A novel theorem on motion stability

TL;DR

This paper introduces a new theorem utilizing two functionals to determine the stability or instability of singular points in nonlinear dynamical systems without needing explicit solutions, and extends to a novel linearization method.

Contribution

The work presents a novel theorem for stability analysis using two functionals and extends it to a superior linearization technique based on averaging, applicable to complex systems.

Findings

Effective in determining equilibrium stability without analytical solutions

Applicable to both linear and nonlinear systems for stability and instability

Extends to higher-order systems with improved linearization approach

Abstract

Determination of stability and instability of singular points in nonlinear dynamical systems is an important issue that has attracted considerable attention in different fields of engineering and science. So far, different well-defined theories have been presented to study the stability of singular points among which the Lyapunov theory is well-known. However, the instability problem of singular points has been neglected to some extent in spite of its application in oscillator design. Besides, it is often difficult to achieve a proper Lyapunov function for a given complex system. This work presents a novel theorem based on defining two distinct functionals and some straightforward criteria to study motion stability that significantly facilitate the determination of equilibrium status at singular points without the requirement to analytical solution. Indeed, this method is applicable to both stability and instability problems of linear and nonlinear dynamical systems. In addition, the presented theorem is further extended to achieve a new linearization approach of dynamical systems based on averaging technique, which is superior to the Jacobian approach. Lastly, the proposed linearization method is generalized to study the stability/instability of higher-order liner/nonlinear systems. The obtained results clearly show the effectiveness of the proposed theorem to assess the motion stability/instability.