A new approach to computing two-dimensional manifolds

We propose an approach to computing two-dimensional unstable and stable manifolds of three-dimensional vector fields. The main idea is to estimate normal direction on each point around the boundary of current loop of manifold and normalize the normal growth rate during a settled time step to counter the disequilibrium in different directions. In order to enhance the reliability of our approach, linear and nonlinear conditions are considered. It is necessary to state that the time step should be appropriately small to meet the adjacent intervals of points on the boundary of manifold. As example we compute the two-dimensional stable manifold of the origin in Lorenz system. Both successes and shortcomings of our method are presented.