An alternative method to estimate the splitting of separatrices for the discretized pendulum equation

Article type: Research Article

Authors: Sellama, Hocine

Affiliations: INRIA Bordeaux-Sud Ouest, Talence, France. E-mail: hocine.sellama@inria.fr

Abstract: We consider the discretization q(t+ε)+q(t−ε)−2q(t)=ε2sin(q(t)), ε>0 a small parameter, of the pendulum equation q″=sin(q); in system form, we have the discretization q(t+ε)−q(t)=εp(t+ε),p(t+ε)−p(t)=εsin(q(t)) of the system q′=p,p′=sin(q). The latter system of ordinary differential equations has two saddle points at A=(0,0), B=(2π,0) and near both, there exist stable and unstable manifolds. It also admits a heteroclinic orbit connecting the stationary points B and A parametrized by q0(t)=4arctan(e−t) and which contains the stable manifold of this system at A as well as its unstable manifold at B. We prove that the stable manifold of the point A and the unstable manifold of the point B do not coincide for the discretization. More precisely, we show that the vertical distance between these two manifolds is exponentially small but not zero and in particular we give an asymptotic estimate of this distance. For this purpose we use a method adapted from the article of Schäfke and Volkmer [J. Reine Angew. Math. 425 (1992), 9–60] using formal series and accurate estimates of the coefficients. Our result is a variant of the results of Gelfreich [Comm. Math. Phys. 201 (1999), 155–216], Lazutkin et al. [Physica D 40 (1989), 235–248] for the pendulum problem and our method of proof, however, is quite different. This method will be useful for other problems of this type.

Keywords: difference equation, manifolds, linear operator, formal solution, Gevrey asymptotic, quasi-solution

DOI: 10.3233/ASY-151326

Journal: Asymptotic Analysis, vol. 95, no. 3-4, pp. 279-324, 2015