自治Birkhoff系统Herglotz方程的组合梯度表示及其解的稳定性OA

As a class of differential equation systems with favorable properties,gradient systems can be utilized for stability analysis if a mechanical system can be changed into a gradient system.This paper adopts a gradient approach to study the stability of solutions to the Herglotz equations for autonomous Birkhoffian systems.First,the Herglotz equations of the system were established and expressed in contravariant algebraic forms.According to six types of combined gradient systems,the conditions under which the Herglotz equations of the autonomous Birkhoffian system can be transformed into these six types of combined gradient systems were given,and the solutions and their stability were subsequently analyzed.Finally,through six examples,the specific computational process of transforming the Herglotz equations of Birkhoffian systems into six types of combined gradient systems was demonstrated,verifying the validity of the obtained results.