Invariance Principle Research Articles

Wasserstein convergence rates in the invariance principle for sequential dynamical systems

Wasserstein convergence rates in the invariance principle for sequential dynamical systems

An invariance principle for the 1D KPZ equation

An invariance principle for the 1D KPZ equation

Analysis of COVID-19’s Dynamic Behavior Using a Modified SIR Model Characterized by a Nonlinear Function

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.

UNRAVELING THE IMPACT OF THE MEMORY, THE COMPETITION, AND THE LINEAR HARVESTING ON A LOTKA-VOLTERRA MODEL

The harvesting of population has a dominant influence in balancing the ecosystem. In this manuscript, the impact of harvesting in addition to competition, and memory effect on a prey-predator interaction following the Lotka-Volterra model is studied. The mathematical validation is provided by proofing that all solutions of the model are always exist, non-negative, and bounded. Obeying Matignon condition, Lyapunov function, and generalized LaSalle invariance principle, the local and global stability are investigated. To complete the analytical results, some numerical simulations are given to show the occurrence of forward bifurcation and the impact of the memory index. All results state that three possible circumstances may occur namely the extinction of both populations, the prey-only population, and the co-existence of both populations.

Invariance Principle and Berry–Esseen Bound for Error Variance Estimator for Pth-Order Nonlinear Autoregressive Models

The invariance principle and Berry–Esseen bound for an error variance estimator based on the residuals are established by using a Taylor expansion and the classical invariance principle and Berry–Esseen bound for independent random variables. Some examples are given to illustrate their applications.

Multiscale scanning with nuisance parameters

Abstract We develop a multiscale scanning method to find anomalies in a d-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic, which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate.

Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

Simulation Research on the Control Method of Bow-Collapse in Gear Cold Roll-Beating

Bow-collapse is a type of geometric defect in the gear cold roll-beating process. In order to effectively control bow-collapse and improve material utilization, this paper calculates the cross-section radius of the blank according to the volume invariance principle of metal plastic deformation, and then performs FE simulation of the cold roll-beating process by using the cyclic beating of adjacent tooth spaces and intermittent blank feeding method. The flow state of metal particles is investigated, revealing that the movement of metal particles along the axial direction of the blank is the main reason for the formation of bow-collapse. This paper proposes a method to correct the cross-section radius of the blank and thus control for bow-collapse, where the loss coefficient K characterizes the state of metal particle loss in an infinitesimal thickness region on the cross-section. The analytical equation for calculating the corrected value of the cross-section radius is derived, and the corrected value is calculated. Conducted on the modified blank, cold roll-beating FE simulation results show that bow-collapse is effectively controlled, and the loss coefficient K correctly reflects the loss state of metal particles on the cross-section. The simulation results also show that after cold roll-beating, the gear teeth with standard face width and tooth depth can be obtained after two turning end faces and turning tip circle processes. The bow-collapse control method proposed in this paper effectively improves material utilization.

A universal drag model for liquid-solid fluidization: Experiment, data-driven modeling, CFD modeling and simulation

Various industrial applications are performed in liquid-solid fluidized beds where drag force plays a significant role in affecting the expansion characteristics of granular-bed, and then determines the mass/heat transfer performance. This study employs dimensionless learning data-driven modeling method, which is derived from the principle of dimensional invariance, to automatically discover the relationship between the drag coefficient and hydraulic dimensionless numbers from the liquid-solid fluidization data. It is found that the Fr number (=ul2/gds) also plays important role in improving the prediction accuracy of drag model except for Re number (=dsulρl/μl). The proposed data-driven modeling method has desired robustness, and the yielded drag model can be applicable to other liquid-solid systems, such as water-polystyrene spheres and water-coal particles, although it is derived from the fluidization of spherical glass beads in rising tap-water. The proposed drag model can also provide good CFD simulation results that agree very well with the experiment data with the relative error less than 5 %.

Dynamics of a Delayed Prey-Predator System With Crowley-Martin Functional Response

ln this paper, an ecological predator-prey model with time delay is proposed and studied analytically and numerically. The stability and bifurcation analysis of the proposed model with a Crowley-Martin response function are covered in this article. In the beginning, equilibrium points of the proposed model are identified. Secondly, the local asymptotic stability of the equilibria and Hopf-bifurcation are discussed by the characteristic equations of the system. Finally, by creating a suitable Lyapunov function and LaSalle’s invariance principle, the equilibria's global asymptotic stability is examined.

Anti-swing position control of super-twisting sliding mode for underactuated tower crane based on nonlinear marine predator algorithm

Due to the nonlinear, underactuated, and strongly coupled characteristics of tower cranes, issues such as inaccurate load positioning and severe swinging are prone to occur during transportation, making the control problem extremely complex. To address these issues, this paper has designed a super-twisting robust sliding mode controller based on the nonlinear marine predator algorithm (NMPA) to ensure the accurate positioning of the desired location during the transportation process of the tower crane while reducing the oscillation of the load. First, a nonlinear dynamic model of a four-degree-of-freedom tower crane considering friction is constructed. Second, to address the difficulty of effectively controlling the underactuated tower crane system model due to its nonlinearity and strong coupling, the super-twisting algorithm is introduced, and a super-twisting sliding mode controller for anti-swing positioning during the crane transportation process is designed. Then, the parameters of the super-twisting sliding mode controller are optimized using the NMPA to suppress sliding mode chattering and quickly achieve the system’s steady state. Finally, the stability of the nonlinear tower crane system is analyzed using Lyapunov’s theory and LaSalle’s invariance principle to ensure the closed-loop stability of the error dynamics. The simulation results show that the proposed control method exhibits excellent load anti-sway performance and robustness.

Bifurcation Analysis and Optimal Control of an SEIR Model with Limited Medical Resources and Media Impact on Heterogeneous Networks

The spread of epidemic diseases is often considered to be bound up with human behavior. Both treatment and awareness-raising are seen as important preventive measures during an outbreak. In this paper, an SEIR model on heterogeneous network containing saturated treatment function with media information variable is constructed. Through mathematical analysis, the basic reproduction number and equilibriums of the system are derived. The global stability of disease-free equilibrium is further proved based on LaSalle’s invariant principle. It is also found that with a gradual increase of the delayed treatment parameter, the system exhibits a transition from forward bifurcation to backward bifurcation with the variation of the basic reproduction number. Finally, an optimal control problem is formulated with both treatment rate and media information dissemination rate as control variables and solved using the Pontryagin’s maximum principle. All the above results are verified by numerical experiments.

Electron surface scattering kernel for a plasma facing a semiconductor.

Employing the invariant embedding principle for the electron backscattering function, we present a scheme for constructing an electron surface scattering kernel to be used in the boundary condition for the electron Boltzmann equationof a plasma facing a semiconducting solid. The scheme takes the solid's microphysics responsible for electron emission and backscattering from the interface within a randium-jellium model into account and is applicable to dielectrics and metals as well. As an illustration, we consider silicon and germanium, describing the interface potential by a Schottky barrier and including impact ionization across the energy gap as well as scattering on phonons and ion cores. The emission yields deduced from the kernel agree well enough with measured data to support its use in the electron boundary condition of a plasma facing silicon or germanium.

Modeling of KSHV/HHV-8 and HIV-1 Co-Dynamics in Vivo

Human immunodeficiency virus kind 1 (HIV-1) compromises the immune system by infecting and damaging CD4+ T cells. Infection can progress to the ultimate stage, acquired immune deficiency syndrome (AIDS), if HIV-1 therapy is not received. People living with HIV/AIDS are more vulnerable to infections that they otherwise wouldn’t develop. Opportunistic infections or malignancies are the terms used to describe them. Kaposi sarcoma (KS) is an AIDSrelated malignancy caused by Kaposi’s sarcoma-associated herpesvirus (KSHV) (also known as human herpesvirus 8 (HHV-8)). HIV-1 and KSHV co-infection cases has been shown in several studies. Using a system of ODEs, we develop a new mathematical model to study the co-dynamics of HIV-1 and KSHV in vivo. The model includes interactions between healthy CD4+ T cells, HIV-1-infected CD4+ T cells, HIV-1 particles, healthy B cells, KSHV-infected B cells, and KSHV particles. By analyzing the boundedness and nonnegativity of the solutions, we prove the mathematical well-posedness and biological compatibility of the model. The existence and stability of the model’s steady states are established by four threshold values that we identify. We prove that steady states are globally asymptotically stable by using Lyapunov’s method and LaSalle’s invariance principle. Numerical simulations are used to display the results. For both basic reproduction ratios of HIV-1 mono-infection (R1) and KSHV mono-infection (R2), sensitivity analysis is carried out. A comparison between HIV-1 or KSHV mono-infections and co-infections with HIV-1 and KSHV is given. Empirical evidence indicates that co-infection results in higher KSHV and HIV-1 concentrations compared to mono-infection cases. This result is in line with a number of findings found in the literature.

Fault geometry invariance and dislocation potential in antiplane crustal deformation: physics-informed simultaneous solutions

Earthquake-induced crustal deformation provides valuable insights into the mechanisms of tectonic processes. Dislocation models offer a fundamental framework for comprehending such deformation, and two-dimensional antiplane dislocations are used to describe strike-slip faults. Previous earthquake deformation analyses observed that antiplane dislocations due to uniform fault slips are influenced predominantly by fault tips. Here, we state a general principle of fault geometry invariance in antiplane dislocations and exploit its theoretical consequence to define dislocation potentials that enable a streamlined crustal deformation analysis. To demonstrate the benefits of this theory, we present an analytical example and construct a rapid numerical solver for crustal deformation caused by variable fault slip scenarios using physics-informed neural networks, whose mesh-free property is suitable for modeling dislocation potentials. Fault geometry invariance and the dislocation potential may further the analysis of antiplane crustal deformation, particularly for uncertainty quantification and inversion analysis regarding unknown fault geometries in realistic crustal structures.

Rates in Almost Sure Invariance Principle for Nonuniformly Hyperbolic Maps

We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski’ (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai’s trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past.

Modeling the Impacts of Diagnosis and Treatment of Snakebite Envenoming Victims with Anti-Snake Venom in a Community

Snakebite envenoming (SBE) is one of the major causes of deaths and disabilities in tropical and subtropical countries especially in the continents of Asia and Africa. Report revealed that on average, 5-8 and about 3,000 cases of SBE are recorded daily and yearly respectively in northeast Nigeria which population are largely agriculturists. Thus, snakebite (SB) led to economy setback and slows down productivity among the workforce as a result of disabilities induced by SBE. Consequently, the revenue generation even including income facing declined. This paper proposes population-based mathematical model for studying the dynamic impacts of diagnosis and treatment of snakebite envenoming victims with anti-snake venom in a community. The model is rigorously analyzed using relevant techniques and theorems in dynamical systems. The model was validated using monthly reported data on snakebite envenoming from 2020–2022, collected from snakebite treatment and research hospital (STRH) Kaltungo in northeast Nigeria. The qualitative study revealed that the model has snakebite-envenoming-free and snakebite-envenoming-endemic equilibrium points. It is shown via the Lyapunov function in conjunction with the Lassale invariance principle that these two equilibrium points are globally asymptotically stable. Furthermore, numerical simulations of the model were carried out to investigate the impact of the diagnosis of snakebite envenoming victims and the treatment of the same on the burden of snakebite envenoming. The results of the simulations revealed that appropriate diagnosis and treatment of snakebite envenoming victims with anti-snake venom could significantly reduce the number of snakebite-induced deaths and disabilities. This work can serve as a framework for making policy and decision in addressing the issues of SBE in the public healthcares in the study region and beyond.

Hybrid feedback control of PMSG-based WECS with multilevel flying-capacitor inverter enhanced by fractional order extremum seeking

This paper proposes an original hybrid control strategy for a three-phase multilevel flying capacitor inverter (FCI) used in wind energy conversion system (WECS). A state feedback law, computed using a Lyapunov function and an invariance principle, is designed to manage grid current control and capacitor voltage balancing, guaranteeing the asymptotic stability of the closed-loop system. Additionally, a fractional-order extremum seeking control (FOESC) algorithm for Maximum Power Point Tracking (MPPT) is introduced, which optimizes power capture by adjusting turbine speed from the DC side without requiring a detailed system model (model-free approach). The use of fractional-order calculus leads to improved convergence and more seamless performance while preserving the robust characteristics of the system. The proposed control strategies are validated through simulation, demonstrating improved transient response and stability under varying wind conditions and parameter uncertainties. These results highlight the potential of advanced control algorithms to enhance the efficiency and reliability of wind energy systems, contributing to global efforts to achieve sustainable energy goals.

Ground-reflected light: The invariance principle and the effect of luminaire height, emission pattern, and non-uniform albedo

The weighted contribution of ground-reflected component of light emissions from artificial sources gradually increases as the transitioning from bad-shielded to modernized light sources with low or zero direct uplight takes place in cities or towns. In this work, we demonstrate that the modeling of reflected light on a large domain for Lambertian flat surfaces does not require information about the height of the light sources and directional distribution of photons their produce. This kind of "invariance principle" becomes invalid when the homogeneity condition for the surface albedo is violated. However, we have shown that an analytical solution exists also for position-dependent albedo and even for angle-dependent reflectance which is the effect we now include to the light pollution models for the first time. This effect is known from the observation of sunbeams entering uneven surfaces at different zenith angles. Here in analogy with that daylight model we derive formulae for ground surfaces illuminated by artificial lights located at different heights above the surrounding terrain.

Stability and Hopf Bifurcation of a Cytokine-Enhanced HIV Infection Model with Saturation Incidence and Three Delays

To discuss the influence of cytokine-enhanced rate and antibody delay on viral infection, in this paper, we propose a cytokine-enhanced HIV model with saturation incidence, antibody immune response and time delays. Multiple time delays, namely intracellular delay, virus replication delay and antibody delay, are included. First, we prove that the model is well-posed. Then, we obtain two basic reproductive numbers of the model. Next, the stability of the equilibria is investigated by using Lyapunov functions and LaSalle’s invariance principle. The results show that intracellular delay and virus replication delay do not impact on the stability of the three equilibria. However, if antibody immune response delay is positive, we determine some conditions for stability switches of the endemic equilibrium by using antibody immune response delay as a bifurcation parameter. It is concluded that with the increase of antibody immune response delay, the endemic equilibrium loses its stability and the system generates Hopf bifurcation. Finally, the numerical simulation results also show the correctness of the theoretical results.