Global topological classification and coefficient conditions of a kind of plane homogeneous fifth system with five special direction is studied.

This paper studies the global topological structure of a kind of plane homogeneous fifth system with three special direction dx[]dt=a_(50)x~5+a_(41)x~4y+a_(32)x~3y~2+a_(23)x~2y~3+a_(14)xy~4+a_(05)y~5,dy[]dt=b_(50)x~5+b_(41)x~4y+b_(32)x~3y~2+b_(23)x~2y~3+b_(14)xy~4+b_(05)y~5.

We discuss the global topological structure of the homogeneous fifth system with one and two special directions ,and give their coefficient conditions.

Globally uniform stability is obtained for a class of nonlinear continuous cascaded systems by using homo- geneous properties of homogeneous systems.

The nonlinear H ∞ control problem of homogeneous systems via output feedback is considered.

A class of fifth system with one twenty-one order singularity;

Infinity is used as a base point in homemorphic transformation to study infinity for a class of fifth system and isochronous center problems.

In this article,the center conditions,isochronous center conditions and bifurcation of limit cycles at infinity for a class of fifth system are investigated.

A class of planar quintic system x=λx-y+yR_2+xR_4,y=x+λy-xR_2+yR_4,is studied,where R_2=b_1x~2+b_2xy+b_3y~2,R_4=a_4x~4+a_2x~2y~2+a_0y~4,and the focus value with order k,the necessary and sufficient conditions are obtained with O(0,0) being a center of the system.

The topological classification of higher order singular point for the quintic system with one zero characteristic root is discussed, and a criterion by the coefficients of polynomials is given.

In this paper, we discuss the global topological structure of thehomogeneous cubic systems, and give their coefficient conditions.