Comparative study of several variational chaos indicators on maps

MAFFIONE, N. P.; DARRIBA, L. A.; CINCOTTA, P. M.; GIORDANO, C. M.

Sao José dos Campos, Brasil

Conferencia; International Conference on Chaos and Non Linear Dynamics, Dynamical Days South America 2010; 2010

IINPE, Brasil

The reader can find in the literature a lot of different techniques to study the dynamics of a givensystem and also, many suitable numerical integrators to compute them. Nevertheless, a detailedcomparison among the widespread indicators of chaos 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. The present work deals with both problems. We first compare on two completely different scenarios, a 4D symplectic mapping and the Henon & Heiles potential, several variational indicators: the Lyapunov Indicator, the Mean Exponential Growth Factor of Nearby Orbits, the Smaller Alignment Index and its generalized version, the Generalized Alignment Index, the Fast Lyapunov Indicator and its variant, the Orthogonal Fast Lyapunov Indicator, the Spectral Distance and the Spectras of Stretching Numbers. We also include in the record the Relative Lyapunov Indicator, which is not a variational indicator as the others, since it uses two different but very close orbits instead of the solution of the first variational equations. Then wetest a numerical technique to integrate Ordinary Differential Equations based on the Taylor method, implemented by Jorba & Zou, and we compare its performance with other two well1 known efficient integrators: the Prince & Dormand implementation of a Runge-Kutta of order 7-8 (DOPRI8) and a Bulirsch-St ̈er implementation. These tests are run under two very different systems from the point of view of the complexity of their equations: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We find that a combination of the OFLI, the GALI and the MEGNO succeeds in describing in detail most of the dynamical characteristics of a general problem. The OFLI shows a rich performance over the regular component, thus it turns out to be the first option to study big samples of regular orbits. The GALI has the fastest convergence for chaotic orbits which makes it a suitable choice to study extended regions of chaos. Lastly, the MEGNO shows very detailed descriptions when studying the time evolution of orbits, so its implementation to singular analysis is strongly advisable. In the second part we show that the precision of the Taylor integrator 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 some of the preceeding indicators of chaos. The result that induces us to come to this conclusion is that the computing times spent by Taylor are far greater than the times insumed by the DOPRI8 and the Bulirsch-Stoer integrator 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-Stoer, indeed.