 V, compared to 0.1645 V obtained by using the direct LSTM approach. It is also much lower than the RMS value of 0.1835 V obtained by using the attention-based LSTM network, another comparison method. Leveraging a deep learning strategy, the validated approach opens up opportunities for accurately modeling energy harvesting systems of high complexities and/or strong nonlinearities.

On dynamical system modeling of learned primal-dual with a linear operator K : stability and convergence properties

Learned Primal-Dual (LPD) is a deep learning based method for composite optimization problems that is based on unrolling/unfolding the primal-dual hybrid gradient algorithm. While achieving great successes in applications, the mathematical interpretation of LPD as a truncated iterative scheme is not necessarily sufficient to fully understand its properties. In this paper, we study the LPD with a general linear operator. We model the forward propagation of LPD as a system of difference equations and a system of differential equations in discrete- and continuous-time settings (for primal and dual variables/trajectories), which are named discrete-time LPD and continuous-time LPD, respectively. Forward analyses such as stabilities and the convergence of the state variables of the discrete-time LPD to the solution of continuous-time LPD are given. Moreover, we analyze the learning problems with/without regularization terms of both discrete-time and continuous-time LPD from the optimal control viewpoint. We prove convergence results of their optimal solutions with respect to the network state initialization and training data, showing in some sense the topological stability of the learning problems. We also establish convergence from the solution of the discrete-time LPD learning problem to that of the continuous-time LPD learning problem through a piecewise linear extension, under some appropriate assumptions on the space of learnable parameters. This study demonstrates theoretically the robustness of the LPD structure and the associated training process, and can induce some future research and applications.

Optimization of cooling rate of Q-P treated 42SiCr steel using AI digital twinning

In the continuously advancing field of mechanical engineering, digitalization is bringing a major transformation, specifically with the concept of digital twins. Digital twins are dynamic digital models of real-world systems and processes, crucial for Industry 4.0 and the emerging Industry 5.0, which are changing how humans and machines work together in manufacturing. This paper explores the combination of physics-based and data-driven modeling using advanced Artificial Intelligence (AI) and Machine Learning (ML) techniques. This approach provides a comprehensive understanding of mechanical systems, improving materials design and manufacturing processes. The focus is on the advanced 42SiCr alloy, where AI-driven digital twinning is used to optimize cooling rates during Quenching and Partitioning (Q-P) treatments. This leads to significant improvements in the mechanical properties of 42SiCr steel. Given its complex properties influenced by various factors, this alloy is perfect for digital twinning. The Q-P heat treatment process not only restores the material's deformability but also gives it advanced high-strength steel (AHSS) properties. The findings show how AI and ML can effectively guide the development of high-strength steels and enhance their treatment processes. Additionally, integrating digital twins with new technologies like the Metaverse offers exciting possibilities for simulated production, remote monitoring, and collaborative design. By establishing a clear workflow from physical to digital twins and presenting empirical results, this paper connects theoretical modeling with practical applications, paving the way for smarter manufacturing solutions in mechanical engineering. Furthermore, this paper analyzes how digital twins can be integrated into advanced technologies like the Metaverse, opening up new possibilities for simulated production, remote monitoring, design collaboration, training simulations, analytics, and complete supply chain visibility. This integration is a crucial step toward realizing the full potential of digitalization in mechanical engineering.

Learning about structural errors in models of complex dynamical systems

Complex dynamical systems are notoriously difficult to model because some degrees of freedom (e.g., small scales) may be computationally unresolvable or are incompletely understood, yet they are dynamically important. For example, the small scales of cloud dynamics and droplet formation are crucial for controlling climate, yet are unresolvable in global climate models. Semi-empirical closure models for the effects of unresolved degrees of freedom often exist and encode important domain-specific knowledge. Building on such closure models and correcting them through learning the structural errors can be an effective way of fusing data with domain knowledge. Here we describe a general approach, principles, and algorithms for learning about structural errors. Key to our approach is to include structural error models inside the models of complex systems, for example, in closure models for unresolved scales. The structural errors then map, usually nonlinearly, to observable data. As a result, however, mismatches between model output and data are only indirectly informative about structural errors, due to a lack of labeled pairs of inputs and outputs of structural error models. Additionally, derivatives of the model may not exist or be readily available. We discuss how structural error models can be learned from indirect data with derivative-free Kalman inversion algorithms and variants, how sparsity constraints enforce a “do no harm” principle, and various ways of modeling structural errors. We also discuss the merits of using non-local and/or stochastic error models. In addition, we demonstrate how data assimilation techniques can assist the learning about structural errors in non-ergodic systems. The concepts and algorithms are illustrated in two numerical examples based on the Lorenz-96 system and a human glucose-insulin model.

Probabilistic grammars for modeling dynamical systems from coarse, noisy, and partial data

Ordinary differential equations (ODEs) are a widely used formalism for the mathematical modeling of dynamical systems, a task omnipresent in scientific domains. The paper introduces a novel method for inferring ODEs from data, which extends ProGED, a method for equation discovery that allows users to formalize domain-specific knowledge as probabilistic context-free grammars and use it for constraining the space of candidate equations. The extended method can discover ODEs from partial observations of dynamical systems, where only a subset of state variables can be observed. To evaluate the performance of the newly proposed method, we perform a systematic empirical comparison with alternative state-of-the-art methods for equation discovery and system identification from complete and partial observations. The comparison uses Dynobench, a set of ten dynamical systems that extends the standard Strogatz benchmark. We compare the ability of the considered methods to reconstruct the known ODEs from synthetic data simulated at different temporal resolutions. We also consider data with different levels of noise, i.e., signal-to-noise ratios. The improved ProGED compares favourably to state-of-the-art methods for inferring ODEs from data regarding reconstruction abilities and robustness to data coarseness, noise, and completeness.

Hausdorff metric based training of kernels to learn attractors with application to 133 chaotic dynamical systems

Kernel methods have been successfully used to learn surrogate models for dynamical systems by regressing the vector field of such systems. However, choosing an appropriate kernel for a specific problem is an essential challenge in practice. Kernel Flows (KFs) have emerged as an effective tool for learning data-adaptive kernels used to interpolate dynamical systems. In this paper, we introduce Hausdorff Metric based Kernel Flows (HMKFs) to discover a’good’ kernel from time series when a system under consideration has an attractor. First, under the premise that a kernel is good if there is no significant loss in accuracy when half of the data is used to reconstruct the attractor, we design an objective function based on the distance between true and forecasted attractors. Then, we optimize the proposal by adding an ℓ1 penalty term when the base kernel contains many terms, thereby generating sparse HMKFs. Furthermore, we apply HMKFs and its sparse version to a library of 133 chaotic systems from various fields such as biochemistry, fluid mechanics, and astrophysics. The results showcase the potential of HMKFs in accurately modeling complex dynamical systems using data-driven kernels.

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium

Based on relative-distance-measure fuzzy interval arithmetic (RDM-FIA), this paper presents several new concepts for fuzzy functions in fractional granular calculus with Mittag-Leffler (ML) kernels, such as Atangana–Baleanu (AB) fractional granular integrals, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. Then, we present formulas for the fuzzy granular Laplace transform on Caputo fractional granular derivatives, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. In addition to the relationship between AB fractional granular derivatives and AB Caputo fractional granular derivatives established using the granular Laplace transform, new fractional Newton–Leibniz-type formulas for both derivatives and integrals are proved. As applications, we obtain the fundamental solutions for the fuzzy economic models in the framework of Caputo fractional granular derivatives and AB Caputo fractional granular derivatives by utilizing the fuzzy granular Laplace transform. Several graphical explanations are given to show the dynamic behavior of the fundamental solutions. Moreover, we illustrate the advantages of using RDM-FIA to study the mathematical models of uncertain dynamical systems. The results of this study are illustrated using several numerical examples.

Form, function, mind: What doesn't compute (and what might)

The applicability of computational and dynamical systems models to organisms is scrutinized, using examples from developmental biology and cognition. Developmental morphogenesis is dependent on the inherent material properties of developing animal (metazoan) tissues, a non-computational modality, but cell differentiation, which utilizes chromatin-based revisable memory banks and program-like function-calling, via the developmental gene co-expression system unique to the metazoans, has a quasi-computational basis. Multi-attractor dynamical models are argued to be misapplied to global properties of development, and it is suggested that along with computationalism, classic forms of dynamicism are similarly unsuitable to accounting for cognitive phenomena. Proposals are made for treating brains and other nervous tissues as novel forms of excitable matter with inherent properties which enable the intensification of cell-based basal cognition capabilities present throughout the tree of life. Finally, some connections are drawn between the viewpoint described here and active inference models of cognition, such as the Free Energy Principle.

Nonlinear modeling of oral glucose tolerance test response to evaluate associations with aging outcomes.

As people age, their ability to maintain homeostasis in response to stressors diminishes. Physical frailty, a syndrome characterized by loss of resilience to stressors, is thought to emerge due to dysregulation of and breakdowns in communication among key physiological systems. Dynamical systems modeling of these physiological systems aims to model the underlying processes that govern response to stressors. We hypothesize that dynamical systems model summaries are predictive of age-related declines in health and function. In this study, we analyze data obtained during 75-gram oral-glucose tolerance tests (OGTT) on 1,120 adults older than 50 years of age from the Baltimore Longitudinal Study on Aging. We adopt a two-stage modeling approach. First, we fit OGTT curves with the Ackerman model-a nonlinear, parametric model of the glucose-insulin system-and with functional principal components analysis. We then fit linear and Cox proportional hazards models to evaluate whether usual gait speed and survival are associated with the stage-one model summaries. We also develop recommendations for identifying inadequately-fitting nonlinear model fits in a cohort setting with numerous heterogeneous response curves. These recommendations include: (1) defining a constrained parameter space that ensures biologically plausible model fits, (2) evaluating the relative discrepancy between predicted and observed responses of biological interest, and (3) identifying model fits that have notably poor model fit summary measures, such as [Formula: see text], relative to other fits in the cohort. The Ackerman model was unable to adequately fit 36% of the OGTT curves. The stage-two regression analyses found no associations between Ackerman model summaries and usual gait speed, nor with survival. The second functional principal component score was associated with faster gait speed (p<0.01) and improved survival (p<0.01).

New deterministic model for calculating mesh stiffness and damping of rough-surface gears considering elastic–plastic contact and energy-dissipation mechanism

For a long time, meshing parameter calculations and dynamic modeling of gear systems have assumed smooth tooth surfaces, disregarding the impact of actual three-dimensional (3D) microscopic topography. Based on elastic–plastic contact and energy-dissipation mechanisms, this work developed a comprehensive deterministic model that integrates 3D rough surface modeling, reconstruction, and contact analysis, for calculating the time-varying mesh stiffness (TVMS) and mesh damping (TVMD) of deterministic rough-surface gears. Distinct from fractal and statistical models, the deterministic model analyzes deterministic, real 3D rough surfaces rather than simple surfaces defined by a few parameters. Utilizing stochastic process theory, different-topography surfaces are obtained using height and spatial features and reconstructed employing watershed algorithm and ellipsoidal asperity fitting. Subsequently, a deterministic elastic–plastic contact model based on equivalent rough curved-surface contact is constructed, emphasizing local contact states and energy dissipation for calculating contact stiffness (CS) and damping (CD) of gears. Accordingly, a meshing characteristics analysis model is established for calculating TVMS, TVMD, and load sharing ratio (LSR). Experimental validation and comparisons with several statistical and gear contact models demonstrate the effectiveness, reliability, and superiority of the proposed deterministic model. Multi-parameter impact analysis reveals small Sq (meaning low roughness Sa) and Ssk and high load correlate with increased CS and TVMS, while CD and TVMD depend on their interaction and hardness. Overall, by bridging the micro-surface topography and macro-meshing parameters, our model provides essential insights for the design and manufacturing of high-performance gear to reduce vibration and noise.

Dynamic modeling of flexible multibody systems with complex geometry via finite cell method of absolute nodal coordinate formulation

Dynamic modeling of flexible multibody systems with complex geometry via finite cell method of absolute nodal coordinate formulation

The impact of ozone on Earth-like exoplanet climate dynamics: the case of Proxima Centauri b

ABSTRACT The emergence of the JWST and the development of other advanced observatories (e.g. ELTs, LIFE, and HWO) marks a pivotal moment in the quest to characterize the atmospheres of Earth-like exoplanets. Motivated by these advancements, we conduct theoretical explorations of exoplanetary atmospheres, focusing on refining our understanding of planetary climate and habitability. Our study investigates the impact of ozone on the atmosphere of Proxima Centauri b in a synchronous orbit, utilizing coupled climate chemistry model simulations and dynamical systems theory. The latter quantifies compound dynamical metrics in phase space through the inverse of co-persistence (θ) and co-dimension (d), of which low values correspond to stable atmospheric states. Initially, we scrutinized the influence of ozone on temperature and wind speed. Including interactive ozone [i.e. coupled atmospheric (photo)chemistry] reduces the hemispheric difference in temperature from 68 °K to 64 °K, increases (∼+7 °K) atmospheric temperature at an altitude range of ∼20–50 km, and increases variability in the compound dynamics of temperature and wind speed. Moreover, with interactive ozone, wind speed during highly temporally stable states is weaker than for unstable ones, and ozone transport to the nightside gyres during unstable states is enhanced compared to stable ones (∼+800 DU). We conclude that including interactive ozone significantly influences Earth-like exoplanets' chemistry and climate dynamics. This study establishes a novel pathway for comprehending the influence of photochemical species on the climate dynamics of potentially habitable Earth-like exoplanets. We envisage an extension of this framework to other exoplanets.

Dynamic system with correlated noise

Developing mathematical models for dynamic systems presents challenges in selecting state variables, crucial for minimizing computational operations during filter implementation. The paper addresses active identification of dynamic systems amid noise and measurement errors, using the Fisher information matrix. Additionally, it introduces Cholesky decomposition for generating correlated noise, demonstrating its applicability in practical problem-solving.