Gradientization of Herglotz-Type Equations and Stability of Their Solutions

The stability of solutions of Herglotz-type equations for non-conservative systems is studied by converting them into gradient systems. Firstly, Herglotz-type equations for non-conservative systems are given and expressed in contravariant algebraic form. Secondly, four kinds of basic gradient systems are introduced. Thirdly, the conditions for transforming Herglotz-type equations of non-conservative systems into basic gradient systems are given. Then the solution of Herglotz-type equations and its equilibrium stability are analyzed. Finally, four examples are presented to illustrate the calculation process in detail for each case. The results show that the gradient method is effective.