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The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model

Yavuz Mehmet, Boulaasair Lahcen, Bouzahir Hassane, Diop Mamadou Abdoul, Benaid Brahim

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

121-134

EN

The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results.

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2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

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2023

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Vol. 29, No. 1-4

65--69

EN

In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e −p 2/2mkT and the corresponding potential distribution, ∝ e −Φ(q)/kT . (For convenience here we choose Boltzmann’s constant k and the particle mass m both equal to unity.) Soon William Hoover pointed out that Nosé’s approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e −ζ 2/2 / √ 2π, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Poznan Supercomputing ´ and Networking Center.

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Cn - pseudo almost automorphic solutions of class r for neutral partial functional differential equations under the light of measure theory

Napo Miailou, Zabsonre Issa, Bayili Gilbert

Silesian Journal of Pure and Applied Mathematics

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2022

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Vol. 12, iss. 1

1--32

EN

The aim of this work is to present new approach to study Cn-(µ,v)-pseudo almost automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed. We use the variation of constants formula and the spectral decomposition of the phase space.

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Pseudo almost periodic solutions of infinite class under the light of measure theory

Votsia Djokata, Zabsonre Issa

Silesian Journal of Pure and Applied Mathematics

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2020

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vol. 10, iss. 1

15--43

EN

The aim of this work is to present new approach to study weighted pseudo almost periodic functions with infinite delay using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of (μ, ν)-pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed on ]−∞, 0] using the spectra decomposition of the phase space developed in Adimy and co-authors.

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Ergodicity of One-dimensional Oscillators with a Signum Thermostat

Sprott J. C.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 3

169--176

EN

Gibbs’ canonical ensemble describes the exponential equilibrium distribution f(q; p; T) α e--H(q;p)/kT for an ergodic Hamiltonian system interacting with a ‘heat bath’ at temperature T. The simplest deterministic heat bath can be represented by a single ‘thermostat variable’ ζ. Ideally, this thermostat controls the kinetic energy so as to give the canonical distribution of the coordinates and momenta fq; pg. The most elegant thermostats are time-reversible and include the extra variable(s) needed to extract or inject energy. This paper describes a single-variable ‘signum thermostat.’ It is a limiting case of a recently proposed ‘logistic thermostat.’ It has a single adjustable parameter and can access all of Gibbs’ microstates for a wide variety of one-dimensional oscillators.

6

Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

155--158

EN

Considerable research has led to ergodic isothermal dynamics which can replicate Gibbs’ canonical distribution for simple (small) dynamical problems. Adding one or two thermostat forces to the Hamiltonian motion equations can give an ergodic isothermal dynamics to a harmonic oscillator, to a quartic oscillator, and even to the “Mexican-Hat” (doublewell) potential problem. We consider here a time-reversible dynamical approach to Gibbs’ “microcanonical” (isoenergetic) distribution for simple systems. To enable isoenergetic ergodicity we add occasional random rotations to the velocities. This idea conserves energy exactly and can be made to cover the entire energy shell with an ergodic dynamics. We entirely avoid the Poincaré-section holes and island chains typical of Hamiltonian chaos. We illustrate this idea for the simplest possible two-dimensional example, a single particle moving in a periodic square-lattice array of scatterers, the “cell model”.

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Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize Award

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

5--8

EN

The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper “Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat”. They introduced a relatively-stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs’ canonical distribution for three one-dimensional problems, the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials: {(q2=2); (q4=4); (q4=4) 􀀀 (q2=2)}. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal, “finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat”. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.

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Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat

Tapias D., Bravetti A., Sanders D. P.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

11--18

EN

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual “holes” in two-dimensional Poincaré sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.

9

Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 3

127--131

EN

For a harmonic oscillator, Nosé’s single-thermostat approach to simulating Gibbs’ canonical ensemble with dynamics samples only a small fraction of the phase space. Nosé’s approach has been improved in a series of three steps: [1] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodic fashion; [2] sets of single-thermostat motion equations, exerting “weak control” over both forces and momenta, have been shown to be ergodic; and [3] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat.

10

Time-Reversible Ergodic Maps and the 2015 Ian Snook Prizes

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 3

161--164

EN

The time reversibility characteristic of Hamiltonian mechanics has long been extended to nonHamiltonian dynamical systems modeling nonequilibrium steady states with feedback-based thermostats and ergostats. Typical solutions are multifractal attractor-repellor phase-space pairs with reversed momenta and unchanged coordinates, (q; p)↔(q;--p). Weak control of the temperature, α p2 and its fluctuation, resulting in ergodicity, has recently been achieved in a threedimensional time-reversible model of a heat-conducting harmonic oscillator. Two-dimensional cross sections of such nonequilibrium flows can be generated with time-reversible dissipative maps yielding æsthetically interesting attractorrepellor pairs. We challenge the reader to find and explore such time-reversible dissipative maps. This challenge is the 2015 Snook-Prize Problem.

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Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 1

5--10

EN

Nosé and Hoover’s 1984 work showed that although Nosé and Nosé-Hoover dynamics were both consistent with Gibbs’ canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs’ Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs’ canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of twothermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.

12

Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2014

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Vol. 20, No. 3

87--92

EN

Shuichi Nosé opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs’ canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nosé’s dynamical Hamiltonian bridge requires the “ergodic” support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé’s dynamical approach failed to agree with Gibbs’ sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated “thermostated" equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive (contradictory) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.

13

A hybrid optimization algorithm based on population migration algorithm and chaos theory

Liu X., Lu Y.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 3b

116--118

EN

A new hybrid optimization algorithm based on population migration algorithm (PMA) and chaos theory is proposed by introducing the logistic mapping of chaos theory into PMA. The proposed algorithm aims to improve solution accuracy and convergence, and avoid the prematurity of PMA. Experimental results show that the solution accuracy and convergence of the new algorithm can be effectively improved, and prematurity can be avoided by introducing ergodicity, randomicity, and regularity of the chaos theory into PMA.

PL

W artykule przedstawiono hybrydowy algorytm optymalizacji, bazujący na algorytmie migracyjnym (PMA) i teorii chaosu. Proponowane rozwiązanie ma na celu zwiększenie dokładności, zbieżności oraz unikanie „wcześniactwa” PMA. Wyniki badań eksperymentalnych potwierdzają skuteczność proponowanego algorytmu.

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Ergodicity examined by the Thirumalai-Mountain metric for Taiwanese seismicity

Li H.-C., Chen C.-C.

Acta Geophysica

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2012

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Vol. 60, no. 3

769-793

EN

Ergodicity is a behavior generally limited to equilibrium states and is here defined as the equivalence of ensemble and temporal averages. In recent years, effective ergodicity is identified in simulated earthquakes generated by numerical fault models and in real seismicity of natural fault networks by using the Thirumalai-Mountain metric. Although the effective ergodicity is already reported for Taiwanese seismicity, an immediate doubt is the unrealistic gridded sizes for discretizing the seismic data. In this study, we re-examined the effective ergodicity in Taiwanese seismicity by using reasonable gridded sizes which corresponded with the location errors in the real earthquake catalogue. Initial time and magnitude cut-off were examined for the validity of ergodic behavior. We found that several subsets extracted from Taiwanese seismicity possessed effectively ergodic intervals and all terminations of these ergodic intervals temporally coincided with the occurrences of large earthquakes (ML < 6.5). We thus confirm the ergodicity in the crustal seismicity by using the Thirumalai-Mountain metric.