Bifurcation of an invariant torus from an equilibrium under commensurability conditions (Q1778261)

This paper considers a \((2n+r)\)-dimensional autonomous ordinary differential equation with a parameter which generates a bifurcation when its value crosses through the value zero. The matrix of the linear approximation has \(n\) pairs, \(n>1\), of pure imaginary eigenvalues, and the zero eigenvalue with the multiplicity \(r\). The pure imaginary eigenvalues are supposed to be distinct and commensurable. The author studies the bifurcation of an \(s\)-dimensional torus (\(s<n\) being defined by two relations satisfied by the eigenvalues) from the equilibrium supposed to be at the origin of the phase space when the parameter is equal to zero. For this study the autonomous ordinary differential equation is normalized in a classical way (in particular used by Malkin in the past). A theorem is proved, and an example of a seven-dimensional equation is given.