On the limit cycles for a class of perturbed fifth-order autonomous differential equations (Q2065436)

Summary: We study the limit cycles of the fifth-order differential equation \(\overset{\cdot \cdot \cdot \cdot \cdot}{x}-e\overset{\ddddot{}}{x}-\text{d}\overset{\dddot{}}{x}-c\ddot{x}-b\dot{x}-ax=\varepsilon F(x, \dot{x}, \ddot{x} , \overset{\cdots}{x}, \overset{\ddddot{}}{x})\) with \(a=\lambda\mu\delta\), \(b=-(\lambda \mu + \lambda \delta + \mu \delta)\), \(c=\lambda+\mu+\delta+\lambda\mu\delta\), \(d=-(1 + \lambda \mu + \lambda \delta + \mu \delta)\), \(e=\lambda+\mu+\delta\), where \(\varepsilon\) is a small enough real parameter, \(\lambda,\mu\), and \(\delta\) are real parameters, and \(F\in C^2\) is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.