An Effective Numerical Procedure to Determine Saddle-Type Unstable Invariant Limit Sets in Nonlinear Systems

A new technique that can efficiently approximate the attracting set of a nonlinear dynamical system is proposed under the framework of point mapping with the cell reference method. With the aid of the approximated attracting set, the difficulties encountered by the PIM-triple method and bisection procedure in finding trajectories on the stable manifolds of chaotic saddles in basins of attraction and on basin boundaries can be overcome well. On the basis of this development, an effective method to determine saddle-type invariant limit sets of nonlinear dynamical systems can be devised. Examples are presented for the purposes of illustration and to demonstrate the capabilities of the proposed method.