Drift, Memory, and Hysteresis in Learning Systems: Insights from Deterministic Non-Invertible Dynamics

This work studies how memory, hysteresis, and drift interact in non-invertible dynamical systems, with direct implications for artificial intelligence and learning architectures. Using a mathematically exact discrete system as a controlled laboratory, we identify the conditions under which memory either stabilizes or is inevitably erased over time. The core theoretical result establishes a sharp Lyapunov-based convergence criterion: in systems with negative average drift, long-term divergence is impossible unless a measurable form of memory accumulation persists. This isolates a precise boundary between transient memory (which naturally resets) and true hysteresis (which can accumulate and alter long-term behavior). Extensive computational analysis shows that even when local low-information regimes persist temporarily, the system consistently recovers toward higher-information states, exhibiting rapid decorrelation and finite memory depth. Apparent feedback loops observed in coarse or modular representations fail to trap the full system, demonstrating that hysteresis does not accumulate in the absence of inversion or explicit reinforcement mechanisms. These findings suggest a general design principle for AI systems: Memory accumulation requires inversion, reinforcement, or external forcing; otherwise, information loss dominates and the system resets. This principle applies broadly to learning algorithms, optimization dynamics, recurrent architectures, and adaptive systems, where stability, forgetting, and recovery must be carefully balanced. The work provides a rigorous mathematical foundation for understanding when memory is structural versus when it is ephemeral—offering guidance for building AI systems that either suppress hysteresis (for robustness) or deliberately sustain it (for long-term adaptation).