On the stability of the equilibrium position in critical and near-critical cases

Abstract

Autonomous systems of ordinary differential equations are examined. The stability of the equilibrium position is studied in nearly degenerate (critical) cases. Estimates are given for the magnitude of the small attracting domain under stability in the critical case and for the magnitude of the “dangerous” perturbations under instability. Standard theorems are formulated and examples given of the proof of appropriate theorems. A list is given of all critical cases of degeneration degree no higher than three and of the corresponding stability criteria.