Relaxation Oscillations and Jacobian Continuation in Barrier Systems

We analyze relaxation oscillations arising in nonlinear iterative schemes and demonstrate that such oscillatory behavior encodes geometric obstruction rather than numerical failure. By introducing Jacobian continuation in parameterized spaces, we show that oscillations can be regularized, revealing their structural origin. A charge-like interpretation of barrier interactions is proposed to model effective repulsion and attraction induced by energy-gradient misalignment. This standalone note interfaces with broader barrier and metastability frameworks while remaining mathematically self-contained.