On small perturbations of a nonlinear periodic system with degenerate equilibrium (Q405598)

This paper considers a small perturbation of a class of periodic systems with degenerate equilibrium: \(\dot{x}=N(x)+h(x, t)+f(x, t)\) where \(N(x)=(x_1^{2l_1+1}, ..., x_n^{2l_n+1})\) with \(l_1,..., l_n\) being non-negative integers, \(h=(h_1,..., h_n)^T\) denotes a higher-order term with \(h_j=O(x_j^{2l_j+2})\) for \(1\leq j\leq n\) as \(x\) tends to \(0\) and \(f\) is a small perturbation. The main aim is to prove that there exists a small constant \(\varepsilon >0\) such that if \(\|f(x, t)\|_{r_0, s_0}<\varepsilon \) then the above system can be reduced to a suitable form with zero as an equilibrium.