Delayed Nonlinear Energy Regulation Model (DHE): Unified Analytical and Numerical Study

This work presents a delayed nonlinear energy regulation model (DHE) describing a bio-dynamic system with active energy, stored energy, periodic forcing, nonlinear delayed feedback, and saturation effects. The model is formulated as a system of delay differential equations and analytically investigated for linear stability and Hopf bifurcation. Analytical conditions for Hopf bifurcation are derived and numerically validated. The system exhibits stable oscillations, period-doubling transitions, weak chaos characterized by a positive Lyapunov exponent, and a thin fractal attractor with Kaplan–Yorke dimension slightly above one. A two-parameter stability map (γ, τ) is computed to identify stable and chaotic regions. The results demonstrate that delayed nonlinear feedback combined with saturation produces structurally robust yet dynamically rich behavior, including bounded chaotic regimes. The model is relevant for bio-dynamic systems with memory, metabolic regulation, and nonlinear feedback control.