scholarly journals Fourier Series Approximations toJ2-Bounded Equatorial Orbits

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei Wang ◽  
Jianping Yuan ◽  
Yanbin Zhao ◽  
Zheng Chen ◽  
Changchun Chen
Keyword(s):  
Fourier Series ◽  
Initial Conditions ◽  
Polar Angle ◽  
Elliptic Functions ◽  
Dynamical Analysis ◽  
Fourier Series Expansion ◽  
Elementary Functions ◽  
Mathematical Functions ◽  
Jacobian Elliptic Functions ◽  
Closed Form Solutions

The current paper offers a comprehensive dynamical analysis and Fourier series approximations ofJ2-bounded equatorial orbits. The initial conditions of heterogeneous families ofJ2-perturbed equatorial orbits are determined first. Then the characteristics of two types ofJ2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.

Hilbert–Huang Transform Approach to Lorenz Signal Separation

2015 ◽  
Vol 07 (01n02) ◽  
pp. 1550004 ◽  
Author(s):  
Gregori J. Clarke ◽  
Samuel S. P. Shen
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Lorenz System ◽  
Initial Conditions ◽  
Fourier Series Expansion ◽  
Finite Collection ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Hilbert Huang Transform ◽  
Time Frequency

This study uses the Hilbert–Huang transform (HHT), a signal analysis method for nonlinear and non-stationary processes, to separate signals of varying frequencies in a nonlinear system governed by the Lorenz equations. Similar to the Fourier series expansion, HHT decomposes a data time series into a sum of intrinsic mode functions (IMFs) using empirical mode decomposition (EMD). Unlike an infinite number of Fourier series terms, the EMD always yields a finite number of IMFs, whose sum is equal to the original time series exactly. Using the HHT approach, the properties of the Lorenz attractor are interpreted in a time–frequency frame. This frame shows that: (i) the attractor is symmetric for [Formula: see text] (i.e. invariant for [Formula: see text]), even though the signs on [Formula: see text] and [Formula: see text] are changed; (ii) the attractor is sensitive to initial conditions even by a small perturbation, measured by the divergence of the trajectories over time; (iii) the Lorenz system goes through “windows” of chaos and periodicity; and (iv) at times, a system can be both chaotic and periodic for a given [Formula: see text] value. IMFs are a finite collection of decomposed quasi-periodic signals, starting from the highest to lowest frequencies, providing detection of the lower frequency signals that may have otherwise been “hidden” by their higher frequency counterparts. EMD decomposes the original signal into a family of distinct IMF signals, the Hilbert spectra are a “family portrait” of time–frequency–amplitude interplay of all IMF members. Together with viewing the IMF energy, it is easy to discern where each IMF resides in the spectra in relation to one another. In this study, the majority of high amplitude signals appear at low frequencies, approximately 0.5–1.5. Although our numerical experiments are limited to only two specific cases, our HHT analyses of time–frequency, marginal spectra, and energy and quasi-periodicity of each IMF provide a novel approach to exploring the profound and phenomena-rich Lorenz system.

scholarly journals Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem

2021 ◽  
pp. 50-68
Author(s):  
Alessio Bocci ◽  
Giovanni Mingari Scarpello
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Satellite Motion ◽  
Keplerian Motion ◽  
Zonal Harmonic ◽  
Closed Form Solutions ◽  
Bounded Motion ◽  
New Feature

This report provides some closed form solutions -and their inversion- to a satellite’s bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the J2 spherical zonal harmonic. The equatorial track of satellite motion- assuming the co-latitude φ fixed at π/2- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse t = t(c) we get the period T of some functions c(t) of mechanical interest and then we construct the relevant c(t) expansion in Fourier series, in such a way performing the inversion. Such approach-which led to new formulations for time laws of a J2 problem- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.

scholarly journals Dynamical symmetry breaking through AI: The dimer self-trapping transition

2021 ◽  
Author(s):  
G. P. Tsironis ◽  
G. D. Barmparis ◽  
D. K. Campbell
Keyword(s):  
Initial Conditions ◽  
Elliptic Functions ◽  
Dynamical Symmetry ◽  
Classical Particle ◽  
Jacobian Elliptic Functions ◽  
Nonlinear Localization ◽  
Additional Information ◽  
Self Trapping ◽  
General Dynamics ◽  
Interacting Systems

The nonlinear dimer obtained through the nonlinear Schrödinger equation has been a workhorse for the discovery the role nonlinearity plays in strongly interacting systems. While the analysis of the stationary states demonstrates the onset of a symmetry broken state for some degree of nonlinearity, the full dynamics maps the system into an effective [Formula: see text] model. In this later context, the self-trapping transition is an initial condition-dependent transfer of a classical particle over a barrier set by the nonlinear term. This transition that has been investigated analytically and mathematically is expressed through the hyperbolic limit of Jacobian elliptic functions. The aim of this work is to recapture this transition through the use of methods of Artificial Intelligence (AI). Specifically, we used a physics motivated machine learning model that is shown to be able to capture the original dynamic self-trapping transition and its dependence on initial conditions. Exploitation of this result in the case of the nondegenerate nonlinear dimer gives additional information on the more general dynamics and helps delineate linear from nonlinear localization. This work shows how AI methods may be embedded in physics and provide useful tools for discovery.

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