Unique Equilibrium State Research Articles

On the ergodic theory of impulsive semiflows

In this work, we established a simpler criterion for the existence of invariant measures for impulsive semiflows. In addition, we exhibit sufficient conditions to obtain a Variational Principle and the existence and uniqueness of equilibrium states. Some examples are provided to illustrate the theory developed.

Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity (ω,Ω) may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus Ω (precisely limx→0+supdΩ(dx)/Ω(d)=0 ) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having ω as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with Ω as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville–Pomeau family.

MATHEMATICAL ANALYSIS OF COVID-19 INFECTION MODEL WITH DEMOGRAPHIC DYNAMICS

This study developed a deterministic mathematical model of COVID-19 infection by incorporating asymptomatically and symptomatically infectious individuals, the vital dynamics such as birth rate and mortality rate. Face mask use, diagnosis of asymptomatic infectious individuals, and isolation of infected individuals as control strategies are also incorporated. The model is shown to have two unique equilibrium states, namely: the disease-free equilibrium points and the endemic equilibrium point. The result from the stability analysis of the critical points is shown to be local asymptotic stable and also, globally asymptotically stable provided the basic reproduction number is less than one (, and the endemic equilibrium state is local asymptotic stable and also, globally asymptotically stable provided . Furthermore, results of the sensitivity index on for the different parameters of the model show that the recruitment rate and the effective contact rate are the most sensitive parameters and thus critical in disease management and eradication. Thus, efforts geared at reducing the recruitment of susceptible individuals and infection transmission rate will significantly eliminate the disease burden.

Nonessentiality of Reservoir's Fading Memory for Universality of Reservoir Computing.

This article describes a novel sufficient condition concerning approximations with reservoir computing (RC). Recently, RC using a physical system as the reservoir has attracted attention. Because many physical systems are modeled as state-space systems, it is necessary to guarantee the approximations given by reservoirs represented as nonlinear state-space systems. There are two problems with existing approaches: a reservoir must have a property called fading memory and must be represented as a set of maps between input and output signals on the bi-infinite-time (BIT) interval. These two conditions are too strict for reservoirs represented as nonlinear state-space systems as they require the reservoir to have a unique equilibrium state for the zero input. This article proposes an approach that employs operators from right-infinite-time (RIT) inputs to RIT outputs. Furthermore, we develop a novel extension of the Stone-Weierstrass theorem to handle discontinuous functions. To apply the extended theorem, we define functionals corresponding to operators and introduce a metric on the domain of the functionals. The resulting sufficient condition does not require the reservoir to have fading memory or continuity with respect to inputs and time. Therefore, our result guarantees the approximations with very common reservoirs and provides a rationale for physical RC. We present an example of a physical reservoir without fading memory. With the example reservoir, the RC model successfully approximates NARMA10, a benchmark task for time series predictions.

Almost Surely Exponential Convergence Analysis of Time Delayed Uncertain Cellular Neural Networks Driven by Liu Process via Lyapunov-Krasovskii Functional Approach.

As with probability theory, uncertainty theory has been developed, in recent years, to portray indeterminacy phenomena in various application scenarios. We are concerned, in this paper, with the convergence property of state trajectories to equilibrium states (or fixed points) of time delayed uncertain cellular neural networks driven by the Liu process. By applying the classical Banach's fixed-point theorem, we prove, under certain conditions, that the delayed uncertain cellular neural networks, concerned in this paper, have unique equilibrium states (or fixed points). By carefully designing a certain Lyapunov-Krasovskii functional, we provide a convergence criterion, for state trajectories of our concerned uncertain cellular neural networks, based on our developed Lyapunov-Krasovskii functional. We demonstrate under our proposed convergence criterion that the existing equilibrium states (or fixed points) are exponentially stable almost surely, or equivalently that state trajectories converge exponentially to equilibrium states (or fixed points) almost surely. We also provide an example to illustrate graphically and numerically that our theoretical results are all valid. There seem to be rare results concerning the stability of equilibrium states (or fixed points) of neural networks driven by uncertain processes, and our study in this paper would provide some new research clues in this direction. The conservatism of the main criterion obtained in this paper is reduced by introducing quite general positive definite matrices in our designed Lyapunov-Krasovskii functional.

Optimally analyzed fractional Coronavirus model with Atangana–Baleanu derivative

In this article, we develop a nonlinear SEIQHR fractional model with Atangana–Baleanu (ABC) derivative for the Corona virus disease (Covid-19). It is significant to mention that using a fractional-order derivative can provide more intricate insights into the complex dynamics of underlying models. We provide two unique equilibrium states of the model in order to analyze the problem. To explore the long-term dynamics of a disease, a threshold parameter for the model utilizing next-generation approach is computed. Local and global asymptotic behaviors of the proposed model at both the equilibrium states are established by imposing some necessary conditions on threshold parameter. To validate our obtained analytical results and to examine the importance of arbitrary order derivative, we implement a recently proposed Toufik–Atangana numerical technique. To reach our conclusions, we investigate a thorough quantitative analysis of the model through the adjustment of quarantine and hospitalization rates as a first constant control technique. Through numerical experiments, it is asserted that the Covid-19 pandemic may be eliminated more quickly if a human community selfishly adopted both of the necessary control measures at various coverage levels with appropriate awareness. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. In addition, the bifurcation nature of the Covid-19 model is examined. Furthermore, we develop an optimal control problem along with the associated optimality conditions of Pontryagin type to discover the most effective controls for the both strategies, one for exposed and another for infected individuals. The goal is to reduce both the financial burden of executing these strategies as well as the number of exposed and infected people. The extremals are obtained numerically. The effectiveness and efficiency of the optimal control strategy are finally demonstrated by numerical simulations before and after the optimization. The importance of the current research work is the use of developed structure preserving Toufik-Atangana numerical scheme, backward in time, for the first time to analyze optimally the epidemic models, for example the proposed SEQIHR Covid-19 model.

A remark on the phase transition for the geodesic flow of a rank one surface of nonpositive curvature

For any rank 1 nonpositively curved surface M , it was proved by Burns-Climenhaga-Fisher-Thompson that for any q < 1 , there exists a unique equilibrium state μ q for q φ u , where φ u is the geometric potential. We show that as q → 1 − , the weak-* limit of μ q is the restriction of the Liouville measure to the regular set.

Framework for solving dynamics of Ca2+ ion concentrations in liver cells numerically: Analysis of a non‐negativity‐preserving non‐standard finite‐difference method

In this article, we consider the numerical approximation of dynamics of calcium ion concentrations in liver cells by a non‐linear first‐order system of ordinary differential equations. At first, we summarize some analytical results of the time‐continuous problem formulation which are of importance to numerically solving this system. We continue with the presentation of our numerical framework which contains two ingredients—an optimization approach for determining the unique equilibrium state and, as our main contribution, a non‐standard finite‐difference scheme for numerical approximation of the time‐discretized dynamical system. We prove that our proposed scheme conserves non‐negativity, and we show convergence towards the time‐continuous unique solution. Conclusively, we stress our theoretical findings by numerical examples.

Transmission dynamics of a novel HIV/AIDS model through a higher-order Galerkin time discretization scheme

There are numerous contagious diseases caused by pathogenic microorganisms, including bacteria, viruses, fungi, and parasites, that have the propensity to culminate in fatal consequences. A communicable disease is an illness caused by a contagion agent or its toxins and spread directly or indirectly to a susceptible animal or human host by an infected person, animal, vector, or immaterial environment. Human immunodeficiency virus (HIV) infection, hepatitis A, B, and C, and measles are all examples of communicable diseases. Acquired immunodeficiency syndrome (AIDS) is a communicable disease caused by HIV infection that has become the most severe issue facing humanity. The research work in this paper is to numerically explore a mathematical model and demonstrate the dynamics of HIV/AIDS disease transmission using a continuous Galerkin–Petrov time discretization of a higher-order scheme, specifically the cGP(2)-scheme. Depict a graphical and tabular comparison between the outcomes of the mentioned scheme and those obtained through other classical schemes that exist in the literature. Further, a comparison is performed relative to the well-known fourth-order Ruge–Kutta (RK4) method with different step sizes. By contrast, the suggested approach provided more accurate results with a larger step size than RK4 with a smaller step size. After validation and confirmation of the suggested scheme and code, we implement the method to the extended model by introducing a treatment rate and show the impact of various non-linear source terms for the generation of new cells. We also determined the basic reproduction number and use the Routh-Hurwitz criterion to assess the stability of disease-free and unique endemic equilibrium states of the HIV model.

Investigating day-to-day route choices based on multi-scenario laboratory experiments, Part I: Route-dependent attraction and its modeling

In the area of urban transportation networks, a growing number of day-to-day (DTD) traffic dynamic theories have been proposed to describe the network flow evolution, and an increasing amount of laboratory experiments have been conducted to observe travelers' behavior regularities. However, the "communication" between theorists and experimentalists has not been made well. This paper devotes to 1) detecting unanticipated behavior regularities by conducting a series of laboratory experiments, and 2) improving existing DTD dynamics theories by embedding the observed behavior regularities into a route choice model. First, 312 subjects participated in one of the eight decision-making scenarios and make route choices repeatedly in congestible parallel-route networks. Second, three route-switching behavior patterns that cannot be fully explained by the classic route-choice models are observed. Third, to enrich the explanation power of a discrete route-choice model, behavioral assumptions of route-dependent attractions, i.e., route-dependent inertia and preference, are introduced. An analytical DTD dynamic model is accordingly proposed and proven to steadily converge to a unique equilibrium state. Finally, the proposed DTD model could satisfactorily reproduce the observations in various datasets. The research results can help transportation science theorists to make the best use of laboratory experimentation and to build network equilibrium or DTD dynamic models with both real behavioral basis and neat mathematical properties.

Empirical Validation of Free Vibration of Sagging Cables

Sagging cables lack unique natural state, but possess unique equilibrium state under sustained loads. However, for the purpose of predicting their dynamic response, the static equilibrium state of heavy cables under self-weight is assumed to be their initial state. The theory of sagging elastic cables presented earlier by the Authors pertains to weightless cables carrying nodal masses apart from static and inertial forces. For applying the proposed theory for sagging heavy cables, the cable is divided into ten segments of equal length. The initial equilibrium configuration of the cable fixed at both the ends is determined by the weights of these lumped nodal masses applied at ten nodes with one end fixed. The stiffness matrix of the cable is determined from tangent configurational flexibility matrix and tangent elastic flexibility matrix of twenty degree of freedom system. The cable tied at both the ends in an equilibrium state is set into motion by releasing it at one end. The theoretically predicted response of the cable is compared with the experimental data from the two investigations. The first experimental investigation by Fried pertains to the planar undamped vibrations of a sagging elastic cable, in which cable is released from its right end and its configurations are plotted at regular time intervals of 0.001s. In the second investigation by Koh, Zhang and Quek, the configurations of a different cable at regular time intervals of 0.125s during half the vibration cycle is measured. The dynamic tensile force at the fixed end has also been predicted in the latter case. The theoretical prediction of the evolving cable configuration have been found to be compatible with experimental data.

Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional

We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue corresponding to strictly positive eigenfunction which is expressed as . By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for . For , and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function.

A new interpretation of TCM pulse diagnosis based on quantum physical model of the human body

Following the quantum theory-based physical model of the human body, a new interpretation of the traditional Chinese medicine (TCM) principle of “Cunkou reads viscera” is presented. Then, a Gaussian pulse wave model as a solution to the Schrodinger equation is shown to accurately describe 19 different pulse shapes, and to quantitatively capture the degree of Yin-Yang attributes of 13 pulse shapes. Furthermore, the model suggests using pulse depth and strength as leading-order quantity and pulse shape as first-order quantity, to characterize the hierarchical resonance between the human body and the environment. The future pulse informatics will focus on determining an individual’s unique quantum human equilibrium state, and diagnose its health state according to the pulse deviation from its equilibrium state, to truly achieve the high level of TCM: “knowing the normal state and reaching the change”.

Relative Entropy, Gaussian Concentration and Uniqueness of Equilibrium States.

For a general class of lattice spin systems, we prove that an abstract Gaussian concentration bound implies positivity of the lower relative entropy density. As a consequence, we obtain uniqueness of translation-invariant Gibbs measures from the Gaussian concentration bound in this general setting. This extends earlier results with a different and very short proof.

Existence of the zero-temperature limit of equilibrium states on topologically transitive countable Markov shifts

AbstractConsider a topologically transitive countable Markov shift $\Sigma $ and a summable locally constant potential $\phi $ with finite Gurevich pressure and $\mathrm {Var}_1(\phi ) &lt; \infty $ . We prove the existence of the limit $\lim _{t \to \infty } \mu _t$ in the weak $^\star $ topology, where $\mu _t$ is the unique equilibrium state associated to the potential $t\phi $ . In addition, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.

GLOBAL STABILITY OF HYBRID SMOKING MODEL WITH NONLOCAL DIFFUSION

Smoking has many harmful effects due to its toxic chemicals, which cause serious diseases. Our major interest in this research is to formulate a spatiotemporal mathematical model that predicts the evolution of smoking in society using a mathematical model that includes the large mobility of individuals. Indeed, we are concerned to study the global asymptotic stability of the unique positive steady state related to this model. The principal objective of this paper is to show that the smoking has no threshold dynamics as it has been shown for a large sample of epidemic models, and the consumption of cigarettes is always persistent, and based on the assumption of the parameters that verify the Lipschitz condition. We proved that the investigated model is always persistent and the unique positive equilibrium state is always globally stable. The mathematical finding is supported numerically using numerical simulations.

Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities

Abstract Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi $. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson, we prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not support the full pressure. Moreover, we show that the pressure gap holds for any potential that is locally constant on a neighborhood of the singular set. Finally, we establish that the corresponding equilibrium states have the $K$-property and closed regular geodesics equidistribute.

Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs

The classical Kubo–Martin–Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin–Lanford–Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross–Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrödinger, Hartree, and wave (Klein–Gordon) equations.

Steepest entropy ascent solution for a continuous-time quantum walker.

We consider the steepest entropy ascent (SEA) ansatz to describe the nonlinear thermodynamic evolution of a quantum system. Recently this principle has been dubbed the fourth law of thermodynamics [Beretta, Phil. Trans. R. Soc. A 378, 20190168 (2020)10.1098/rsta.2019.0168]. A unique global equilibrium state exists in this context, and any other state is driven by the maximum entropy generation principle towards this equilibrium. We study the SEA evolution of a continuous-time quantum walker (CTQW) on a cycle graph with N nodes. SEA solutions are difficult to find analytically. We provide an approximate scheme to find a general single-particle evolution equationgoverned by the SEA principle, whose solution produces dissipation dynamics. We call this scheme the fixed Lagrange's multiplier (FLM) method. In the Bloch sphere representation, we find trajectories traced out by the Bloch vector within the sphere itself. We have discussed these trajectories under various initial conditions for the case of a qubit. A similar dissipative motion is also observed in the case of CTQW, where probability amplitudes have been used to characterize decoherence. Our FLM scheme shows good agreement with numerical results. As we report, in CTQW, a strong delocalization exists for low system relaxation time.

Multiagent Interval Consensus With Flocking Dynamics

In this article, we investigate the interval consensus for a network of agents with flocking dynamics, i.e., second-order multiagent systems, where each agent imposes an interval constraint on its preferred consensus values, with the aim of driving the agent into a favorable interval. Specifically, we work on two different frameworks of interval constraints, viz., the first one that the node states are constrained in their own constraint intervals and the second one that the node states are constrained in their neighbors’ constraint intervals. For both of the frameworks, we provide a complete solution to the equilibrium seeking problem by resolving a system of nonlinear equations. It is proved that if the underlying graph is strongly connected and the intersection of constraint intervals is empty, then there exists a unique equilibrium point; and if the intersection is nonempty, then there exist multiple equilibrium points, all of which lead to state consensus. We also establish several conditions for the local stability of the unique equilibrium point (corresponding to the case with empty intersection of constraint intervals) or local constraint consensus (corresponding to the case with nonempty intersection of constraint intervals) by invoking Lyapunov’s indirect method. Characterization of the global behavior of such a second-order multiagent system is technically rather challenging at this stage. As a first step toward this end, we show in two special cases that global convergence to the unique equilibrium point or state consensus can be guaranteed by employing Lyapunov stability theory and robust analysis techniques. Finally, some numerical examples are provided to illustrate the theoretical findings.