This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincar\'e method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincar\'e) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.

翻译：本文首次提供了在普通Bogdanov- takens两面点附近使用美元维值的三等同临床预测器。 要达到这个目的, Lindstedt-Poincar\'e 方法中需要更高层次的时间近似非线性时间变异。 此外,需要将近似转换为正常形式的解决方案,而将近似转换为参数依赖中心元的解决方案。在应用不同的正常形式(mooth和轨道)、不同阶段条件和不同的扰动方法(常规和Lindstedt-Poincar\'e)以接近Bogdanov-Takes 点附近的同道解方法(常规和Lindsted-Poincar\'e)之间进行了详细比较。 提供了一些实例,表明预测器的正确性。 在开放源 MATLAB/GNU Octave连续包 MatCont 中实施了新的同源预测器。