Dynamical Systems Theory Research Articles

Role of informal sector to combat unemployment in developing economy: A modeling study

In developing countries, informal sector is the primary job provider for a significant portion of the workforce. This study aims to analyze how jobs in the informal sector affect the unemployment dynamics of developing nations. To achieve this goal, we formulate a nonlinear mathematical model by categorizing the considered workforce into three distinct classes: unemployed, employed, and self-employed, and include a separate dynamic variable to represent vacancies within the informal sector. The proposed model is analyzed using the qualitative theory of dynamical systems. A threshold quantity known as the reproduction number is derived and using this, one can compute the job generation rate necessary to stabilize the system. It is observed that variations in the reproduction number lead to qualitative changes, such as transcritical (forward or backward) and saddle-node bifurcations in the formulated system. Moreover, we propose an optimal control problem to determine an optimal strategy for government policy implementation in enhancing the employment rate of unemployed individuals and promoting the self-employment of informal employees inside the informal sector. Further, the analytical findings are validated numerically. The obtained results suggest that promoting the self-employment of informal employees for job generation effectively reduces unemployment.

Dark energy and dark matter interaction: A nonlinear dynamical system study

This paper presents a methodical dynamical analysis of the Big Bang model, taking into account dark matter and an arbitrary form of dark energy. Nevertheless, why dynamic analysis? The primary advantage of the dynamical analysis approach is that it obviates the necessity to precisely solve the system for every individual dependent variable. Instead, it enables the prediction of the overall evolutions. In this regard, there are some powerful tools of non-dynamical systems (theorems, perturbation, etc.) which can help us a lot. The model under consideration is a nonlinear system with two dimensions (2D). In order to investigate this, we employ a sophisticated form of nonlinear dynamical systems theory in our research. We have delved into the full potential of nonlinear dynamical system theory known until now in our work. This analysis has produced very interesting solutions, which are in quite a contrast with the linear analysis approach. The stability of the system is considered at length, and the corresponding evolutions of the universe are also discussed consequently. A handful number of figures are given to visualize the behavior of the evolutions. The plots that are presented here are vector field plots and a new plotting technique initiated by us and used in our earlier papers. Interestingly, our work indicates the universe might resemble “phantom” evolution. Also, among the fixed points of our model, one is able to escape the singular situation called “Big Rip”.

A Stochastic Landscape Approach for Protein Folding State Classification.

Protein folding is a critical process that determines the functional state of proteins. Proper folding is essential for proteins to acquire their functional three-dimensional structures and execute their biological role, whereas misfolded proteins can lead to various diseases, including neurodegenerative disorders like Alzheimer's and Parkinson's. Therefore, a deeper understanding of protein folding is vital for understanding disease mechanisms and developing therapeutic strategies. This study introduces the Stochastic Landscape Classification (SLC), an innovative, automated, nonlearning algorithm that quantitatively analyzes protein folding dynamics. Focusing on collective variables (CVs) - low-dimensional representations of complex dynamical systems like molecular dynamics (MD) of macromolecules - the SLC approach segments the CVs into distinct macrostates, revealing the protein folding pathway explored by MD simulations. The segmentation is achieved by analyzing changes in CV trends and clustering these segments using a standard density-based spatial clustering of applications with noise (DBSCAN) scheme. Applied to the MD-based CV trajectories of Chignolin and Trp-Cage proteins, the SLC demonstrates apposite accuracy, validated by comparing standard classification metrics against ground-truth data. These metrics affirm the efficacy of the SLC in capturing intricate protein dynamics and offer a method to evaluate and select the most informative CVs. The practical application of this technique lies in its ability to provide a detailed, quantitative description of protein folding processes, with significant implications for understanding and manipulating protein behavior in industrial and pharmaceutical contexts.

A Dynamic Entropy Approach Reveals Reduced Functional Network Connectivity Trajectory Complexity in Schizophrenia.

Over the past decade and a half, dynamic functional imaging has revealed low-dimensional brain connectivity measures, identified potential common human spatial connectivity states, tracked the transition patterns of these states, and demonstrated meaningful transition alterations in disorders and over the course of development. Recently, researchers have begun to analyze these data from the perspective of dynamic systems and information theory in the hopes of understanding how these dynamics support less easily quantified processes, such as information processing, cortical hierarchy, and consciousness. Little attention has been paid to the effects of psychiatric disease on these measures, however. We begin to rectify this by examining the complexity of subject trajectories in state space through the lens of information theory. Specifically, we identify a basis for the dynamic functional connectivity state space and track subject trajectories through this space over the course of the scan. The dynamic complexity of these trajectories is assessed along each dimension of the proposed basis space. Using these estimates, we demonstrate that schizophrenia patients display substantially simpler trajectories than demographically matched healthy controls and that this drop in complexity concentrates along specific dimensions. We also demonstrate that entropy generation in at least one of these dimensions is linked to cognitive performance. Overall, the results suggest great value in applying dynamic systems theory to problems of neuroimaging and reveal a substantial drop in the complexity of schizophrenia patients' brain function.

Student Engagement with Teacher Written Corrective Feedback among Chinese Private College Students of Varying Language Proficiencies

While the efficacy of teacher written corrective feedback (WCF) has been extensively explored, a research gap exists in examining the disparities in how low-proficiency (LP) and high-proficiency (HP) students receive such feedback in second language (L2) writing. Through an analysis of five writing tasks distributed over a 16-week course, this research explored the affective, behavioral, and cognitive dimensions of student engagement with WCF. Six Chinese EFL sophomores, three designated as LP and three as HP students, were selected through purposive sampling. Data collection methods included analysis of students’ L2 writing tasks and stimulated recall sessions. The findings illustrate varying engagement patterns, highlighting LP students’ frustrations and HP students’ reflective and constructive interactions with feedback. These patterns are interpreted through the lens of sociocultural, social cognitive, student engagement, and complex dynamic systems theories, offering a multifaceted framework for understanding the influence of WCF on L2 writing proficiency. These findings also contribute to language learning pedagogy by highlighting the importance of tailored feedback strategies that address the comprehensive dimensions of student engagement to enhance the pedagogical effectiveness of WCF in fostering language proficiency and engagement in EFL settings.

Causalized Convergent Cross Mapping and Its Implementation in Causality Analysis.

Rooted in dynamic systems theory, convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting linear and nonlinear causal coupling in both random and deterministic settings. One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis.

A dynamical systems approach to WKB-methods: The simple turning point

In this paper, we revisit the classical linear turning point problem for the second order differential equation ϵ2x″+μ(t)x=0 with μ(0)=0,μ′(0)≠0 for 0<ϵ≪1. Written as a first order system, t=0 therefore corresponds to a turning point connecting hyperbolic and elliptic regimes. Our main result is that we provide an alternative approach to WBK that is based upon dynamical systems theory, including GSPT and blowup, and we bridge – perhaps for the first time – hyperbolic and elliptic theories of slow-fast systems. As an advantage, we only require finite smoothness of μ. The approach we develop will be useful in other singular perturbation problems with hyperbolic–to–elliptic turning points.

Control of HPV infection leading to cervical cancer

HPV is the most common sexually transmitted infection and can lead to cancer. Vaccination has been considered as a very effective measure against HPV and along with regular screenings is recommended for the prevention of cervical cancer. In this study we propose and analyze a model for the transmission dynamics of Human Papilloma Virus (HPV). We discuss the optimal vaccination strategy in the case when multiple vaccines are available. We have considered an ODE based compartmental model, incorporating sex structure, we also consider HPV leading to cervical cancer by including pre-cancerous and cancerous compartments in the model. Adding the pre-cancerous and cancerous compartments will help us better understand the role of vaccination in prevention of cancers due to HPV. Using standard techniques from dynamical systems theory, we determine the disease free (DFE) and endemic steady states (EE). We determine a threshold quantity, the basic reproductive number R0 in terms of the model parameters, such that, the DFE is stable when R0<1 and the EE is stable in the case R0>1. The goal of control measures is to try an reduce R0 to be below the threshold value of 1, the regime in which the disease will ultimately be driven to extinction. We also prove the existence of a backward bifurcation in the model, epidemiologically this means that the long time behavior of the model depends on the initial infected population and that the disease may be harder to control, and the condition R0<1 may not suffice to eradicate the disease. Finally, using optimal control theory we consider the effective vaccination strategies using multiple vaccines, the goal is to study the best vaccination strategy using multiple vaccines. Our results show that a higher coverage with vaccines that provide protection against multiple strains, although more expensive, is best suited to control HPV transmission and subsequent development of cancer.

DynGMA: A robust approach for learning stochastic differential equations from data

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Physics-Informed Particle-Based Reinforcement Learning for Autonomy in Signalized Intersections

In this paper, we develop a framework to enhance the control of autonomous vehicles within signalized intersections by integrating system dynamics with imperfect sensor data. Although sensor data for autonomous vehicles is often partial, noisy, or missing, its alignment with underlying physics has the potential to significantly improve prediction accuracy. Our methodology involves a physics-based representation of system dynamics, accounting for stochastic elements originating from differences between real-world scenarios and model-based representations, using a Markov decision process (MDP) model that embraces system physics while accommodating uncertainties. Leveraging partial sensor data, we aim to diminish uncertainty associated with segments of the system model reflected in the data and better estimate the other physical variables that are not measurable through data. This framework develops particle-based reinforcement learning action policies, seamlessly integrating data and physics for controlling autonomous vehicles approaching signalized intersections. Numerical experiments showcase the effectiveness of these policies in ensuring safe control and decision-making in different scenarios of missing state variables and varying levels of data sparsity.

Purported Self-Organized Criticality of the Cardiovascular Function: Methodological Considerations for Zipf's Law Analysis.

Self-organized criticality is a universal theory for dynamical systems that has recently been applied to the cardiovascular system. Precise methodological approaches are essential for understanding the dynamics of cardiovascular self-organized criticality. This study examines how the duration and quality of data recording affect the analysis of cardiovascular self-organized criticality, with a focus on the beat-by-beat heart rate variability time series obtained from seven healthy subjects in a standing position. Drawing a Zipf diagram, we evaluated the distribution of cardiovascular events of bradycardia and tachycardia. We identified tipping points for the distribution of both bradycardia and tachycardia events. By varying the recording durations (1, 2, 5, 10, 20, 30, and 40 min) and sampling frequencies (500, 250, and 100 Hz), we investigated their influence on the observed distributions. While shorter recordings can effectively capture cardiovascular events, they may underestimate the variables describing their distribution. Additionally, the tipping point of the Zipf distribution differs between bradycardia and tachycardia events. Comparisons of the distribution of bradycardia and tachycardia events should be conducted using long data recordings. Utilizing devices with lower sampling frequencies may compromise data fidelity. These insights contribute to refining experimental protocols and advancing our understanding of the complex dynamics underlying cardiovascular regulation.

Traveling wave solution and qualitative behavior of fractional stochastic Kraenkel–Manna–Merle equation in ferromagnetic materials

The main purpose of this article is to investigate the qualitative behavior and traveling wave solutions of the fractional stochastic Kraenkel–Manna–Merle equations, which is commonly used to simulate the zero conductivity nonlinear propagation behavior of short waves in saturated ferromagnetic materials. Firstly, fractional stochastic Kraenkel–Manna–Merle equations are transformed into ordinary differential equations by using the traveling wave transformation. Secondly, the phase portraits, sensitivity analysis, and Poincaré sections of the two-dimensional dynamic system and its perturbation system of ordinary differential equations are drawn. Finally, the traveling wave solutions of fractional stochastic Kraenkel–Manna–Merle equations are obtained based on the analysis theory of planar dynamical system. Moreover, the obtained three-dimensional graphs of random solutions, two-dimensional graphs of random solutions, and three-dimensional graphs of deterministic solutions are drawn.

Criticality and chaos in auditory and vestibular sensing

The auditory and vestibular systems exhibit remarkable sensitivity of detection, responding to deflections on the order of angstroms, even in the presence of biological noise. The auditory system exhibits high temporal acuity and frequency selectivity, allowing us to make sense of the acoustic world around us. As the acoustic signals of interest span many orders of magnitude in both amplitude and frequency, this system relies heavily on nonlinearities and power-law scaling. The vestibular system, which detects ground-borne vibrations and creates the sense of balance, exhibits highly sensitive, broadband detection. It likewise requires high temporal acuity so as to allow us to maintain balance while in motion. The behavior of these sensory systems has been extensively studied in the context of dynamical systems theory, with many empirical phenomena described by critical dynamics. Other phenomena have been explained by systems in the chaotic regime, where weak perturbations drastically impact the future state of the system. Using a Hopf oscillator as a simple numerical model for a sensory element in these systems, we explore the intersection of the two types of dynamical phenomena. We identify the relative tradeoffs between different detection metrics, and propose that, for both types of sensory systems, the instabilities giving rise to chaotic dynamics improve signal detection.

A two-species diffusion-advection competition model with protection zones

In this paper, we study a two-species reaction-diffusion-advection competition model imposed the Danckwerts boundary conditions and with boundary protection zones. Some special cases have been extensively studied, which include reaction-diffusion-advection competition models without protection zones [25,29] and reaction-diffusion competition models with protection zones in non-advective environments [7]. If the product of the two competition rates in the unprotected zone is less than a given value, by establishing a prior estimate and applying the monotone dynamical system theory, we completely characterize the global dynamics, which extends some existing ones. If the product is bigger than the above mentioned value, we find that the size of the protection zone plays an important role in determining the global dynamics. We also discuss the problem of optimal protection zone setting and investigate the effect of advection on the critical size of the protection zone.

Hamilton–Jacobi theory for nonholonomic and forced hybrid mechanical systems

A hybrid system is a system whose dynamics is given by a mixture of both continuous and discrete transitions. In particular, these systems can be utilized to describe the dynamics of a mechanical system with impacts. Based on the approach by Clark [Invariant measures, geometry and control of hybrid and nonholonomic dynamical systems, Ph.D thesis, University of Mirchigan (2020)], we develop a geometric Hamilton–Jacobi theory for forced and nonholonomic hybrid dynamical systems. We state the corresponding Hamilton–Jacobi equations for these classes of systems and apply our results to analyze some examples.

Teacher Perceptions on Using Translanguaging for Teaching Second and Foreign Languages in Sri Lanka

Abstract The linguistic landscape of Sri Lanka underwent dramatic transformations owing to the influence of Indic languages and the colonization which resulted in increased language contact and hybridization of local languages. Advancements in digital and commercialization of language learning increased the demand for second and foreign language education. Whether language planning and education have undergone changes in par with the transformations in the linguistic landscape is questionable. Approaches and methodologies of second and foreign language learning in Sri Lanka remain entrenched in hegemonic practices of post-colonial language education. Taking on a convergent parallel design, the present study explored the potentials of incorporating translanguaging into second and foreign language education in Sri Lanka. Teachers of English as a second language and three other foreign languages were selected as the key informants of the study. Recognizing language as a transformative entity from a sociolinguistic stance, the study employed dynamic systems theory to evaluate the significant transformations of the linguistic landscape in Sri Lanka juxtaposed with developments in language studies. The findings have revealed that although the teachers have identified the need for shifting towards inclusive language education that recognizes the significance of learners’ native language repertoire, the language education system is still entrenched in the conservative approaches. Teaching learning material and ideologies attached to second and foreign languages education have become ‘leg irons’ that constrain adaptation into the new language environment. National scale linguistic landscape surveys, transformations in methodologies, localized learning material and long-term language policy and planning initiatives are key requirements for ensuring inclusive, equitable and decolonized language education.

Residual Echo State Networks: Residual recurrent neural networks with stable dynamics and fast learning

Residual connections have been established as a staple for modern deep learning architectures. Most of their applications are cast towards feedforward computing. In this paper, we study the architectural bias of residual connections in the context of recurrent neural networks (RNNs), specifically in the temporal dimension. We frame our discussion from the perspective of Reservoir Computing and dynamical system theory, focusing on important aspects of neural computation like memory capacity, long-term information processing, stability, and nonlinear computation capability. Experiments corroborate the striking advantage brought by temporal residual connections for a plethora of different time series processing tasks, comprehending memory-based, forecasting, and classification problems.

Exact solutions and bifurcations for the (3+1)-dimensional generalized KdV-ZK equation

In this paper, a class of (3+1)-dimensional generalized Korteweg–de Vries-Zakharov-Kuznetsov (KdV-ZK) equation is studied by utilizing the bifurcation theory of the planar dynamical systems and the Fan sub-function method. This model can be used to explain the effects of magnetic fields on weakly nonlinear ion-acoustic waves investigated in plasma fields composed of cold and hot electrons. Under the different parameter conditions, the phase portraits and bifurcations are derived, and new exact solutions including soliton, periodic, kink and breaking wave solutions for the model are constructed. Moreover, some exact solutions, which contain soliton, kink, trigonometric function, hyperbolic function, Jacobi elliptic function solutions, are derived via the improved Fan sub-function method. The types of solutions obtained completely correspond to the types of the orbits acquired above, which verifies the validity of the method. Finally, the physical structures of some exact solutions are analyzed in graphical forms.

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]