How a System’s Own History Builds Internal Gain and Activates New Dynamical Behavior
Nonlinear systems often change behavior only after coupling, control gain, or another internal influence becomes strong enough to cross a critical threshold. But how can this transition occur when the gain is not changed from outside? This paper develops a general framework in which the trajectory’s own history generates the transition: repeated exposure to a region of state space accumulates hidden internal variables, which progressively amplify an effective gain until it persistently crosses the required threshold. Under the appropriate geometric conditions, this crossing can then activate capture, stabilization near a compatibility manifold, or tracking of a slowly moving coherent structure. A two-channel phase-oscillator model demonstrates the complete sequence from cumulative exposure to gain crossing, capture, and deep phase locking.
Ori Manzur
reBORN Research Initiative
Working Manuscript / Research Preprint: This manuscript is shared by the reBORN Research Initiative for open visibility, discussion, and technical feedback. It has not been peer-reviewed. Mathematical, numerical, and conceptual feedback is welcome.