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We study the limit cycles of the fifth-order differential equation  x ⋅ ⋅ ⋅ ⋅ ⋅ − e x ⃜ − d x ⃛ − c x ¨ − b x ˙ − a x = ε F x , x ˙ , x ¨ , x ⋯ , x ⃜ with  a = λ μ δ , b = − λ μ + λ δ + μ δ , c = λ + μ + δ + λ μ δ , d = − 1 + λ μ + λ δ + μ δ , e = λ + μ + δ , where  ε is a small enough real parameter,  λ , μ , and  δ are real parameters, and  F ∈ 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.

On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations