Dynamical Systems Theory Research Articles

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

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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.

Исследование многоуровневых моделей управления вагонопотоками в адрес припортовых станций с применением теории дифференциальных уравнений

Purpose: To consider the issue of sustainability of multi-level control systems for car flows to port railway stations in the conditions of intellectualization of transportation management processes. To assess the influence of three-level and two-level control systems for the operational work of port roads on the quality of planning the supply of trains to port stations under various modes of operational operation of railways. Determine the conditions for the stability of the car flow control system by constructing and studying the corresponding hard and soft mathematical models described by autonomous systems of ordinary linear differential equations. Methods: In this study, the methods of the theory of stability of solutions of differential equations, as well as the theory of control and dynamical systems are used. A phase space is used in a visual representation, a qualitative and quantitative analysis of the behavior of phase trajectories in it is performed, corresponding to undisturbed and perturbed solutions of systems of differential equations describing the constructed models of control systems. Results: A comparison of the characteristics of multistage car traffic control systems is given. A study of the stability of multilevel models of car traffic management in the context of the intellectualization of management functions is proposed. In the environment of the analytical computing system, the conditions of asymptotic stability in time of a two-level wagon traffic control system are found and studied. Practical importance: The results obtained by computational experiment make it possible to assess the stability of the functioning of port-side transport and technological systems in the context of changes in the organization of production and the intellectualization of transportation management processes. Computer mathematics systems make it possible to implement the heuristic component of research and obtain theoretically sound and visual solutions to problems of optimizing cargo and wagon traffic control modes.

Challenges and Approaches in the Study of Neural Entrainment.

When exposed to rhythmic stimulation, the human brain displays rhythmic activity across sensory modalities and regions. Given the ubiquity of this phenomenon, how sensory rhythms are transformed into neural rhythms remains surprisingly inconclusive. An influential model posits that endogenous oscillations entrain to external rhythms, thereby encoding environmental dynamics and shaping perception. However, research on neural entrainment faces multiple challenges, from ambiguous definitions to methodological difficulties when endogenous oscillations need to be identified and disentangled from other stimulus-related mechanisms that can lead to similar phase-locked responses. Yet, recent years have seen novel approaches to overcome these challenges, including computational modeling, insights from dynamical systems theory, sophisticated stimulus designs, and study of neuropsychological impairments. This review outlines key challenges in neural entrainment research, delineates state-of-the-art approaches, and integrates findings from human and animal neurophysiology to provide a broad perspective on the usefulness, validity, and constraints of oscillatory models in brain-environment interaction.

On thermally driven fluid flows arising in astrophysics

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak-strong uniqueness principle

Tipping prediction of a class of large-scale radial-ring neural networks

Understanding the emergence and evolution of collective dynamics in large-scale neural networks remains a complex challenge. This paper seeks to address this gap by applying dynamical systems theory, with a particular focus on tipping mechanisms. First, we introduce a novel (n+mn)-scale radial-ring neural network and employ Coates’ flow graph topological approach to derive the characteristic equation of the linearized network. Second, through deriving stability conditions and predicting the tipping point using an algebraic approach based on the integral element concept, we identify critical factors such as the synaptic transmission delay, the self-feedback coefficient, and the network topology. Finally, we validate the methodology’s effectiveness in predicting the tipping point. The findings reveal that increased synaptic transmission delay can induce and amplify periodic oscillations. Additionally, the self-feedback coefficient and the network topology influence the onset of tipping points. Moreover, the selection of activation function impacts both the number of equilibrium solutions and the convergence speed of the neural network. Lastly, we demonstrate that the proposed large-scale radial-ring neural network exhibits stronger robustness compared to lower-scale networks with a single topology. The results provide a comprehensive depiction of the dynamics observed in large-scale neural networks under the influence of various factor combinations.

An approach to formalization of WEEE informal collectors: Collection model under a membership-based community

Informal collectors’ formalization has become a key challenge that developing countries such as China face in governing informal recycling. This study establishes an innovative collection model of waste electrical and electronic equipment (WEEE) under the membership community operation system, which integrates waste merchants into the formal recycling system in the form of members. Based on a Bass model of membership community development and an evolutionary game model between recyclers and waste merchants, respectively, this study uses system dynamics theory to simulate and analyze the model’s implementation effectiveness and influencing factors. We find that the membership community’s interpersonal promotion and customer relationship management strategies can increase waste merchants’ membership rate and membership loyalty, which is an effective way to incorporate waste merchants into the formal recycling system. Waste merchants’ main interest demands in the formalization process are to gain economic benefits and identity recognition, which motivate them to become members. Membership’s transaction cost establishes a long-term cooperative relationship between waste merchants and recyclers in the form of an economic contract. However, the potential losses from information sharing are the key factor restricting waste merchants from enrolling as members, which has also become the main obstacle to their formalization. This study provides a new theoretical and empirical basis for informal collectors’ formalization, which has important practical significance for WEEE recycling and management in developing countries.