Analytical and symbolic characterization of nonlinear dynamics for a class of energy supply demand system versus time delay

This paper investigates the nonlinear dynamic behavior of a three-dimensional energy supply–demand system with government regulation and time delay. A mathematical model is first established. By employing the τ -decomposition strategy, the exact stability boundary associated with the timing of government regulation is determined, and analytical expressions in terms of the system parameters are derived. The conditions for the existence of bifurcation and its direction are then identified. Unlike traditional analytical methods, the proposed approach avoids computing the derivative of the real part of the eigenvalues with respect to the time delay when determining the bifurcation direction. The center manifold theorem and normal form theory are subsequently used to analyze the Hopf bifurcation. In contrast to conventional approaches that require complicated integral calculations, a novel bilinear form is constructed to reduce inner-product operations in infinite-dimensional space to pure matrix algebra. Numerical simulations verify the theoretical predictions, and symbolic computation makes the dependence of key quantities on system parameters explicit. These results reveal the stability boundary of government regulation and the transition from steady states to periodic oscillations, while also providing a more efficient theoretical framework for analyzing other three-dimensional complex systems.