Structural Memory via Delayed Feedback in Linear Dynamical Systems

This paper analyzes a minimal class of linear delayed-feedback systems and establishes precise conditions under which structural memory arises at the operator level. Memory is defined in terms of non-instantaneous input–output dependence, and we show that delayed recurrence generates explicit nonlocal expansions in both continuous and discrete time. Within the single-delay model, delay and feedback are proven jointly necessary for structural memory, and bounded input–output mappings exist if and only if the system satisfies classical stability conditions. In discrete time, the admissibility condition |α| < 1 characterizes the regime in which recurrence produces geometrically decaying persistence. A compact frequency-domain analysis demonstrates how recurrence structure manifests through pole placement and approximately 2π/N resonance spacing in the single-delay case. Numerical illustrations confirm consistency between analytic impulse expansions, stability boundaries, and spectral behavior. The results isolate delayed feedback as a minimal structural generator of memory in linear systems, providing a rigorous foundation for subsequent extensions.