Characterization of the Phase Space Structure of Circular Restricted Three-Body Problem: An Alternative Approach
09 Mar 2016-International Journal of Bifurcation and Chaos (World Scientific Publishing Company)-Vol. 26, Iss: 02, pp 1650029
TL;DR: Time-Frequency Analysis and Poincare Surface of Section are considered for the study of the phase space structure of nonlinear dynamical system and with the help of ridge-plots, the phenomenon of transient chaos is visualize.
Abstract: In this paper, we have considered Time-Frequency Analysis (TFA) and Poincare Surface of Section (PSS) for the study of the phase space structure of nonlinear dynamical system. We have examined a sample of orbits taken in the framework of Circular Restricted Three-Body Problem (CRTBP). We have computed ridge-plots (i.e. time-frequency landscape) using the phase of the continuous wavelet transform. Clear visualization of resonance trappings and the transitions is an important feature of this method, which is presented using ridge-plots. The identification between periodic and quasi-periodic, chaotic sticky and nonsticky and regular and chaotic orbits are done in comparatively less time and with less computational effort. The spatial case of Circular Restricted Three-Body problem is considered to show the strength of Time-Frequency Analysis to higher dimensional systems. Also, with the help of ridge-plots, we can visualize the phenomenon of transient chaos.
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19 Apr 2020
TL;DR: In this paper, the authors considered the Sun-Mercury-satellite in the model of restricted three body problem with zero eccentricity and used continuation method to obtain the halo orbits around the Libration points L1 and L2.
Abstract: Mercury is the planet which gains maximum radiation pressure from the Sun. On 20th October 2018 Bepicolombo was launched to do the comprehensive study of the magnetic field, magnetosphere, surface and internal structure of the Mercury. For this we need to maintain the nominal multi-revolution halo orbits. In this paper, we have considered the Sun-Mercury-satellite in the model of restricted three body problem with zero eccentricity. Here continuation method have been used to obtain the halo orbits around the Libration points L1 and L2 . We observe that the frequencies remain constant throughout the time interval using wavelet transform. The ridge plot at the initial guess confirms the periodicity of the halo orbits.
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Book•
01 Jan 1998
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
17,693 citations
TL;DR: In this article, a measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable, and the relation of this measure to fractal dimension and information-theoretic entropy is discussed.
Abstract: A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.
4,323 citations
TL;DR: The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated and it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws.
Abstract: The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed. >
592 citations
TL;DR: A survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as numerical techniques developed for the computation of the maximal, of few and of all of them, can be found in this article.
Abstract: We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec (102), which pro- vides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been exten- sively used as an indicator of chaos, and the algorithm of the so-called standard method, developed by Benettin et al. (14), for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although we are mainly interested in finite-dimensional conservative systems, i.e., autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.
259 citations