Lyapunov stability of equilibrium states of reversible systems
The problem of the stability of the zero solution of reversible systems of differential equations, either autonomous or periodic in time, is considered. The main result is concerned with four-dimensional autonomous systems with purely imaginary eigenvalues of the matrix of coefficients of the system of linear approximation. It is proved that if a certain number \(A\), say, defined by the normal form of the system up to order five is positive, then the trivial solution is stable in the sense of Lyapunov. (I don't mention other restrictions imposed on the system which are usual for the considered problem). The proof is based on the fact that the four-dimensional autonomous system corresponds in a sense to a two-dimensional periodic one. The condition \(A> 0\) guarantees the ``elliptic'' character of the origin. Systems of dimensions higher than 4 are also considered.
scientific article; zbMATH DE number 846461
scientific article