Graph-Based Analysis of Chaotic Dynamics in the Double Pendulum System

Understanding the interplay between spatial and kinematic properties in chaotic systems is crucial for advancing nonlinear dynamics, yet remains a challenging problem. The double pendulum, as a classic example of deterministic chaos, provides a rich platform for exploring these dynamics, making its study highly relevant to researchers in nonlinear systems. Previous works have extensively analyzed the trajectories of the double pendulum using traditional numerical and analytical methods, offering insights into its chaotic behavior. However, these approaches often fail to quantitatively link spatial relationships with physical motion under varying initial conditions, leaving gaps in our understanding of how geometry and dynamics interact. This study addresses these challenges by introducing a novel graph-theoretic framework to analyze the motion of the double pendulum. By representing trajectories as undirected graphs where nodes correspond to discrete positions of the pendulum bob and edges are defined based on a tunable threshold distance parameter ϵ we systematically explore the sensitivity of graph connectivity to ϵ. This methodology emphasizes the importance of careful parameter tuning to preserve trajectory continuity and ensure valid graph-based metrics. Through this framework, we hypothesize that graph representations will reveal meaningful correlations between spatial relationships (encoded as edge weights) and kinematic properties, providing a quantitative link between geometric and physical features. Additionally, sensitivity analysis is expected to uncover power-law relationships between ϵ and trajectory resolution, demonstrating the robustness of graph-based methods in capturing both regular and chaotic dynamics. These findings underscore the potential of graph theory as a versatile tool for analyzing the complex behavior of nonlinear systems, offering fresh insights into the intricate dynamics of chaotic phenomena such as those in the double pendulum.