Multistability Of Delayed Differential Equations With Non-monotonic Feedback

This investigation aims at developing a methodology to establish multistability dynamics generated by the scalar delay differential equations of the form(?)(t) =-g(x(t)) + f(x(t- ?)). With the help of the method of domain decomposition,and some techniques of monotone dynamical system and one-dimensional map, positively invariant regions for the semiflows and basins of attraction for these stationary are estimated. Furthermore, in great detail, the global dynamics of the Allee-Type model are proved. The Hopf bifurcation and the existence of heteroclinic orbits(including two types of heteroclinic orbits: from one equilibria to another one and from one equilibria to a periodic orbit oscillating around the largest positive equilibrium) are also established. An example with a series of numerical simulations is arranged to illustrate these general results. This thesis is divided into four parts.In chapter one, we briefly introduce nonlinear time-delay system at first, then we give an outline of the background of the models and the main research contents.In chapter two, We present the basic definitions and notations of the background theory.In chapter three, we will concern with the invariant intervals and the multistability properties of equilibria of model (?)(t) =-g(x(t)) + f(x(t- ?)).In chapter four, in great detail, the multistability issue of Allee-Type model is addressed.In particular, the existence of heteroclinic orbits from the equilibrium point to another equilibrium point or a periodic oscillating around the largest positive equilibrium point are established.In the end,examples with numerical simulations is arranged to illustrate the main results in this paper.