Did you receive my email ?@Carl Love 

 I could not sed email to you@shatha 

@Carl Love 

Thanks for your help , I did not study the case when it is resonance . I tried to solve when it is non reasonant in this case it becomes linear equation ,but in the case of resonant it does not . I try to solve aparticular system ,but it you can explain to me in general case  in maple ,then I try in the system which I have to solve , as I know in each stpes the degree will become higher.  

Step2: change the ode above by the coordinate change

Step3:apply step 1 on ode

Step2: change the ode above by the coordinate change

Step3:apply step 1 on ode

 , Thank you so much for your reply . actully, I  try to solve non-linear ordinary differential equation and its solution is symbolic .the idea behand the poincare theorm is to solve nonlinear equation  by reduce it to linear equation . by formal change of veriable  @Carl Love 

the thorem is to solve   diff(x,t)=Ax+sum(vr,i=2..infinity) where vr(x) is homogenous polynomial of degree r in the veriable x1,x2,..,xn   and A is a matrix the thorem the idea behanid the thorem is to  reudced the nonlinear equation to a linear equation  diff(y)=Ay    by a formal chang of veriable x=y+h(y) where h(y) denote series starting  with  term of degree greater then one  so it is consider of an algorith because it depend on how many dgree would like  you like to solve up to . the question which I have is .
diff(x,t)=Matrix([[ 1, 0],
        [ 0, i ]]) x+Vector[column]([ x^2 ,
                 0])x; 

so h2(x)=x^2 v1=Vector[column]([ x^2 ,
                 0])   
substituting x=y+h2(y) in the original equation 
(I+dh/dy)diff(y)=A(y+h2(y)+v2(y+h2(y))
diff(y)=(I+dh/dy)^-1 [Ay+h(y)+v2(y)+
[[[2 y^(3)+y^(4)],[0]]]]  ;
                                                                                                                                                                                                                          diff(y)=(I-dh/dy+(dh/dy)^2 the expend of the series is depend on how many degree it work up to here i write three term because I worked up to order 3 [Ay+Ah2(y)+v2(y)+v3(y)]           diff(y)=Ay+Ah2(y)+v2(y)+v3(y)-dh2/dyAy-dh2/dy Ah2(y)-dh2/dy v2(y)-dh2/dy v3(y)                                                                                                                                                             diff(y)=Ay+[Ah2(y)-dh2/dy Ay)+v2(y)] all this term =0 +w3(y)                                                                                                                                                                                                w3=Vector[column]([ -2y^3-3y^4-2y^5+4y^3+8y^4+8y^5+y^6 ,
                 0])                                                                                                                                                                                                                                                                                      there is an equation in the form diff(y)=Ay+Vector[column]([ 2y^3 ,
                 0 ])                                                                                                                                                                                                                                                                                    and now it is beuase deuation of dgree three and the same proceudre used to get up to more degree .
restart;
with(LinearAlgebra);
ode; transformfunction(x, y); _local(A, h(x), x); ode := diff(x, t) = Ax+v(x); A := Matrix([[1, 0], [0, i]]); v(x) := Vector[column]([x^2, 0]); x := y+h(y); pdsolve(ode, series, order = 3); solve; endproc;

how can I write this procedure this my attemt ,but it is not correct