A Time-Step-Based Neural Network Framework for Modeling Chaotic Dynamics in Multi-Pendulum and Biological Systems

Chaotic systems are highly sensitive to initial conditions and external influences, making them difficult to predict yet critically important across disciplines such as prediction of brainwaves in medicine, modulating protein production in biology, gene regulation networks, and robotics in bioengineering. In this study, we explore the predictability of a classical chaotic system, the multi-pendulum, intending to establish it as a basis for future explorations into specialized cases of the results of this paper, using ten different machine learning models and neural networks. We begin by generating synthetic data representing the angles of the pendulum over time using the Runge-Kutta method for solving fourth-order differential equations. Initially, a single-step sliding window approach was employed, where the model predicted the 50th time step after training on steps 0 through 49. To better capture the chaotic nature of the system, we later adopted a time-step-based approach, training on a wide range of initial angles and testing on unseen, interpolated initial conditions to evaluate generalization. We also assessed system stability using Lyapunov exponents. Our results showed that for the double pendulum, the Long Short-Term Memory network outperformed other models in both the sliding window and time-step methods, under both frictional and frictionless conditions. For the triple pendulum, the Vanilla Recurrent Neural Network was most effective in the sliding window setup, while the Gated Recurrent Unit excelled in the time-step approach. However, in scenarios with a predefined frictional constant, the Long Short-Term Memory model again proved to be the most robust.