Bifurcations of periodic solutions of nonlinear autonomous differential systems with a condition and resolution of singularities arising from bifurcations

This paper deals with bifurcations of 4 omega $-periodic solutions ($ omega $ is unknown) of a parameter-dependent nonlinear autonomous differential system $ \frac{dx}{d\tau} = f(x,λ) (x,f(x, lambda) in R^n, lambda in R) $ whose right member $ f(x,lambda) $ satisfies the condition $ f(Px, lambda)= Pf(x,lambda)(x in R^n, lambda in R) $ for a real matrix $ P (\ eq I_n $; a unit matrix) satisfying $ P^q = I_n $ for a positive even integer $ q $