Detecting all Homoclinic Points in Nonlinear Discrete Dynamical Systems Via Resurgent Analysis

We present a novel and completely deterministic method to model chaotic orbits in nonlinear discrete dynamics, taking the quadratic map as an example. This method is based on the resurgent analysis developed by {\'E}calle to perform the resummation of divergent power series given by asymptotic expansions in linear differential equations with variable coefficients. To determine the long-term behavior of the dynamics, we calculate the zeros of a function representing the unstable manifold of the system using Newton's method. By use of the obtained zeros, we visualize the full set of homoclinic points. This set corresponds to the Julia set in one-dimensional complex dynamical systems. Our method is easily extendable to two-dimensional nonlinear dynamical systems such as H{\'e}non maps.