Improved generalized cell mapping for global analysis of dynamical systems

Abstract

Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that correspond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable invariant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make generalized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.