The Global Structure Of Several Types Of Planar Polynomial Systems, And Bifurcation

In this paper, we maily discuss a c]ass of codirnention-two higher order nonlinear vector field = y ?P(x, y) { z~ Q(r~ y) XVI tere x , y ) , (J ( .r, y ) are polv toni mis of ([egree not less titan 7. 憊Ve study their global strucl tire , local aii(l glol.)fe bifurcattons. First, we reduce the vector ficl(l by normal form theory, then the study is e(luiva.lent to the below { q2 ~ + /t2Y + ciX~ + /3x~抷, o/3 0, 慽i 7 for n.:2 or n=3, T3ogdauov, laketts, Cart etc have been studied it抯 Lo? cal bifurcation, after that, Wangminshu, Luodingj un, Lijibin, Wangxian etc studied global bifurcations for n=3. for n =5, Chenfangyue obtai tied the homoc I iti~c. hyl)ercLi tue hi fu teat ion curves by using Pi card- Fuchs equation met. hod. Secondly, we stti(Iy the changing of fixed ponds arid orbit structure with the changing of the parameters of the reduced vector field. We t.ttilize the typica.l ways to studying local bilurcations. There are two cases iii (liscussing it: .r 1 and ??i Stuclyi ng time possibly local l.)i lurca.t ions and l)i furcat.ion (tirVeS. ci yawing the Orl)jt structure diagram. When we study its glol)Le bifurcation, first, we transform it~ to the~norma1?[orm to use Meltukov method. For n~7, Melnikov function is very (iifficnlt to work out. We use 1~icard-Fuchs ecluation method, we introduce a. new judgment function, then discuss it抯 characters, obtain it抯 homoclinic bifurcation curve. At last, we draw the orbit structure diagram in the parameter 1)lane.