Geometry Controlled Instability in a Nonlinear Dual-Planar Field System

Nonlinear dynamical systems frequently exhibit abrupt transitions from stable dynamics to catastrophic amplification when interaction strengths exceed critical thresholds. Understanding the mechanisms controlling such transitions remains an important problem across many areas of physics and applied mathematics. In this work we investigate a nonlinear dual–planar field model in which two interacting spectral components couple through a transverse feedback channel. The system is controlled by a nonlinear coupling parameter $\alpha$ and a geometric alignment parameter $\theta$ governing the interaction structure between the two planar modes. A large ensemble of GPU–accelerated simulations was performed, consisting of $2400$ independent realizations spanning the parameter space $(\alpha,\theta)$. The simulations reveal a clear geometry–dependent instability transition characterized by three distinct dynamical regimes: stable dynamics, bounded amplification, and runaway growth. For aligned configurations $(\theta = 0)$ runaway behavior emerges above a critical coupling $\alpha_c \approx 0.156$, while partially aligned systems $(\theta = \pi/4)$ exhibit a delayed transition near $\alpha_c \approx 0.218$. Orthogonal configurations $(\theta = \pi/2)$ remain stable across the explored parameter range. These results demonstrate that geometric coupling strongly regulates nonlinear feedback and instability formation in the system. The findings highlight the importance of interaction geometry in controlling amplification processes within nonlinear multi–mode dynamical systems.