Theory and Applications of the Fast Lyapunov Indicator (FLI) Method

Abstract

In the last 20 years numerical experiments have allowed to study dynamical systems in a new way providing interesting results. The development of tools for the detection of regular and chaotic orbits has been one of the key points to access the global properties of dynamical systems. In many cases the visualization of suitably chosen sections of the phase space has been determinant for the comprehension of the fascinating and complex interplay between order and chaos. The Fast Lyapunov Indicator introduced in Froeschle et al. (Celest Mech Dyn Astron 67:41–62, 1997) and further developed in Guzzo et al. (Physica D 163(1–2):1–25, 2002), is an easy to implement and sensitive tool for the detection of order and chaos in dynamical systems. Closely related to the computation of the Largest Lyapunov Exponent, the Fast Lyapunov Indicator relies on the idea that the computation of tangent vectors contains a lot of information even on short integration times, while for the Largest Lyapunov Indicator large integration times are required in order to accurately approximate a limit value. The aim of this Chapter is to provide the definition of the Fast Lyapunov indicator and some simple examples of applications for readers that would like to implement and use the indicator for the first time. We associate to each example of application the references to more specific papers that we have published during these years.