Homoclinic orbits to a saddle-center in a fourth-order differential equation (Q1881139)

This paper proves the existence of homoclinic orbits for the fourth-order ordinary differential equation \[ u^{(\text{iv})} + q u^{(\text{ii})} + f(u) = 0, \] where \(u \in \mathbb R\), \(q \in \mathbb R\) and \(f (u)\) is a piecewise, odd, continuous function. This equation is reduced to an autonomous Hamiltonian system with \(2\)-degrees-of-freedom, where the origin is a saddle-center equilibrium (a pair of eigenvalues of the linearized system has negative real part and a pair of eigenvalues is purely imaginary). The authors look for even homoclinics since \(f\) is odd. The one-dimensional stable and unstable manifolds to the origin intersect to form even homoclinic orbits for an infinite sequence \(q_k \to +\infty \) of values of the parameter \( q \). Thanks to the specific form of the nonlinearity \(f\), very precise estimates are obtained.