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The objective of the research is to develop a general method of constructingLyapunov functions for non-linear non-autonomous differential inclusionsdescribed by ordinary differential equations with parameters. The goal has beenattained through the following ideas and tools. First, three-point Poincarestrategy of the investigation of differential equations and manifolds has beenused. Second, the geometric-topological structure of the non-linearnon-autonomous parametric differential inclusions has been presented andanalyzed in the framework of hierarchical fiber bundles. Third, a specialcanonizing transformation of the differential inclusions that allows to presentthem in special canonical form, for which certain standard forms of Lyapunovfunctions exist, has been found. The conditions establishing the relationbetween the local asymptotical stability of two corresponding particularintegral curves of a given differential inclusion in its initial and canonicalforms are ascertained. The global asymptotical stability of the entire freedynamical systems as some restrictions of a given parametric differentialinclusion and the whole latter one per se has been investigated in terms of theclassificational stability of the typical fiber of the meta-bundle. There havediscussed the prospects of development and modifications of the Lyapunov secondmethod in the light of the discovery of the new features of Lyapunov functions.

The Lyapunov Concept of Stability from the Standpoint of Poincare  Approach: General Procedure of Utilization of Lyapunov Functions for  Non-Linear Non-Autonomous Parametric Differential Inclusions