nonlinear dynamical systems

Nonlinear dynamical system research is essential in science and engineering because it could be potentially employed to represent real-world phenomena.  Traditional methods rely on pre-defined models or computationally expensive simulations, limiting their applicability to only numerical data. In the present research, without any prior knowledge of the system, we suggest a novel way to build a linear representation of the Duffing oscillator by fusing the capabilities of deep neural networks with the Koopman operator. This recently established methodology makes it easier to estimate system parameters effectively and accurately predict the oscillator's future behavior. Our approach incorporates a modified training procedure that restricts the Koopman operator to a single linear layer within the neural network, improving interpretability and potentially reducing training complexity. This methodology not only simplifies nonlinear system analysis but also paves the way for advancements in predictive modeling across various fields. Notably, our method yields distinctive eigenvalues of the Koopman generator matrix, enabling the Koopman operator to exhibit robustness against noise and capture a spectrum of Duffing equation behaviors. This includes the precise prediction of periodic oscillations and the capturing of period-doubling bifurcation, all while maintaining tractability within the neural network framework.