Attraction and Stability of Nonlinear Ode's using Continuous Piecewise Linear Approximations

TL;DR

This paper introduces a simple, broadly applicable method using continuous piecewise linear approximations to determine the attraction and stability of nonlinear ordinary differential equations, extending classical results and the Markus-Yamabe conjecture.

Contribution

It presents a new sufficient condition for stability of nonlinear ODEs that unifies and extends existing linearization and stability criteria, including the Markus-Yamabe conjecture.

Findings

The main theorem simplifies stability analysis for nonlinear ODEs.

Application examples demonstrate the method's advantages and limitations.

Comparison with existing techniques shows improved or comparable results.

Abstract

In this paper, several results concerning attraction and asymptotic stability in the large of nonlinear ordinary differential equations are presented. The main result is very simple to apply yielding a sufficient condition under which the equilibrium point (assuming a unique equilibrium) is attractive and also provides a variety of options among them the classical linearization and other existing results are special cases of the this main theorem in this paper including and extension of the well known Markus-Yamabe conjecture. Several application examples are presented in order to analyze the advantages and drawbacks of the proposed result and to compare such results with successful existing techniques for analysis available in the literature nowadays.