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This work aims to investigate a nonlinear energy regulation model that incorporates delayed feedback and saturation effects in dynamic systems.

February 17, 2026Open Access

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

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Analytical and numerical study reveals complex dynamics in bio-dynamic systems with nonlinear feedback.

This work aims to investigate a nonlinear energy regulation model that incorporates delayed feedback and saturation effects in dynamic systems.
Developed a system of delay differential equations for modeling
Conducted analytical stability analysis including Hopf bifurcation
Numerically validated analytical findings
Computed two-parameter stability map to identify dynamic behaviors
Identified stable oscillations and period-doubling transitions
Observed weak chaos characterized by a positive Lyapunov exponent
Produced a thin fractal attractor with Kaplan–Yorke dimension slightly above one
Demonstrated stable and chaotic regions in the stability map

Kujtim Gjoka (2026) studied this question.

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