Dynamical Systems Theory Research Articles

Bowen’s Formula for a Dynamical Solenoid

More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections (fn:X→X)n∈N defined on a compact metric space (X,d). Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection f:X→X of a compact

The Diffusive Lotka–Volterra Competitive Model with Advection Term: Exclusion, Coexistence and Multiplicity

Earlier, the research was mainly focused on the classic diffusive two-species Lotka–Volterra competitive system (without advection). Averill et al. [2017] regarded a diffusive two-species Lotka–Volterra competitive system with advection term (with one species having a combination of random dispersal and biased movement upward along the resource gradient, while the other species adopts purely random dispersal). In this paper, we consider a more general diffusive two-species Lotka–Volterra competitive system with advection term (involving both species with a combination of random dispersal and biased movement upward along their respective resource gradients). By utilizing the theory of monotone dynamical systems, spectral analysis, methods of super- and sub-solutions and some nontrivial analytic skills, we finally obtain a rather deep understanding on the global dynamics. Moreover, our results reveal that the dynamical behavior of the system is very complex: exclusion, coexistence and bistability all may happen depending closely on the interspecific competition intensities. More interestingly, by bifurcation theory, we find that the system always admits multiple coexistence steady states when the interspecific competition intensities are near the critical point.

САМОДЕТЕРМИНАЦИЯ КАК ФАКТОР ПОЗИТИВНОЙ ТРАНСФОРМАЦИИ ПРОФЕССИОНАЛЬНО-ТРУДОВОЙ Я-КОНЦЕПЦИИ ЛИЦ С НАРУШЕНИЯМИ ОПОРНО-ДВИГАТЕЛЬНОГО АППАРАТА

The article is devoted to the problem of professional and personal development of persons with musculoskeletal disorders. The purpose of the study was to identify the characteristics of the development of the image of the professional “I” in this category of students and its transformation in the process of studying at the university. According to the developed system-dynamic theory of the professional-labor self-concept of persons with musculoskeletal disorders, the process of its transformation occurs during the period of professional training. The types of such transformation are identified: negative, neutral and positive. The study involved 106 students with musculoskeletal disorders studying in educational institutions of secondary vocational and higher education in various areas of training. It is hypothesized that the dominant factor in the positive transformation of the professional and labor self-concept of students with musculoskeletal disorders is self-determination as their ability to independently manage their professional and personal development, make informed choices, make meaningful decisions taking into account real opportunities and potential abilities. The hypothesis was verified using a self-determination scale and a methodology for identifying students’ self-esteem as subjects of future professional activity. It has been revealed that many students with congenital disorders of the musculoskeletal system are characterized by an orientation towards external reinforcement and a desire for super-achievement, which leads to the formation of maladaptive types of professional and labor self-concept, which impede their successful integration into society. It is concluded that students with musculoskeletal disorders require psychological support aimed at developing their subjective position, developing the need for autonomy and overcoming the syndrome of “learned helplessness” through correction of their motivational sphere and assistance in understanding the internal motives of their future professional career. activities.

Expected First Occurrence Time of Uncertain Future Events in One-Dimensional Linear Systems

The rapid advancement of machine learning algorithms has significantly enhanced tools for monitoring system health, making data-driven approaches predominant in Prognostics and Health Management (PHM). In contrast, model-based approaches have seen modest progress, as they are often constrained by the need for prior knowledge of specific governing equations, limiting their applicability to a wide range of problems. Recently, rigorous theoretical foundations have been established to extend dynamical systems theory by incorporating prognosis of uncertain events. This article leverages this formal framework to introduce and demonstrate a fundamental mathematical result for one-dimensional linear dynamical systems. The presented theorem offers an exact expression for calculating the expected time at which an event will first occur in the future. Unlike typical thresholds, this event is triggered by a hazard zone, defined as an uncertain event likelihood function over the system's state space. Applications of this theorem can be found in implementing real-time prognostic frameworks, where it is crucial to quickly estimate the magnitude of impending failures. Emphasis is placed on minimizing computational burden to facilitate prognostic decision-making.

Stable isotopy connectivity of gradient-like diffeomorphisms of 2-torus

One of the most important problems in the theory of dynamical systems (mentioned in the Palis-Pugh list) is the construction of a stable arc between structural stable diffeomorphisms in the space of diffeomorphisms. The paper considers the gradient-like diffeomorphisms of 2-torus that induce an isomorphism of fundamental groups determined by a matrix (−100−1). We prove that all such diffeomorphisms are stable isotopy connected.

Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions

Compared with time-independent Hamiltonians, the dynamics of generic quantum Hamiltonians H(t) are complicated by the presence of time ordering in the evolution operator. In the context of digital quantum simulation, this difficulty prevents a direct adaptation of many time-independent simulation algorithms for time-dependent simulation. However, there exists a framework within the theory of dynamical systems that eliminates time ordering by adding a “clock” degree of freedom. In this work, we provide a computational framework, based on this reduction, for encoding time-dependent dynamics as time-independent systems. As a result, we make two advances in digital Hamiltonian simulation. First, we create a time-dependent simulation algorithm based on performing qubitization on the augmented clock system and, in doing so, provide the first qubitization-based approach to time-dependent Hamiltonians that goes beyond Trotterization of the ordered exponential. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials and then propose and analyze an algorithm based on these formulas. Unlike other algorithms of similar accuracy, the multiproduct approach achieves commutator scaling, meaning that this method outperforms existing methods for physically local time-dependent Hamiltonians with sufficient smoothness. Our work reduces the disparity between time-dependent and time-independent simulation and indicates a step toward optimal quantum simulation of time-dependent Hamiltonians. Published by the American Physical Society 2024

Notes from the field: Practical considerations for agent‐based and continuous representations of system dynamics

Notes from the field: Practical considerations for agent‐based and continuous representations of system dynamics

Harvesting strategies in ecological systems: Evaluating their efficiency in infection dominance

In population biology, the interplay between prey and predators in the presence of infection can give rise to complex dynamics. On the flip side, implementing harvesting is an infection control measure. In the present work, we use the dynamical system theory to discuss the dynamics of the harvested prey–predator system in the presence of infection in prey species. Detailed mathematical and numerical evaluations have been presented to discuss the susceptible‐free state, infection‐free state, predator‐free state, species coexistence, stability, and occurrence of various bifurcations (saddle‐node, transcritical, and Hopf bifurcation). The study reveals the impact of harvesting parameters on the dynamics. Interestingly, we observe that an infection‐free state could be achieved by varying the harvesting parameter under all three harvesting schemes (linear, quadratic, and nonlinear). Moreover, with the help of reproduction number, we claim that linear harvesting is more effective in controlling the infection than quadratic and nonlinear harvesting provided the half‐saturation constant for nonlinear harvesting is greater than a threshold value ; otherwise, nonlinear harvesting is more effective. Also, the system can support more susceptible prey in the presence of harvesting. The present theoretical study suggests different threshold values of implemented harvesting to control the disease.

Bifurcation analysis, phase portraits and optical soliton solutions of the perturbed temporal evolution equation in optical fibers

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

Leveraging the Interplanetary Superhighway for Propellant–Optimal Orbit Insertion into Saturn–Titan System

This paper presents an innovative approach using Dynamical Systems Theory (DST) for interplanetary orbit insertion into Saturn−Titan three−body orbits. By leveraging DST, this study identifies invariant manifolds guiding a spacecraft into Titan−centered Distant Retrograde Orbits (DROs), strategically selected for their scientific significance. Subsequently, Particle Swarm Optimization (PSO) is employed to fine−tune the insertion parameters, thereby minimizing ΔV. The results demonstrate that the proposed method allows for a reduction in ΔV of over 70% compared to conventional approaches like patched conics−based flybys (2.68 km/s vs. 9.23 km/s), albeit with an extended time of flight, which remains notably faster than weak stability boundary transfers. This paper serves as an interplanetary mission planning methodology to optimize spacecraft trajectories for the exploration of the Saturn−Titan system.

On the Existence of Extreme Coherent Distributions with No Atoms

The paper is devoted to the study of extremal points of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}, the family of all bivariate coherent distributions on [0,1]2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,1]^2$$\\end{document}. It is well-known that the set C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} is convex and weak∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {weak}^*$$\\end{document} compact, and all extreme points of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} must be supported on sets of Lebesgue measure zero. Conversely, examples of extreme coherent measures with a finite or countably infinite number of atoms have been successfully constructed in the literature. The main purpose of this article is to bridge the natural gap between those two results: We provide an example of extreme coherent distribution with an uncountable support and with no atoms. Our argument is based on classical tools and ideas from dynamical systems theory. This unexpected connection can be regarded as an independent contribution of the paper.

Random Walks and Lorentz Processes

Random walks and Lorentz processes serve as fundamental models for Brownian motion. The study of random walks is a favorite object of probability theory, whereas that of Lorentz processes belongs to the theory of hyperbolic dynamical systems. Here we first present an example where the method based on the probabilistic approach led to new results for the Lorentz process: concretely, the recurrence of the planar periodic Lorentz process with a finite horizon. Afterwards, an unsolved problem—related to a 1981 question of Sinai on locally perturbed periodic Lorentz processes—is formulated as an analogous problem in the language of random walks.

Stationary correlation pattern in highly non-stationary MEG recordings of healthy subjects and its relation to former EEG studies.

In this study, we analyze magnetoencephalographic (MEG) recordings from 48 clinically healthy subjects obtained from the Human Connectome Project (HCP) while they performed a working memory task and a motor task. Our results reveal a well-developed, stable interrelation pattern that spans the entire scalp and is nearly universal, being almost task- and subject-independent. Additionally, we demonstrate that this pattern closely resembles a stationary correlation pattern (SCP) observed in EEG signals under various physiological and pathological conditions (the distribution of Pearson correlations are centered at about 0.75). Furthermore, we identify the most effective EEG reference for studying the brain's functional network derived from lag-zero cross-correlations. We contextualize these findings within the theory of complex dynamical systems operating near a critical point of a phase transition.

Changes in Language Learners’ Affect: A Complex Dynamic Systems Theory Perspective

AbstractThis study investigated changes in motivation, self‐efficacy beliefs, and a range of emotions, including enjoyment, hope, pride, curiosity, anxiety, boredom, apathy, confusion, and shame, from a complex dynamic systems theory (CDST) perspective over a 2‐year period in the Hungarian English as a Foreign Language (EFL) context. Using the same questionnaire, we collected data four times throughout 4 semesters from 101 participants studying English in two Hungarian high schools. For data analysis, we used latent growth curve modeling (LGCM) to detect the group‐level changes in learners’ motivation, self‐efficacy, and emotions. We also employed dynamic cluster analysis to identify trends in learners’ trajectories regarding these variables. In our panel data, linear models described the data well concerning the ought‐to second language (L2) self, language learning experience, boredom, apathy, and confusion, and for enjoyment, curiosity, anxiety, and shame, nonlinear models had the best fit. We could also identify trajectories depicting attractor states and learner paths that featured influences of perturbations.

Low-dimensional projection of reactivity classes in chemical reaction dynamics using supervised dimensionality reduction.

Transition state theory (TST) provides a framework to estimate the rate of chemical reactions. Despite its great success with many reaction systems, the underlying assumptions such as local equilibrium and nonrecrossing do not necessarily hold in all cases. Although dynamical systems theory can provide the mathematical foundation of reaction tubes existing in phase space that enables us to predict the fate of reactions free from the assumptions of TST, numerical demonstrations for large systems have been yet one of the challenges. Here, we propose a dimensionality reduction algorithm to demonstrate structures in phase space (called reactive islands) that predict reactivity in systems with many degrees of freedom. The core of this method is the application of supervised principal component analysis, where a coordinate transformation is performed to preserve the dynamical information on reactivity (i.e., to which potential basin the system moves from a region of interest) as much as possible. The reactive island structures are expected to be reflected in the transformed, low-dimensional phase space. As an illustrative example, the algorithm is scrutinized using a modified Hénon-Heiles Hamiltonian system extended to many degrees of freedom, which has three channels leading to three different products from one stable potential basin. It is shown that our algorithm can predict the reactivity in the transformed, low-dimensional coordinate system better than a naïve coordinate system and that the reactivity distribution in the transformed low-dimensional space is considered to reflect the underlying reactive islands.

On the Question of the Concept of “Historical Time” in the Psychology

The article examines the theoretical and methodological problems of historical psychology. The problem of forming one’s own conceptual and categorical apparatus of historical psychology is substantiated, and a scheme for highlighting the essential characteristics of the concept of “historical time” is proposed on the basis of V.I. Razumov’s categorical-system methodology and theory of dynamic information systems. The primary categorical triad allowed us to identify three main characteristics that are essential, necessary and sufficient to describe the content of a particular historical time. It is shown that “historical time” in psychological research can be considered taking into account the entire time context: historical events, historical personalities, worldview, lifestyle. The article identifies another significant aspect of the manifestation and reflection of “historical time” — material material history. The task of considering material history in a specific historical time is set.

Research on the recycling decision of WEEE in rural China under dual supervision

Against the backdrop of promoting green, low-carbon, and high-quality development, this paper aims to promote efficient recycling of waste electrical and electronic equipment (WEEE) in rural China. In this paper, considering the integrated development of urban and rural areas and the standardization of industrial recycling system, under the joint action of the extension of producer responsibility and “dual regulation” of supply and marketing cooperatives, the evolutionary game model and system dynamics model of three-level recycling network of farmers, supply and marketing cooperatives and retailers are established. The mechanism of the participants to promote the recycling of WEEE in rural areas is discussed and the strategic choices and interactive relationships of various entities in the evolutionary process were used to analyze through the evolutionary game method. Meanwhile, using the theory of system dynamics, the main influencing factors of different evolution stages and the dynamic change process of the system are analyzed. The results show that: (1) supply and marketing cooperatives, retailers, and farmers can initially tend to participate in and supervise the recycling of WEEE; however, (2) they can finally achieve strong supervision, actively undertake and participate in the recycling and stabilization stable strategy of rural WEEE depends on their benefits and cost of expenditure expenditures are reasonable. (3) The strategic choice of supply and marketing cooperatives has the most significant impact on the strategic choice of retailers and the strategic choice of retailers has the most significant impact on the strategic choice of farmers.

Study abroad experiences in homestay: where complexity, dynamicity, and individuality stay

Abstract Informed by Complex Dynamic Systems Theory, this study examines the nature of homestay as a situated language learning site, with an eye toward the development of American student guests and Chinese host families’ homestay experiences within the larger context of the study abroad (SA) program and local community. Drawing on data collected from host families and student guests, this study discovers benefits, conflicts, and co-adaptations in homestay; the evolution of host families’ dispositions toward homestay; student guests’ proficiency gains and interactions with hosts and locals; and contextual factors that influence the proficiency gains and interactions. The findings reveal that homestay is a dynamic, self-evolving subsystem, which constantly interacts with other dynamic subsystems, such as the SA program and local community, within the complex social ecosystem of the entire SA environment. This study also provides suggestions for the design and management of the homestay component in the SA program and future research.

The Effects of Constraints on the Variability of Throwing Patterns in Young Children

This research examined how changes in task constraints impacted the throwing patterns of children. The study involved 24 children, with an equal number of males and females, aged 5 and 6. The primary task constraints were the orientation of the target (horizontal or vertical hoops) and the size of the ball (diameters of 6 cm or 12 cm). We observed throwing patterns and analyzed kinematic changes in the preferred throws’ components. Initially, some children transitioned from using two hands to using one hand, and from underhand to overarm throws, particularly when using the larger balls. However, the preferred pattern for most children was one-hand overarm throwing. The kinematic analysis revealed that the participants adapted their throwing technique based on the size of the ball and the orientation of the hoop. The most significant adjustments occurred in the forearm component in response to changes in the target orientation. Notably, when aiming for a vertical hoop, distinct modifications were observed, including elevating the humerus and pulling the hand backward. These findings support the dynamical systems theory, which explains how movement patterns vary during motor development. The study also discussed the potential benefits of using constraints for skill acquisition in physical education settings.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.