Parallel Subdomain Synthesis of Cell Mapping for Capturing Global Invariant Sets in Higher-Dimensional Dynamical Systems

Abstract

Unaffordable computational cost and memory storage induced by the curse of dimensionality has become the bottleneck of numerical methods in different fields. In the global analysis of nonlinear dynamical systems, the capability of numerical methods, like cell mapping methods, are mostly feasible only to a system dimension less than four. Although cell mappings are naturally parallelizable that may be used to greatly enhance the computational efficiency, it is still not enough to release the computational burden on a higher-dimensional system of greater than seven, not to mention the memory in dealing with millions of billion cells. In this paper, the subdomain synthesis method, which partitions the chosen region in state space into subdomains suitable for operating in a computational unit and then synthetizes the so-called virtual invariant sets to get the underlying global invariant sets, is promoted to be parallelizable on the subdomains so as to build a two-layer massively parallel architecture in both cell and subdomain levels. The proposed approach can be implemented by GPU Cluster that can maximize the powerful computation capability of hardwares. Examples with global invariant sets in very fine distances of a Jerk system and in bifurcations of a twelve-dimensional nonsmooth rotor system are presented for the first time to demonstrate then the feasibility of the proposed approach.