Nonlinear Irreversible Energy Cascade and the Emergence of Temporal Structure in a Minimal Dynamical System

抄録

We present a minimal nonlinear dynamical system describing irreversible energy transfer across four coupled modes: temporal, spatial, dynamical, and material components. The model extends a linear cascade framework by incorporating nonlinear interaction and self-dissipation terms under a structural constraint that preserves total energy.

We establish four fundamental properties of the system: conservation of total energy, monotonic growth of an entropy-like quantity, boundedness of internally defined time, and global asymptotic stability of the terminal state. We prove that all physically admissible initial conditions converge to a fully materialized state.

These results demonstrate that irreversible dynamics and directional evolution can arise intrinsically in a closed nonlinear system without externally imposed time asymmetry. Furthermore, we propose an interpretative framework in which temporal behavior emerges as a transient phenomenon generated by internal energy redistribution.