The Qualitative Properties And Application Of Some Differential Systems In R~2and R~3

It is very difficult to analyze the geometric properties of vector fields in R3. In this paper, by establishing a bridge between a class of quasi-homogenous system in R3and its induced planar tangent system, we proved the existence of homoclinic cycle to a class of cubic systems in R3.In the second chapter,We consider a predator-prey system of Leslie type with generalized Holling-III functional response:p(x)=mx2/ax2+bx+1. By allowing b to be negative (b>-2(?)a), p(x) is monotonic for b≥0and nonmonotonic for b<0when x≥0. It is shown that the model has two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of codimension2and the other is a multiple focus of multiplicity one. When parameters vary in a small neighborhood of the values of parameters, we show that the model undergoes the Bogdanov-Takens bifurcation and the subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. The bifurcation portrait and phase portraits show that the model can have a stable limit cycle enclosing two non-hyperbolic positive equilibria, or a stable limit cycle enclosing an unstable homoclinic loop, or two limit cycles enclosing a hyperbolic positive equilibrium, or one stable limit cycle enclosing three hyperbolic positive equilibria. These results show that the dynamical behaviors of this model with b>-2(?)a are more complex and richer than that of this model with b>0..