Basis of the algorithm of asymptotic decomposition for a finite number of approximations (Q1112220)

We consider the problem of the basis of the asymptotic decomposition algorithm. We compare the perturbed system \(dx'/dt=\omega (x')+\epsilon {\tilde \omega}(x'),\) where \(\epsilon\in [0,1]\), \(x=colon\| x_ 1,...,x_ n\|,\) \(\omega =colon\| \omega_ 1(x),...,\omega_ n(x)\|,\) \({\tilde \omega}=colon\| {\tilde \omega}_ n(x),...,{\tilde \omega}_ n(x)\|,\) to the contracted centralized system \[ dx_ j^{(m)}/dt=\omega_ j(x^{(m)})+\epsilon N^{(m)}(x^{(m)})x_ j^{(m)}+\epsilon^{m+1}\Phi_ j^{(m+1)}(x^{(m)},\epsilon),\quad j=1,...,n, \] where \(N^{(m)}(x^{(m)})=N_ 1(x^{(m)})+...+\epsilon^{m-1}N_ m(x^{(m)})\), \(x^{(m)}=colon\| x_ 1^{(m)},...,x_ n^{(m)}\|\), \(\Phi_ j^{(m+1)}(x^{(m)},\epsilon)\) are known analytic functions.