Branching of periodic solutions of quasilinear autonomous systems in the resonance case (Q1177730)

The author obtains sufficient conditions under which the system \(\ddot x=Cx+\epsilon f(x,\dot x,\epsilon)\) has periodic solutions. Here \(\epsilon\) is a small positive parameter, \(x\in R^ n\), \(f\) is a vector function assumed odd in terms of \(x\), and \(C\) is an \(n\times n\) constant matrix such that some eigenvalues are proportional to others with coefficients of proportionality whole numbers (internal resonance). The results are similar to those found by \textit{G. Bradistilov} and the reviewer [C. R. Acad. Bulg. Sci. 22, 1099-1102 (1972; Zbl 0224.34040)].