A METHOD FOR PREDICTING ENERGY-STABILIZED MOTION OF SPACECRAFT BASED ON DIFFERENTIAL TAYLOR TRANSFORMATIONS

2021 ◽  
Vol 2 ◽  
pp. 119-128
Author(s):  
Mikhail Rakushev ◽  
Keyword(s):  
Differential Equation ◽  
Differential Method ◽  
Integration Step ◽  
Rectangular Coordinate System ◽  
Near Space ◽  
Adaptive Schemes ◽  
Internal Adaptation ◽  
Spacecraft Stabilization ◽  
Lyapunov Instability

To predict the motion of spacecrafts, a numerical-analytical method for integrating the differential equation of the orbital motion of a spacecraft stabilized by the Baumgart differential method is proposed. The stabilization of the differential equation of motion by the Baumgart method is carried out according to the energy of the spacecraft. Stabilization is carried out to reduce the influence of the Lyapunov instability on the accumulation of numerical errors in the integration of the differential equation, which is effective when conducting a long-term numerical prediction of the motion of spacecraft. Integration of the stabilized equation is based on differential Taylor transformations. Computational schemes with a constant step and an integration order are considered, as well as schemes with adaptation by an integration step and order. For adaptive schemes, the results of forecasting the motion of spacecraft according to the criterion “accuracy-computational complexity» for a given relative error of integration with respect to integration phase variables and spacecraft energy are presented. It is shown that both options require setting various internal adaptation parameters, but they have comparable efficiency. Recommendations are proposed on the use of the developed method for integrating energy-stabilized equations for predicting the motion of spacecraft in the near space in the Greenwich rectangular coordinate system.

scholarly journals County-Level Social Distancing and Policy Impact in the United States: A Dynamical Systems Model

10.2196/23902 ◽  
2020 ◽  
Vol 6 (4) ◽  
pp. e23902
Author(s):  
Kevin L McKee ◽  
Ian C Crandell ◽  
Alexandra L Hanlon
Keyword(s):  
United States ◽  
Differential Equation ◽  
Population Density ◽  
Household Income ◽  
The United States ◽  
Policy Impact ◽  
County Level ◽  
Median Household Income ◽  
Social Distancing

Background Social distancing and public policy have been crucial for minimizing the spread of SARS-CoV-2 in the United States. Publicly available, county-level time series data on mobility are derived from individual devices with global positioning systems, providing a variety of indices of social distancing behavior per day. Such indices allow a fine-grained approach to modeling public behavior during the pandemic. Previous studies of social distancing and policy have not accounted for the occurrence of pre-policy social distancing and other dynamics reflected in the long-term trajectories of public mobility data. Objective We propose a differential equation state-space model of county-level social distancing that accounts for distancing behavior leading up to the first official policies, equilibrium dynamics reflected in the long-term trajectories of mobility, and the specific impacts of four kinds of policy. The model is fit to each US county individually, producing a nationwide data set of novel estimated mobility indices. Methods A differential equation model was fit to three indicators of mobility for each of 3054 counties, with T=100 occasions per county of the following: distance traveled, visitations to key sites, and the log number of interpersonal encounters. The indicators were highly correlated and assumed to share common underlying latent trajectory, dynamics, and responses to policy. Maximum likelihood estimation with the Kalman-Bucy filter was used to estimate the model parameters. Bivariate distributional plots and descriptive statistics were used to examine the resulting county-level parameter estimates. The association of chronology with policy impact was also considered. Results Mobility dynamics show moderate correlations with two census covariates: population density (Spearman r ranging from 0.11 to 0.31) and median household income (Spearman r ranging from –0.03 to 0.39). Stay-at-home order effects were negatively correlated with both (r=–0.37 and r=–0.38, respectively), while the effects of the ban on all gatherings were positively correlated with both (r=0.51, r=0.39). Chronological ordering of policies was a moderate to strong determinant of their effect per county (Spearman r ranging from –0.12 to –0.56), with earlier policies accounting for most of the change in mobility, and later policies having little or no additional effect. Conclusions Chronological ordering, population density, and median household income were all associated with policy impact. The stay-at-home order and the ban on gatherings had the largest impacts on mobility on average. The model is implemented in a graphical online app for exploring county-level statistics and running counterfactual simulations. Future studies can incorporate the model-derived indices of social distancing and policy impacts as important social determinants of COVID-19 health outcomes.

Optimal reserves growth under a Wiener process

1975 ◽  
Vol 12 (03) ◽  
pp. 457-465
Author(s):  
W. F. Foster
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Wiener Process ◽  
Order Differential Equation ◽  
Optimality Criterion ◽  
First Passage ◽  
First Order ◽  
Bounded Below ◽  
Passage Times

This paper considers a body whose funds accumulate according to a Wiener Process that has parameters which can be controlled at any stage. The process is bounded above by a level at which dividends (or savings) are set aside, and it is bounded below by a level at which a ‘rescue’ policy is invoked to avoid insolvency. Taking long-term dividend maximisation as the optimality criterion, first passage times are used to derive a general first order differential equation for the optimal control of the system at any reserves level, and this equation is solved fully for a certain class of problems. Examples are given of insurance and investment applications.

ON THE RELATION BETWEEN ORGANIZATIONS AND LIMIT SETS IN CHEMICAL REACTION SYSTEMS

2011 ◽  
Vol 14 (01) ◽  
pp. 77-96 ◽  
Author(s):  
STEPHAN PETER ◽  
PETER DITTRICH
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Organization Theory ◽  
Dynamical Behavior ◽  
Chemical Reactor ◽  
Deterministic System ◽  
Discrete Problem ◽  
Limit Sets ◽  
Long Term Behavior

Chemical organization theory has been suggested as a new approach to analyze complex reaction networks. Concerning the long-term behavior of the network dynamics we will study its foundations mathematically. Therefore we consider a chemical reactor containing molecules of different species reacting with each other according to a set of reaction rules. We further assume that the dynamical behavior of the concentration of each species is given by a continuous chemical ordinary differential equation. Abstracting from dedicated concentration values we consider a discrete problem: Which species can appear in the reactor after a long time? We define the limit set abstraction, which contains the subsets of species characterizing the long-term behavior. We prove that all these subsets are closed and that for all bounded limit sets, at least one of them is self-maintaining and thus is an organization. This implies for a chemical ordinary differential equation systems that any attractor that does not touch the state space boundaries (in particular, any periodic attractor) lies within one organization, that is, the set of species with positive concentrations in any state of such an attractor is an organization. This in turn explains why in a deterministic system evolving according to reaction rules one can observe species sets that are closed and self-maintaining, thus organizations.

scholarly journals CALCULATION OF THE FLAT BENDING SHAPE STABILITY OF RECTANGULAR CROSS SECTION BEAMS WITH REGARD TO CREEP

2019 ◽  
Vol 46 (1) ◽  
pp. 169-176
Author(s):  
I. M. Zotov ◽  
A. S. Chepurnenko ◽  
S. B. Yazyev
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Euler Method ◽  
Shape Stability ◽  
Concentrated Force ◽  
Flat Shape ◽  
The Stability

Objectives. The article presents the conclusion of the resolving equation for calculating the stability of the flat form of deformation of prismatic beams, taking into account the rheological properties of the material.Method. The problem is reduced to a second-order differential equation for the twist angle, which is solved numerically by the finite difference method in combination with the Euler method.Result. The obtained differential equation allows one to take into account the presence of initial imperfections in the form of the initial deflection of the beam, the initial angle of twist, and also the eccentricity of the applied load. The solution of the test problem for a cantilever polymer beam under the action of a concentrated force is presented. The non-linear Maxwell-Gurevich equation is used as the creep law. The value of the long-term critical load is introduced and it is shown that with a load less than the long-term critical creep is limited. It has been established that, as with the squeezed rods, with a load less than the long-term critical, the growth rate of the displacements with time decays. When F = F_dl, the displacements grow at a constant speed, and when F> F_dl, the growth rate of displacements increases with time. The results obtained confirm the validity of the chosen method.Conclusion. A universal resolving equation is obtained for calculating the stability of a flat shape of bending of rectangular beams, suitable for arbitrary creep laws.

scholarly journals Stability Analysis and Optimization of Semi-Explicit Predictor–Corrector Methods

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2463
Author(s):  
Aleksandra Tutueva ◽  
Denis Butusov
Keyword(s):  
Numerical Stability ◽  
Large Scale ◽  
Integration Step ◽  
Variable Integration ◽  
Predicted Values ◽  
Low Dimensional ◽  
Three Body ◽  
Systems Simulation ◽  
Predictor Corrector

The increasing complexity of advanced devices and systems increases the scale of mathematical models used in computer simulations. Multiparametric analysis and study on long-term time intervals of large-scale systems are computationally expensive. Therefore, efficient numerical methods are required to reduce time costs. Recently, semi-explicit and semi-implicit Adams–Bashforth–Moulton methods have been proposed, showing great computational efficiency in low-dimensional systems simulation. In this study, we examine the numerical stability of these methods by plotting stability regions. We explicitly show that semi-explicit methods possess higher numerical stability than the conventional predictor–corrector algorithms. The second contribution of the reported research is a novel algorithm to generate an optimized finite-difference scheme of semi-explicit and semi-implicit Adams–Bashforth–Moulton methods without redundant computation of predicted values that are not used for correction. The experimental part of the study includes the numerical simulation of the three-body problem and a network of coupled oscillators with a fixed and variable integration step and finely confirms the theoretical findings.

scholarly journals A Fundamental Inconsistency in the SIR Model Structure and Proposed Remedies

2020 ◽  
Author(s):  
Michael Nikolaou
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Compartmental Model ◽  
Sir Model ◽  
Model Structure ◽  
The Novel ◽  
Algebraic Equations ◽  
Exponential Rates ◽  
Delay Differential

AbstractThe susceptible-infectious-removed (SIR) compartmental model structure and its variants are a fundamental modeling tool in epidemiology. As typically used, however, this tool may introduce an inconsistency by assuming that the rate of depletion of a compartment is proportional to the content of that compartment. As mentioned in the seminal SIR work of Kermack and McKendrick, this is an assumption of mathematical convenience rather than realism. As such, it leads to underprediction of the infectious compartment peaks by a factor of about two, a problem of particular importance when dealing with availability of resources during an epidemic. To remedy this problem, we develop the dSIR model structure, comprising a single delay differential equation and associated delay algebraic equations. We show that SIR and dSIR fully agree in assessing stability and long-term values of a population through an epidemic, but differ considerably in the exponential rates of ascent and descent as well as peak values during the epidemic. The novel Padé-SIR structure is also introduced as a approximation of dSIR by ordinary differential equations. We rigorously analyze the properties of these models and present a number of illustrative simulations, particularly in view of the recent coronavirus epidemic. Suggestions for further study are made.

Prediction of long-term groundwater recharge by using hydropedotransfer functions

2013 ◽  
Vol 27 (1) ◽  
pp. 31-37 ◽  
Author(s):  
K. Miegel ◽  
K. Bohne ◽  
G. Wessolek
Keyword(s):  
Differential Equation ◽  
Transfer Function ◽  
Groundwater Recharge ◽  
Standard Error ◽  
Climatic Conditions ◽  
Actual Evapotranspiration ◽  
Groundwater Depletion ◽  
Site Conditions ◽  
Hydrologic Processes

Abstract The investigations to estimate groundwater recharge were performed. Improved consideration of soil hydrologic processes yielded a convenient method to predict actual evapotranspiration and hence, groundwater recharge from easily available data. For that purpose a comprehensive data base was needed, which was created by the simulation model SWAP comprising 135 different site conditions and 30 simulation years each. Based upon simulated values of actual evapotranspiration, a transfer function was developed employing the parameter b in the Bagrov differential equation dEa/dP = 1- (Ea/Ep)b. Under humid conditions, the Bagrov method predicted long-term averages of actual evapotranspiration and groundwater recharge with a standard error of 15 mm year-1 (R = 0.96). Under dry climatic conditions and groundwater influence, simulated actual evapotranspiration may exceed precipitation. Since the Bagrov equation is not valid under conditions like these, a statistic-based transfer function was developed predicting groundwater recharge including groundwater depletion with a standard error of 26mm(R = 0.975). The software necessary to perform calculations is provided online.

Validity of averaging methods for certain systems with periodic solutions

1972 ◽  
Vol 330 (1582) ◽  
pp. 349-371 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Methods ◽  
Small Error ◽  
Time Interval ◽  
Oscillatory Solutions ◽  
Weakly Nonlinear ◽  
Simple Bounds ◽  
Physical Effects ◽  
Uniformly Bounded

Heuristic asymptotic methods are commonly used to study the long-term development of oscillatory solutions of nonlinear ordinary differential equations and wave-type solutions of partial differential equation. For certain classes of weakly nonlinear systems, energy methods are here used to establish the validity of such approximations. There is an overall limitation of the results to a time interval determined by the time scale of significant energy transfer, but this is sufficiently long for interesting physical effects to be discussed. The basic results take the form that, when the heuristic methods yield an approximation giving a uniformly bounded small error in the differential equation, then the error in the solution is small. They do not depend strongly on the properties of the approximation, other than on simple bounds.

Sign in / Sign up

Export Citation Format

Share Document