Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in  a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation.
Barreira, L., Llibre, J., & Valls, C. (2012). Periodic orbits near equilibria via averaging theory of second order. Mathematical Modelling and Analysis, 17(5), 715-731. https://doi.org/10.3846/13926292.2012.736090
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