Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynamical systems

Abstract

The cell mapping method is a prominent one for global analysis of nonlinear dynamical systems, with which multiple invariant sets can be obtained. However, it is a continuous challenge to enhance the efficiency of the cell mapping method, especially when dealing with high-dimensional nonlinear dynamical systems. In this paper, the subdivision technique commonly used in the cell mapping method is incorporated with an interpolation sampling method, which can further enhance the efficiency over the set-oriented method with subdivision for global attractors of nonlinear dynamical systems. In the present method, a new lattice of interpolating nodes is adopted to fit into the nesting structures of the subdivided cells, portions of which are sequentially removed when resolution goes from low to high. An improved interpolation method is developed to obtain one-step sample mappings, instead of integrations when error bounds are met, in the process of subdivision iterations. Furthermore, a Hash table is introduced in order to fast search and locate the coordinates of cells that cover the invariant sets. Three examples of nonlinear dynamical systems are presented to demonstrate the enhancement in efficiency and effectiveness of the proposed method, indicating that a computational cost is one half down to one sixth of the previous methods.