Algorithmic Investigation of Complex Dynamics Arising from High-Order Nonlinearities in Parametrically Forced Systems

The geometric content of chaos in nonlinear systems with multiple stabilities of high order is a challenge to computation. We introduce a single algorithmic framework to overcome this difficulty in the present study, where a parametrically forced oscillator with cubic–quintic nonlinearities is considered as an example. The framework starts with the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which is a self-learned algorithm that extracts an interpretable and correct model by simply analyzing time-series data. The resulting parsimonious model is well-validated, and besides being highly predictive, it also offers a solid base on which one can conduct further investigations. Based on this tested paradigm, we propose a unified diagnostic pathway that includes bifurcation analysis, computation of the Lyapunov exponent, power spectral analysis, and recurrence mapping to formally describe the dynamical features of the system. The main characteristic of the framework is an effective algorithm of computational basin analysis, which is able to display attractor basins and expose the fine scale riddled structures and fractal structures that are the indicators of extreme sensitivity to initial conditions. The primary contribution of this work is a comprehensive dynamical analysis of the DM-CQDO, revealing the intricate structure of its stability landscape and multi-stability. This integrated workflow identifies the period-doubling cascade as the primary route to chaos and quantifies the stabilizing effects of key system parameters. This study demonstrates a systematic methodology for applying a combination of data-driven discovery and classical analysis to investigate the complex dynamics of parametrically forced, high-order nonlinear systems.