A new class of Lyapunov functions for stability analysis of singular dynamical systems. Elements of $p$-regularity theory

A new approach is proposed for studying the stability of dynamical systems in the case when traditional Lyapunov functions are ineffective or not applicable for research at all. The main tool used to analyze degenerate systems is the so-called $p$-factor Lyapunov function, which makes it possible to reduce the original problem to a new one based on constructions of $p$-regularity theory. An example of a meaningful application of the considered method is given.