High order Melnikov method: pendulums
This paper is concerned with the computation of the second order Melnikov function for periodically perturbed pendulum equations given by \[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-\sin x +\varepsilon\cos^2(x/2)\cdot P(t), \] where \(P(t)\) is a periodic function in \(t\) and \(\varepsilon\) is a small parameter. Let \(D(t_0 , \varepsilon)\) be the splitting distance of the stable and unstable manifold of a saddle fixed point of the unperturbed system. Then \(D(t_0 , \varepsilon)\) can be expanded into a formal power series in \(\varepsilon\) as \[ D(t_0 , \varepsilon) = D_0 (t_0 ) + \varepsilon D_1 (t_0 ) + \cdots + \varepsilon^n D_n (t_0 ) + \cdots. \] The authors present an explicit integral formula for \(D_1(t_0)\). This work extends the theory derived in [\textit{F. Chen} and \textit{Q. Wang}, Adv. Nonlinear Stud. 17, No. 4, 793--818 (2017; Zbl 1378.37121)] to cover the pendulum equations.
scientific article; zbMATH DE number 7460669