scholarly journals Asymptotic behaviour of orbit determination for hyperbolic maps

2021 ◽  
Vol 133 (6) ◽  
Author(s):  
Stefano Marò ◽  
Claudio Bonanno
Keyword(s):  
Asymptotic Behaviour ◽  
Orbit Determination ◽  
Initial Conditions ◽  
Confidence Region ◽  
Time Span ◽  
Simultaneous Increase ◽  
Dynamical Parameter ◽  
Rate Of Decay

AbstractWe deal with the orbit determination problem for hyperbolic maps. The problem consists in determining the initial conditions of an orbit and, eventually, other parameters of the model from some observations. We study the behaviour of the confidence region in the case of simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for the solution, distinguishing whether a parameter is added to the estimate of the initial conditions or not. We prove that the inclusion of a dynamical parameter causes a change in the rate of decay of the uncertainties, as suggested by some known numerical evidences.

scholarly journals The Manifold Of Variations: hazard assessment of short-term impactors

2020 ◽  
Vol 132 (10) ◽  
Author(s):  
Alessio Del Vigna
Keyword(s):  
Orbit Determination ◽  
Compact Manifold ◽  
Confidence Region ◽  
Time Span ◽  
Short Term ◽  
Admissible Region ◽  
Short Time Span ◽  
Region Theory ◽  
Range Rate ◽  
Short Time

AbstractWhen an asteroid has a few observations over a short time span the information contained in the observational arc could be so little that a full orbit determination may be not possible. One of the methods developed in recent years to overcome this problem is based on the systematic ranging and combined with the Admissible Region theory to constrain the poorly-determined topocentric range and range-rate. The result is a set of orbits compatible with the observations, the Manifold Of Variations, a two-dimensional compact manifold parametrised over the Admissible Region. Such a set of orbits represents the asteroid confidence region and is used for short-term hazard predictions. In this paper we present the Manifold Of Variations method and make a detailed analysis of the related probabilistic formalism.

Estimation of anthracnose dynamics by nonlinear filtering

2018 ◽  
Vol 11 (01) ◽  
pp. 1850005
Author(s):  
David Jaurès Fotsa-Mbogne
Keyword(s):  
Nonlinear Filtering ◽  
Space Variable ◽  
Initial Conditions ◽  
Confidence Region ◽  
Inhibition Rate ◽  
Deterministic Models ◽  
Incomplete Observations ◽  
Partially Observed ◽  
Uniform Law ◽  
Filtering Problem

In this work, we apply the nonlinear filtering theory to the estimation of the partially observed dynamics of anthracnose which is a phytopathology. The signal here is the inhibition rate and the observations are the fruit volume and the rotted volume. We propose stochastic models based on deterministic models studied previously in the literature, in order to represent the noise introduced by uncontrolled variations on parameters and errors on the measurements. Under the assumption of Brownian noises, we prove the well-posedness of the models in either they take into account the space variable or not. The filtering problem is solved for the nonspatial model giving Zakai and Kushner–Stratonovich equations satisfied respectively by the unnormalized and the normalized conditional distribution of the signal with respect to the observations. A prevision problem and a discrete filtering problem are also studied for the realistic cases of discrete and possibly incomplete observations. We illustrate the filter behavior through figures displaying the average estimation relative error and a 95% confidence region obtained after a hundred of numerical simulations with initial conditions taken randomly with respect to uniform law.

A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations

1966 ◽  
Vol 25 ◽  
pp. 281-287 ◽  
Author(s):  
P. E. Zadunaisky
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Current Theory ◽  
Numerical Process

Let bex′=f(t,x) a system of ordinary differential equations, with initial conditionsx(a) =s, which is integrated numerically by a finite difference method of orderpand constant steph.To estimate the truncation and round-off errors accumulated during the numerical process it is established a method based on the current theory of the asymptotic behaviour (whenh→0) of errors. This method should avoid the main difficulties that arise when the results of the theory must be applied to practical cases. The method has been successfully tested and applied to estimate the errors accumulated in a numerical computation of planetary perturbations on the orbit of a comet.

Stability and Robustness Analysis of Uncertain Nonlinear Systems Using Entropy Properties of Left and Right Singular Vectors

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Young Kap Son ◽  
Gordon J. Savage
Keyword(s):  
Initial Conditions ◽  
Mechanistic Model ◽  
Robustness Analysis ◽  
Life Time ◽  
Small Time ◽  
Time Span ◽  
Singular Vectors ◽  
Design Variables ◽  
Left And Right ◽  
The Stability

This paper presents a novel approach to determine the stability space of nonlinear, uncertain dynamic systems that obviates the traditional eigenvalue approach and the accompanying linearizing approximations. In the new method, any long-term dynamic uncertainty is used in an extremely simple and economical way. First, the variability of the design variables about a particular design point is captured through the design of experiments (DOE). Then, corresponding computer simulations of the mechanistic model, over only a small time span, provide a matrix of discrete time responses. Finally, singular value decomposition (SVD) separates out parameter and time information and the expected uncertainty of the first few left and right singular vectors predicts any instability that might occur over the entire life-time of the dynamics. The singular vectors are viewed as random variables and their entropy leads to a simple metric that accurately predicts stability. The stable/unstable spaces are found by investigating the overall design space using an array of grid points of suitable spacing. The length of the time span needed to capture the nature of the dynamics can be as short as two or three periods. The robustness of the stability space is related to the tolerances assigned to the design variables. Errors due to sampling size, time increments, and number of significant singular vectors are controllable. The method can be implemented with readily available software. A study of two practical engineering systems with different distributions and tolerances, various initial conditions, and different time spans shows the efficacy of the proposed approach.

Chairman’s introduction

1987 ◽  
Vol 413 (1844) ◽  
pp. 3-4
Keyword(s):  
Asymptotic Behaviour ◽  
Frequency Spectrum ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Strange Attractors ◽  
Qualitative Properties ◽  
Differential Topology ◽  
Deterministic Systems ◽  
Sensitive Dependence ◽  
Quantitative Properties

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought. 1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiable changes of coordinates, as opposed to quantitative properties which are invariant only under linear changes of coordinates. To give an example of this influence, I recall a year-long symposium at Warwick in 1979/80, which involved sustained interaction between pure mathematicians and experimentalists, and one of the most striking consequences of that interaction was a transformation in the way that experimentalists now present their data. It is generally in a much more translucent form: instead of merely plotting a frequency spectrum and calling the incomprehensible part ‘noise’, they began to draw computer pictures of underlying three-dimensional strange attractors.

An independent solution for the precise orbit determination of Mercury planetary orbiter (MPO)

2020 ◽  
Author(s):  
Alireza HosseiniArani ◽  
Stefano Bertone ◽  
Daniel Arnold ◽  
Adrian Jäggi ◽  
Nicolas Thomas
Keyword(s):  
Orbit Determination ◽  
Initial Conditions ◽  
European Space Agency ◽  
Precise Orbit Determination ◽  
Independent Solution ◽  
Force Model ◽  
Precise Orbit ◽  
Gravitational Forces ◽  
Imperfect Knowledge

<p>Navigation of deep space probes is most commonly operated using the spacecraft Doppler<br>tracking technique. Orbital parameters are determined from a series of repeated measurements of the frequency shift of a microwave carrier over a given integration time. This study addresses the work that is done on Doppler orbit determination of MPO - one of the two spacecraft of the European Space Agency’s BepiColombo mission- using Bernese software.</p><p>For modelling the orbit of MPO around Mercury, we use a full force model, including Mercury gravity field GGMES-100V07 (up to degree and order 50), solid tides and third body perturbations. We also have an extensive modelling of non-gravitational forces that act on the orbit of spacecraft. This modelling includes the solar radiation pressure and planetary IR and albedo radiation together with a 33-plates macromodel of MPO. We propagate the orbit using this force model. Our simulations of Doppler tracking measurements include 2-way X-band and K-band Doppler measurements, station and planetary eclipses and the relativistic corrections. </p><p>The imperfect knowledge of the non-gravitational forces due to the proximity of Mercury to the Sun, together with the effect of desaturation maneuvers uncertainties, makes the use of the accelerometer necessary. Therefore, in our modelling of the orbit recovery, the models for the non-conservative forces were replaced by the noisy simulated accelerometer measurements. We find out that the modelling of the accelerometer noise has a huge impact on the results of the POD.</p><p>We perform several orbit reconstruction tests using daily arcs with noise modulated Doppler data with different settings on the arc lengths, arcs initial conditions, dynamical model, observation mode and orbit determination process and we solve for the initial state vector of each arc. We also run sensitivity analysis with respect to the different accelerometer model. The final goal of this study is to provide an independent solution for the precise orbit determination of Mercury planetary orbiter (MPO) using the planetary extension of the Bernese GNSS software. We present out latest results and then compare our results with the existing ones from the MORE team.</p>

The effect of jet injection geometry on two-dimensional momentumless wakes

1991 ◽  
Vol 224 ◽  
pp. 29-47 ◽  
Author(s):  
W. J. Park ◽  
J. M. Cimbala
Keyword(s):  
Turbulence Intensity ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Spreading Rate ◽  
Two Dimensional ◽  
Jet Injection ◽  
Mean Velocity ◽  
Rate Of Decay ◽  
The Mean ◽  
Momentumless Wake

It is shown experimentally that a two-dimensional momentumless wake is strongly dependent on the jet injection configuration of the model. Namely, the decay rate of mean velocity overshoot ranged from x−0.92 to x−2.0 for three different configurations, while the spreading rate ranged from x0.3 to x0.46 for those same configurations. The magnitude of axial turbulence intensity was also found to depend on model configuration. On the other hand, the rate of decay of axial turbulence intensity was the same (x−0.81) for all three models. In all cases the mean shear and Reynolds stress decayed rapidly, leaving nearly isotropic turbulence beyond 30 or 40 model diameters.Appropriate length- and velocity scales are identified which normalize the mean velocity profiles into self-similar form. The shape of the normalized profile, however, was different for each configuration, indicating again that the initial conditions are felt very far downstream.

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