Approximating homoclinic orbits of autonomous systems via discrete dichotomies

Abstract. This paper deals with the detection of transversal homoclinic orbits of implicit autonomous difference schemes using concepts and arguments of exponential dichotomy. The difference equations which are examined arise in the studying of first order autonomous systems by discretization. Based on the result in [3] a numerical method is developed for computing the homoclinic orbit of the discretized system. This method is applied to two examples, in which we show that a critical value of a system parameter Hopf bifurcation occurs and a homoclinic orbit borns. The homoclinic orbits are visualized by numerical simulations and by constructing a Matlab code.