Dynamical Analysis Of A Class Of Energy Resources Systems

The energy system is a complex nonlinear system. Nowadays energy supply and demand issue has become a hot topic in society and a major concern of all countries in the world with the continuous development of economy. Solving the energy problem is an important part of realizing the science development view and an important guarantee of the stable, healthy and sustainable development of the country economy. Nonlinear dynamics is widely used in many scientific and makes many achievements at home and abroad with the development of society. Nonlinear dynamics is rarely used to study the energy system although the nonlinear dynamics is developing rapidly. How to make better use of the theory and methods to study complex energy issues to provide strategic choice for sustainable development has become one of the most concerned issue in energy study.In this paper, we firstly study the effect of time delay on a kind of energy system using bifurcation theory of differential equations. We obtain the stability of the equilibrium point and sufficient condition for the existence of the Hopf bifurcation through analyzing the linear equations and characteristic equations in the positive equilibrium point according to the stability of the equilibrium theory and Hopf bifurcation theory. We obtain the precise formula to determine the Hopf bifurcation direction and stability of bifurcating periodic solutions using specification model theory of functional differential equations and center manifold theorem. These systems reflect energy resources better and have more practical significance compared with the energy system without time delay.Secondly, we propose a new energy system with diffusion term by adding diffusion term of space and suitable initial and boundary conditions on the basis of the original model in consideration of the diffusivity of the supply and demand of energy in space. We give the linear system of this new system at equilibrium point and obtain the stability of the positive equilibrium point and the existence of local Hopf bifurcation by analyzing the corresponding characteristic equation of the system.Furthermore, we obtain Hopf bifurcation direction and the formula of the stability of the periodic solutions.At last, in order to conform the theoretical results obtained before, we show the corresponding numerical simulation.