Poincare maps: a modern systematic approach toward obtaining effective sections

Despite its importance and widespread applications, the use of the Poincare map has remained in its rudimentary stages since its proposition in the nineteenth century and there exists no systematic method to effectively obtain Poincare sections. Additionally, and due to its graphical structure, it has previously been very arduous to utilize Poincare maps for high dimensional systems, and two- and three-dimensional systems remain as its sole area of applicability. In this study, a novel systematic geometrical-statistical approach is proposed that is capable of obtaining the effective Poincare sections regardless of the attractor's complexity and provides insight into the entirety of the attractor's structure. The presented algorithm requires no prior knowledge of the attractor's dynamics or geometry and can be employed without any involvement with the governing dynamical equations. Several classical systems such as the Van der Pol, Lorenz, and Rossler's attractor are examined via the proposed algorithm and the results are presented and analyzed.