Chaos And Hopf Bifurcation Analysis On Two Classes Of The Delayed Differential Equations

The delayed differential equations is used to describe the dynamical systems not only depend on the current state, but also rely on the past history status. By fully considering of the influence of history on the current state, it has many important ap-plications in mechanics, mechanical engineering, aerospace, chemical, biology, neural network, laser, electronics and information technology, confidential communications and economics, and many other fields. Due to the development of bifurcation theory and chaos theory, the chaos and Hopf bifurcation analysis of the delayed dynamical system is an important and meaningful topic.In the second chapter of this article, the local Lengyel-Epstein system with delay is investigated. By choosing τ as bifurcating parameter, we show that Hopf bifurca-tions occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold, then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.In the third chapter of this article, a modified Leslie-Gower predator-prey system with two delays is investigated. By choosing τ1and τ2as bifurcation parameters, we show that Hopf bifurcations occur when time delay crosses some critical values. More-over, we derive the equation describing the flow on the center manifold, then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bi-furcation theorem for functional differential equations, we show the global existence of the periodic solutions.