Comparison of chaos detection methods in the circular restricted three‐body problem

Abstract

In this study, a sample of orbits is considered in the framework of the planar circular restricted three-body problem. In order to separate ordered from chaotic orbits three numerical methods are compared: the Largest Lyapunov Characteristic Exponent (LLCE) and the Smaller Alignment Index (SALI) provide a fairly good characterization of the chaotic motions, while the computational time required is of the same order; the Correlation Dimension (CD) has the advantage of correctly classifying sticky orbits, but at the expense of a longer computational time. In order to classify a given orbit, any pair of the three methods can be considered, but LLCE and SALI are recommended due to their speed.

Theory and Applications of the Mean Exponential Growth Factor of Nearby Orbits (MEGNO) Method

“…All the methods are compared only in double and quadruple precision using the Lahey LF 95 compiler (fortran) because the dop853 cannot be directly used in multiple precision. The multiple-precision tests are done using C++ and the GMP and MPFR [75] multiple precision packages. From Fig.…”

“…In [18,88,89] the stability properties of orbits in a particular few-body problem, the so-called the Sitnikov problem, were studied, while in [94] the long term stability of two-planet extrasolar systems initially trapped in the 3:1 mean motion resonance was investigated. The SALI was also used to study the dynamics of the Caledonian symmetric four-body problem [90], as well as the circular restricted three-body problem [75].…”

Time-Frequency Analysis of Asymmetric Triaxial Galaxy Model Including Effect of Spherical Dark Halo Component

“…Time interval considered for the TFA is (1, 32768) except two figures (Figure 3 and Figure 4). For Figure 3 and Figure 4, the interval of time unit is taken as (1,65536) to present the results in a better way. It adds an additional expense of eighteen seconds in the C.P.U time for the execution of the program.…”

“…Moreover, for dynamical system of two degrees of freedom, there are several chaos indicators such as the Poincare Surface of Section (PSS), Largest Lyapunov Characteristic Exponent (LLCE), Smaller Alignment Index (SALI). Fast Liapunov Indicators (FLI), the Generalized Alignment Index (GALI) and the Correlation Dimension (CD) (see [1]- [3]). Visualization of the Poincare surfaces of section (PSS) is very useful for the study of the dynamical system of two degrees of freedom.…”

The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection

“…In [88,89,18] the stability properties of orbits in a particular few-body problem, the so-called the Sitnikov problem, were studied, while in [94] the long term stability of two-planet extrasolar systems initially trapped in the 3:1 mean motion resonance was investigated. The SALI was also used to study the dynamics of the Caledonian symmetric four-body problem [90], as well as the circular restricted three-body problem [75].…”