Comprehensive classification of multistability and Lyapunov exponent with multiple dynamics of nonlinear Schrödinger equation

In this paper a detailed investigation of the non-linear Schrödinger equation is presented. A comprehensive set of tools, such as chaotic attractors, Lyapunov exponents, multistability, Poincaré maps, and phase portraits are employed for the analysis of the non-perturbed and perturbed dynamical systems. In addition, the system power frequency response is analyzed, and the sensitivity to time delays is examined with return maps. To this end, the study introduces a novel classification for multi-stability and Lyapunov exponents to analyze and categorize the complex behavior of the system. The developed algorithm explores the wide parameters and initial state space, providing a complete classification of system behavior. The chaotic, quasi-periodic and periodic behaviors of the perturbed dynamical system are observed. Furthermore, the paper compares the two soliton solutions by analyzing their outcomes across a defined range of parameters to gain valuable insights into the state under which the solutions exhibit identical behavior. The classification algorithms are provided in an organized manner, and complete results are made available at the GitHub repository, ensuring that future researchers can expand upon these results. The proposed classification algorithms enable the engineers to select parameters informed by classification results. Moreover, by analyzing the results of the soliton overlap, engineers can leverage soliton co-existence behaviors to improve the resilience and performance of the system in various operating conditions. The proposed methodology has applications in optical fibers and other engineering fields.