Some Numerical Experiments on the Application of Relative Lyapunov Indicator to Symplectic and Dissipative Mappings

The Relative Lyapunov Indicator (RLI, for short) is a simple and efficient technique that can discriminate with certainty between ordered and chaotic motions in dynamical systems. Based on the evolution of two distinct but very close orbits, it behaves quite different for the two types of orbits. Generally, in the chaotic case the indicator’s value increases rapidly in the first few hundreds time steps and then exhibits a slow decreasing or remains constant. On the other hand, the RLI displays a nearly constant value for ordered motions. One of the purposes of this paper is to check how the indicator behaves not only for ordinary regular or chaotic orbits but for other more awkward to assess, including sticky orbits, orbits possessing long transients or characterized by periodic sequences interrupted by short intermittent chaotic windows. Moreover, the authors want to verify if the RLI realizes a finely distinction between periodic and quasi-periodic orbits. The second goal of the contribution is to provide RLI plots that clearly separate even tiny regions of order and chaos in the phase space or parametric space of the analysed dynamical system. To achieve these, the authors generate data from the two-dimensional (2D) area-preserving standard map and 2D dissipative Tinkerbell map, known for their rich and interesting dynamical behaviour