Splitting of separatrices calculations for the microtron map

Abstract

Initially, a review of some works on separator splitting is made, such as standard map, McMillan map and the Hopf-Zero, in which the measures used for the calculation of the splitting are highlighted: angle, distance or area between transversal intersections of heteroclinical points. The necessary concepts are added for the development of the theory related to the study of the microtron map, after having all the definitions a study of the application of the microtron is carried out, determining which are the fixed points and their type according to the relationship with the parameter. Conditions are established to indicate when the origin is a saddle type hyperbolic fixed point, with which local parameterizations are made that are analytically prolonged until the first intersection occurs on the X, in this intersection the determines the angle formula observing that it has an exponential factor is not analytical at the origin. Thus we arrive at the conclusion that the manifold intersect transversely, forming a non-analytical exponential angle when the parameter is small.