On the periodic solutions of a class of Duffing differential equations (Q1943272)

Consider the differential equation \[ x''+\varepsilon Cx'+ \varepsilon^2A(t)x+ b(t) x^3= \varepsilon^3\Lambda h(t),\tag{\(*\)} \] where \(C\), \(\varepsilon\) and \(\Lambda\) are real parameters, \(C\) is positive, \(\varepsilon\) is small and positive, the functions \(A\), \(b\) and \(h\) are continuous and \(T\)-periodic satisfying \(A_0b_0> 0\) with \[ A_0= {1\over T} \int^T_0 A(t)\,dt,\quad b_0= {1\over T} \int^T_0 b(t)\,dt. \] The authors study the existence of periodic solutions of \((*)\) and their stability for sufficiently small \(\varepsilon\) in dependence on the parameter \(\Lambda\) and the averaged functions \(A\), \(b\) and \(h\). The proof is based on the averaging theory applied to the bifurcation of periodic solutions.