Chaotic Dynamics of the Double Pendulum: A Quantitative Analysis via the AMDCF Kernel

We investigate the chaotic dynamics of the double pendulum system using the Adaptive Memory-Driven Coupling Framework (AMDCF). By integrating a non-Markovian memory field, a Critical Slowing Down (CSD) early warning metric, and an adaptive safety valve, we demonstrate numerical stability across highly chaotic regimes. We generate high-resolution phase space heatmaps to visualize the "Strange Attractor." Furthermore, we perform a quantitative numerical sweep to identify the critical phase transition (separatrix) between local and global chaos. Our results show that around 63.7 degrees, the system undergoes a sharp transition, evidenced by a collapse in both the phase space Fill Ratio and Shannon Entropy. This methodology can be extended to other complex systems in physics and biology.