Manifold Theory Research Articles

Exploring complex dynamics in a Stackelberg Cournot duopoly game model

Abstract In this study, we investigate the dynamics of a Cournot duopoly game model with bounded rational players, incorporating a leader-follower mechanism where the first player acts as a leader and the second as a follower, aware of the leader’s production. We examine the existence and stability of all fixed points in the model and use center manifold and bifurcation theory to analyze the occurrence and direction of period-doubling and Neimark-Sacker bifurcations at the positive fixed point. To control bifurcation and chaos, feedback control and hybrid control methods are applied. Numerical examples are provided to confirm our theoretical results and reveal the model’s complex dynamics. Our results highlight the critical role of the leader’s strategic decisions, particularly the adjustment speed parameter v 1, in driving the system from stability to chaos, affecting both firms and leading to significant shifts in market dynamics.

Manifold geometry optimization and flow distribution analysis in commercial-scale proton exchange membrane fuel cell stacks

The manifold structure in high-power Proton Exchange Membrane Fuel Cell (PEMFC) stacks critically influences airflow distribution, thereby affecting stack performance, temperature uniformity, and longevity. Therefore, this paper explores how manifold geometry affects fluid distribution within the manifold of a fuel cell stack. In addition, a new step-shaped manifold structure is established to improve the manifold flow field structure and enhance the uniformity of gas flow distribution in each single cell. In our study, each cell within the stack is modeled as a porous medium. We examine how varying the height at different positions within the manifold impacts the uniformity of gas flow distribution, focusing on different length ratios and heights. Our results demonstrate that adjusting the manifold shape improves airflow uniformity, with length ratios having a more pronounced impact than height variations. Specifically, the mass flow coefficient of variation decreased by 22.62 % and 23.07 % for cases 9 and 12, respectively. Similarly, the velocity inhomogeneity coefficient in the inlet manifold decreased by 21.22 % and 21.34 % for cases 9 and 12, respectively. Additionally, pressure distribution in case 9 showed greater uniformity. Our findings indicate that a manifold shape with a length ratio of 2/3 and a height of 2.5 mm delivers optimal performance.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Nonlinear dynamics of a Darwinian Ricker system with strong Allee effect and immigration

In this paper, we investigate the complex dynamics of a Darwinian Ricker system through a comprehensive qualitative and dynamical analysis. Our research shows that the system exhibits Neimark–Sacker bifurcation, period-doubling bifurcation, and codimension-two bifurcations associated with 1:2, 1:3, and 1:4 resonances. These findings are derived using bifurcation and center manifold theories. We numerically illustrate all bifurcation results and chaotic features, providing a thorough understanding of the system’s behavior. This detailed examination of the Darwinian Ricker system, with a focus on the interplay between immigration and the strong Allee effect, enhances our understanding of the intricate mechanisms driving population dynamics. Furthermore, it highlights the significant implications for ecological modeling, particularly in predicting ecosystem responses to external perturbations such as climate change and species invasions.

Distortion corrected kernel density estimator on Riemannian manifolds

Manifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data.

A Novel Nonlinear SAZIQHR Epidemic Transmission Model: Mathematical Modeling, Simulation, and Optimal Control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The "infodemic population" refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number (R_0) is below one and becomes unstable when R_0 exceeds one. The model's behavior at R_0=1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R_0>1, with no backward bifurcation observed when R_0≤1. The EE is also shown to be locally asymptotically stable when R_0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

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.

Light fermion masses in partially deconstructed models

Considering a theory space consisting of a large number of five-dimensional Dirac fermion field theories including background abelian gauge fields, we can construct a theory similar to a continuous six-dimensional theory compactified with two-dimensional manifolds with and without magnetic flux or orbifolds as extra dimensions. This method, called dimensional deconstruction, can be used to construct a model with one-dimensional discrete space, which represents general graph structures. In this paper, we propose the models with two extra dimensions, which resemble two-dimensional tori, cylinders, and rectangular regions, as continuum limits. We also try to build a model that mimics one with the two-dimensional orbifold compactification.

Higher-page Hodge theory of compact complex manifolds

Higher-page Hodge theory of compact complex manifolds

Strange nonchaotic attractors and bifurcation of a four-wheel-steering vehicle system with driver steering control

In this paper, a dynamical model is developed for the four-wheel-steering(4WS)vehicles, with nonlinear lateral tyre forces and quasi-periodic disturbances from the steering mechanism and quasi-periodic pavement external deformation. Initially, the stability and Hopf bifurcation of the 4WS vehicle system without disturbance are analyzed employing the central manifold theory and projection method. The effects of parameters on the stability and the types of Hopf bifurcation for the vehicle system are also discussed. It is shown that both Hopf bifurcation and degenerate Hopf bifurcation occur in the vehicle system. The critical speed decreases with increased driver’s perceptual time delay and distance from the vehicle’s center of gravity to the front axles, while it increases with the control strategy coefficient. However, the increase of the frontal visibility distance of the driver leads to an initial increase of the critical speed, decreasing afterwards. Subsequently, the strange nonchaotic dynamical behaviour of the disturbed 4WS system is investigated. Several routes and mechanisms for the birth of strange nonchaotic attractors (SNAs) are identified, including the torus doubling bifurcation, the torus fractalization, and the loss of transverse stability of a torus. Finally, the nonchaotic property is verified through the calculation of the maximum Lyapunov exponent, and the strange property is characterized using power spectra and rational approximation.

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Geometry of K-trivial Moishezon manifolds: decomposition theorem and holomorphic geometric structures

Geometry of K-trivial Moishezon manifolds: decomposition theorem and holomorphic geometric structures

Geometry of Statistical Sequential Warped Product Manifolds.

The main purpose of the present study is to generalize the dualistic structures, specially the statistical structures on warped product manifolds to statistical structures on sequential warped product manifolds. On the one hand we proved that the statistical structure on sequential warped product manifold induces statistical on the factors and conversely. On the other hand we proved that if the statistical warped product is dually at space, then the base manifold is also dually at and ber is a space form, i.e. of constant sectional curvature.

Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.

Rigidity of flat holonomies

Abstract We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension $n \geq 3$ with strongly $1/4$ -pinched or relatively $1/2$ -pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

Organelle landscape analysis using a multiparametric particle-based method.

Organelles have unique structures and molecular compositions for their functions and have been classified accordingly. However, many organelles are heterogeneous and in the process of maturation and differentiation. Because traditional methods have a limited number of parameters and spatial resolution, they struggle to capture the heterogeneous landscapes of organelles. Here, we present a method for multiparametric particle-based analysis of organelles. After disrupting cells, fluorescence microscopy images of organelle particles labeled with 6 to 8 different organelle markers were obtained, and their multidimensional data were represented in two-dimensional uniform manifold approximation and projection (UMAP) spaces. This method enabled visualization of landscapes of 7 major organelles as well as the transitional states of endocytic organelles directed to the recycling and degradation pathways. Furthermore, endoplasmic reticulum-mitochondria contact sites were detected in these maps. Our proposed method successfully detects a wide array of organelles simultaneously, enabling the analysis of heterogeneous organelle landscapes.

An investigation of Susceptible–Exposed–Infectious–Recovered (SEIR) tuberculosis model dynamics with pseudo-recovery and psychological effect

Tuberculosis is one of the most pressing issues of the modern era, posing a severe health risk to humans in recent decades. This study proposes a Susceptible–Exposed–Infectious–Recovered (SEIR) tuberculosis epidemic transmission model with psychological effects and pseudo-recovery. We consider a compartmental mathematical model in which the entire population is divided into four compartments based on their natural features. The model is validated, and parameter values are estimated using Indonesian data from 2002 to 2022. To investigate their epidemiological significance, we proved the positivity and boundedness of solutions, as well as the local and global stability of equilibria. Sensitivity analysis is used to find the most influential parameters with the most significant influence on the basic reproduction number, R0. The bifurcation procedure tools of the center manifold theory are used to conduct a bifurcation study. Mathematical conditions ensure the inferred event of forward bifurcation. We performed numerical simulations that support our theoretical findings.

Scalable imaging-free spatial genomics through computational reconstruction.

Tissue organization arises from the coordinated molecular programs of cells. Spatial genomics maps cells and their molecular programs within the spatial context of tissues. However, current methods measure spatial information through imaging or direct registration, which often require specialized equipment and are limited in scale. Here, we developed an imaging-free spatial transcriptomics method that uses molecular diffusion patterns to computationally reconstruct spatial data. To do so, we utilize a simple experimental protocol on two dimensional barcode arrays to establish an interaction network between barcodes via molecular diffusion. Sequencing these interactions generates a high dimensional matrix of interactions between different spatial barcodes. Then, we perform dimensionality reduction to regenerate a two-dimensional manifold, which represents the spatial locations of the barcode arrays. Surprisingly, we found that the UMAP algorithm, with minimal modifications can faithfully successfully reconstruct the arrays. We demonstrated that this method is compatible with capture array based spatial transcriptomics/genomics methods, Slide-seq and Slide-tags, with high fidelity. We systematically explore the fidelity of the reconstruction through comparisons with experimentally derived ground truth data, and demonstrate that reconstruction generates high quality spatial genomics data. We also scaled this technique to reconstruct high-resolution spatial information over areas up to 1.2 centimeters. This computational reconstruction method effectively converts spatial genomics measurements to molecular biology, enabling spatial transcriptomics with high accessibility, and scalability.

Constrained Model Predictive Control for Generation Power Distribution on Aircraft Engines

Aiming at the increasing demand for electric energy in aircraft in the future, a multi-objective optimization aircraft engine constrained model predictive control method based on generation power distribution is proposed. Firstly, based on the aircraft engine component level model and the equilibrium manifold theory, the aircraft engine equilibrium manifold expansion model is established. Secondly, the influence of the power generation is modeled, and the influence of the low- and high-pressure shaft generators on the normal operation of the aircraft engine is studied and compared. The control variables such as fuel flow and total generation power are taken as the constraint conditions to design the constraint model predictive controller. Furthermore, the multi-objective grey wolf optimization algorithm is introduced to intelligently optimize the parameters of the designed controller. At last, the simulation based on the component level model shows that the high-pressure shaft generator has less influence on the state quantity, including engine thrust, than the low-pressure shaft generator. The proposed control method using the multi-objective gray wolf optimization (MOGWO) algorithm has rapid response and no steady-state error.

Complex dynamical properties and chaos control for a discrete modified Leslie-Gower prey-predator system with Holling II functional response

In this study, the semi-discretization technique is employed to establish a discrete representation of a modified Leslie-Gower prey-predator system that includes a Holling II type functional response. The dynamics of this model are then analyzed through the application of center manifold theory and bifurcation theory. We present comprehensive results for the local stability of the fixed points across the entire parameter space. Additionally, we provide sufficient conditions for the occurrence of flip bifurcation and Neimark-Sacker bifurcation. Besides, the system has experienced a flip bifurcation to chaos controlled using the method of chaos control, viz., state feedback method, pole placement technique, and hybrid control strategy. Furthermore, we provide specific conditions to ensure that bifurcation and chaos can be stabilized. Finally, numerical simulations are conducted to validate theoretical analysis and illustrate several new complex dynamical behaviors between two species.