Existence and stability of limit cycles in a perturbed seventh-order differential equation

This paper investigates the existence and stability of limit cycles in a class of perturbed seventh-order non-autonomous differential equations that model multi-frequency oscillatory behavior with damping, commonly encountered in mechanical and control systems. The high order of the equation presents significant challenges in detecting periodic solutions and analyzing their stability. By applying first-order averaging theory, we reduced the original equation to a lower-dimensional averaged equation. We then derived explicit conditions under which isolated periodic solutions bifurcate from the equilibrium point. Furthermore, the stability of these limit cycles was examined by analyzing the Jacobian matrix of the averaged equation. These results extend the applicability of averaging methods to high-order nonlinear differential equations and provide valuable insights into the dynamics and control of oscillatory phenomena in science and engineering.