COMPARATIVE STUDY OF VARIATIONAL CHAOS INDICATORS AND ODEs' NUMERICAL INTEGRATORS

Abstract

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

Orbital dynamics in the post-Newtonian planar circular restricted Sun–Jupiter system

“…Here it should be noted that the Bulirsch-Stoer integrator is both faster and more accurate than a double precision Runge-Kutta-Fehlberg 7(8) algorithm with Cash/Karp coefficients. 9 Throughout all our computations, the Jacobi constant of Eq. (3) was conserved with fractional accuracy of about 10 −11 , or even better.…”

“…where j = 4, 5, denotes the j-th triangular point. The numerical coefficients entering equations (8) and (9) are given in Appendix B. Additionally, in Table 1 we present the numerical values of the Lagrange points for the Sun-Jupiter system in the framework of the PN-approximations (column 3) compared to the classical Newtonian values (column 2). It deserves mentioning that the numerical results presented in Table 1 satisfy the respective system of algebraic equation, with an accuracy of 10 −4 m, which indicates that the differences presented in column 4 are not numerical artifacts.…”

Orbit classification in the planar circular Pluto-Charon system

“…Over the years, there have been numerous attempts to solve the systems of differential equations with variable time steps, however results can exhibit parametric instabilities associated with resonances between the time step variation and the orbital motion (see e.g., Richardson & Finn (2012)). Here we should emphasize, that our previous numerical experience suggests that the Bulirsch-Stoer integrator is both faster and more accurate than a double precision Runge-Kutta-Fehlberg algorithm of order 7 with Cash-Karp coefficients (e.g., Darriba et al (2012)). Throughout all our computations, the Jacobi integral (Eq.…”

Fugitive stars in active galaxies

“…The equations of motion as well as the variational equations for the initial conditions of all orbits were integrated using a double precision Bulirsch-Stoer FORTRAN 77 algorithm (e.g., [49]) with a small time step of order of 10 −2 , which is sufficient enough for the desired accuracy of our computations. Here we should emphasize, that our previous numerical experience suggests that the Bulirsch-Stoer integrator is both faster and more accurate than a double precision Runge-Kutta-Fehlberg algorithm of order 7 with Cash-Karp coefficients (e.g., [19]). Throughout all our computations, the Jacobian energy integral (Eq.…”