Bifurcation of bounded solutions of systems of ordinary differential equations (Q1397521)

The system of ordinary differential equations with a small parameter \(\varepsilon\) \[ \dot x=A(\varepsilon) x +F(x,\varepsilon) \] is considered. Here, \(x\in \mathbb R^n;\) \(A(\varepsilon)\in C^{m+2};\) \(F(x,\varepsilon)\in C^{m+2},\) \(F(0,\varepsilon)=0,\) \(F^{'}_{x}(0,\varepsilon)=0;\) and the matrix \(A(\varepsilon)\) has eigenvalues \[ \lambda_{k}(\varepsilon)=\alpha_{k}(\varepsilon)+i\beta_{k}(\varepsilon),\;\overline{\lambda_{k}(\varepsilon)},\;\lambda_{r}(\varepsilon)=\alpha_r(\varepsilon), \] \(k={1,\dots, l},\;r={l+1,\dots, n-l},\;l\geq 1,\) such that \(\alpha_s(\varepsilon)=O(\varepsilon^{\rho_s}),\) \(\varepsilon\rightarrow 0,\) \(\rho_s>0,\) \(s={1,\dots, n-l},\) \(\beta_k(0)\not= 0,\) \(\text{rank} A(0)=2l.\) The condition \(\sum_{j=1}^{l} q_j \beta_j(0)\not= 0\) with integers \(q_j, 0<\sum_{j=1}^{l} | q_j| \leq m+2\), is supposed to be satisfied. Bifurcation of periodical solutions for the case \(n\geq 2l\) is investigated. The case \(n=2l\) was studied in [\textit{E. P. Kubyshkin}, Differ. Uravn. 22, 1693--1697 (1986; Zbl 0637.34034)].