Discrete conditions of Lyapunov stability

TL;DR

This paper introduces a novel discrete approach to Lyapunov stability analysis, replacing traditional Lyapunov functions with hypersurfaces and focusing on conditions of the differential equations' right-hand side.

Contribution

It proposes a new method for stability analysis using hypersurfaces and differential equation conditions, offering an alternative to classical Lyapunov functions.

Findings

Provides conditions for stability using hypersurfaces

Offers a discrete approach to Lyapunov stability

Extends classical methods with new geometric considerations

Abstract

We address the classic problem of stability and asymptotic stability in the sense of Lyapunov of the equilibrium point of autonomic differential equations using discrete approach. This new approach includes a consideration of a family of hypersurfaces instead of the Lyapunov functions, and conditions on the right part of the differential equation instead of conditions on a Lyapunov function along trajectories of the equation.