ON THE PERIODIC ORBITS OF CONTINUOUS THIRD-ORDER DIFFERENTIAL EQUATION WITH PIECEWISE PERTURBATIONS

In this paper, we study the sufficient conditions for the existence of periodic solutions of the following differential equation <p class="disp_formula">$ \begin{equation*} \dddot{x}=-\dot{x}+\varepsilon |\ddot{x}|-\varepsilon \left( \alpha x-\beta \dot{x}\right) ^{m}, \end{equation*} $ where $ m $ is a natural number, and $ \alpha $, $ \beta $ and $ \varepsilon $ are real parameters with $ |\varepsilon|&gt;0 $ being small. We apply the averaging method and the Melnikov function method respectively to study the periodic solutions of this type of differential equation. We also provide an example as an application.