A Bifurcation Dynamical Analysis of a Non-Darcy Seepage System in Post-Failure Rock Based on a Novel Truncated Spectral Method

This paper investigates the dynamic behavior of non-Darcy seepage systems in post-failure rock. A one-dimensional non-Darcy seepage evolution equation is established, and a 6-dimensional nonlinear ordinary differential system is derived via the spectral truncation method. Eigenvalue analysis is adopted to determine the instability and bifurcation conditions, with the bifurcation diagram plotted. The fourth-order Runge–Kutta method is used to obtain phase trajectory patterns under different initial values. The results confirm the existence of transcritical bifurcations and fold bifurcations. The dynamic response of the system is discontinuous with control parameters, and phase trajectory symmetry breaking occurs with the increase in nonlinear terms. The reduced-order model shows diverse phase trajectories including equilibrium, periodic, chaotic attractors and unstable states. The system is sensitive to initial values, which significantly affect phase trajectory behaviors. The system may lose stability and trigger water inrush hazards under critical conditions. The bifurcation diagram and critical parameters obtained can provide a theoretical basis for the early warning, risk assessment and prevention of coal mine water inrush hazards.