Resolution-Induced Collapse in Quantized Nonlinear Dynamics: A Finite-Horizon Structural Framework

Abstract

Finite-precision implementation fundamentally changes nonlinear dynamical systems by replacing continuous-state evolution with deterministic dynamics on a finite set of representable states. This study examines when that change becomes structurally important over a finite observation horizon. Quantization is treated as a resolution constraint, and an operational separation scale δsep(T0,T;ε) is introduced to compare the implementation resolution with the attractor detail exposed by a reference trajectory. The ratio η=Δ/δsep and its associated critical bit width bc(T) are used as protocol-dependent measures for precision screening. Experiments on the Hénon map, a Lorenz system integrated by fixed-step fourth-order Runge–Kutta, and the Logistic map show strong system dependence. Hénon exhibits broad, non-monotonic finite-state reshaping across bit width, whereas Lorenz remains in a low-complexity regime over a wider low-bit range before recovering more complex recurrent behavior. Results from 100 selected occupied quantized attractor positions show that entropy alone is insufficient to characterize collapse; recurrence and transient lengths provide complementary information about orbit organization. A 21-horizon Lorenz study produces stepwise changes and a long plateau in bc(T), rather than a smooth linear scaling law. For Hénon, the largest-horizon crossing is resolved at bc=25.457, corresponding to a minimum integer bit width of 26 under the declared estimator. An alternative estimator gives materially different crossing values, while the Logistic map also shows strongly non-monotonic behavior. Overall, η and bc(T) provide useful implementation-oriented screening measures, but they are not universal thresholds or hardware guarantees.