A data-driven framework for continuous parameter-space categorization of nonlinear dynamical systems using discrete simulations

Parameter space categorization in nonlinear dynamical systems is usually based on discrete data, which creates limitations because system responses are sensitive to minor changes in parameters. As a result, an extensive stability examination across the parameter space is computationally costly, and sparse sampling might neglect important behaviors. To overcome these challenges, this work develops a continuous predictive framework, which is presented using the Duffing oscillator as an application. Building upon this framework, machine learning models are employed to predict and continuously map the dynamical response of the system in the parameter space defined by the stiffness parameters ( α , β ) and the damping coefficient ( γ ) . The computations were conducted over a structured parameter space in which the linear and nonlinear stiffness parameters ( α , β ) were varied within the range [ − 2,2 ] and the damping coefficient ( γ ) within [ − 1,1 ] , together with multiple initial states to generate representative trajectories that exhibit stable, unstable, and periodic responses. The resulting data were balanced with the SMOTE method to enhance the reliability and robustness of the classification models. Several supervised learning methods were trained to understand the connection between dynamical responses and system parameters, including Random Forest, Support Vector Machine, Gaussian Process, Neural Network, and gradient-boosting models. Among the evaluated models, the ensemble methods yielded the most consistent results, with the stacking-based ensemble achieving the highest classification accuracy. The trained models allowed for continuous prediction of the system dynamics across the entire parameter space, effectively extending discrete simulation data into a continuous representation of dynamical behavior. The proposed approach provides an efficient alternative to exhaustive numerical simulations and offers a practical framework for analyzing nonlinear dynamical systems.