Existence of Lyapunov functions of variable sign for linear extensions of dynamical systems on a torus

The problem of the existence of quadratic forms that have a positive definite derivative along the solutions of linear extensions of dynamical systems on a torus is considered. Assuming the existence of quadratic forms whose derivative is positive definite only with respect to part of the variables, conditions ensuring the existence of a quadratic form whose derivative is already positive definite with respect to all variables are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1713–1717, December, 1990.