Temporal Integrity Limits in Discrete-Time Delayed Dissipative Systems

Delayed dissipative systems are ubiquitous in physical, biological, and engineered contexts, yet they are almost invariably analyzed, simulated, and controlled using discrete-time implementations. Temporal discretization is typically regarded as a numerical approximation whose effects vanish as the time step decreases. Here, we show that this assumption fails for systems with intrinsic delays. We introduce the Self-mapping Consistency Index (SMCI), a quantitative measure of dynamical identity across temporal resolutions, and apply it to a class of delayed dissipative feedback systems. Systematic parameter sweeps reveal a sharply defined temporal integrity boundary: as the temporal grain ΔT exceeds a critical value ΔT_c, trajectories generated under discrete-time evolution abruptly lose dynamical equivalence with their continuous-time counterparts. This identity collapse occurs discontinuously rather than through gradual degradation and is consistently accompanied by structural reorganization of the attractor and breakdown of predictive consistency. Across a wide range of parameters and numerical solvers, the critical temporal grain scales predominantly with the intrinsic delay time τ, following a quasi-universal relation ΔT_c / τ ≈ O(10⁻¹), with weak sensitivity to dissipation and nonlinearity. The persistence of this boundary under higher-order integration schemes demonstrates that identity collapse is not a numerical artifact but a fundamental constraint imposed by temporal discretization itself. These findings establish temporal resolution as a causal resource and identify an intrinsic limit on the faithful discrete-time representation of delayed dynamical systems.