Production of computational portraits of bounded invariant manifolds

To develop an effective procedure for describing manifold tangencies, an automotive process for describing the invariant manifolds involved is required. The present paper presents such an automotive process. The accurate location of the saddle is achieved by a modified Powell hybrid procedure which finds the local solution to a non-linear system of algebraic equations. The resulting eigenvalue problem, based on the local Jacobian matrix of the Poincare map, is solved and the eigenvectors evaluated. These are then used to extrapolate the initial points on the invariant manifold. By repeated iteration of the Poincare map other points on this manifold can be located. By storing the resultant data in a singly linked list and making use of a stack reference structure and negatively signed pointers, ordering and inserting/deletion of points can be achieved while keeping a bound on computational time and space. The technique provides a basis for investigative procedures for describing global bifurcation events