Bifurcation of invariant tori in a neighborhood of an equilibrium of a system of ordinary differential equations (Q5930632)

An autonomous nonlinear ODE system with a dominant linear part is studied. Both, the eigenvalues and the nonlinear terms of the system depend on a continuous parameter \(\varepsilon\). The system is close, in \(\varepsilon\), to a Lyapunov critical case, devoid of internal resonances (incommensurable eigenfrequencies). A bifurcation is presumed to occur when \(\varepsilon\) crosses a critical value. The argumentation is based on the content of chapter 4 of I. G. Malkin's treatise [Some problems of the theory of nonlinear oscillations, Moscow: Gostekhizdat, (1956; Zbl 0070.08703)], although the references cited go back only to 1979. No quasi-periodic solutions are presented, and unlike in Malkin's treatise, no construction method is proposed.