Appl. Math. Mech. -Engl. Ed. Applied Mathematics and Mechanics (English Edition) https://doi.org/10.1007/s10483-020-2604-6 A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems * Zigang LI 1 , Jun JIANG 2 , Ling HONG 2 , J. Q. SUN 3,† 1. Department of Mechanics, School of Science, Xi’an University of Science and Technology, Xi’an 710054, China; 2. State Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, China; 3. Department of Mechanical Engineering, School of Engineering, University of California, Merced, CA 95343, U. S. A. (Received Jan. 2, 2020 / Revised Feb. 19, 2020) Abstract A subspace expanding technique (SET) is proposed to efficiently discover and find all zeros of nonlinear functions in multi-degree-of-freedom (MDOF) engineering systems by discretizing the space into smaller subdomains, which are called cells. The covering set of the cells is identified by parallel calculations with the root bracketing method. The covering set can be found first in a low-dimensional subspace, and then gradually extended to higher dimensional spaces with the introduction of more equations and variables into the calculations. The results show that the proposed SET is highly- efficient for finding zeros in high-dimensional spaces. The subdivision technique of the cell mapping method is further used to refine the covering set, and the obtained numerical results of zeros are accurate. Three examples are further carried out to verify the applica- bility of the proposed method, and very good results are achieved. It is believed that the proposed method will significantly enhance the ability to study the stability, bifurcation, and optimization problems in complex MDOF nonlinear dynamic systems. Key words spatial discretization, subspace expanding technique (SET), parallel com- puting, subdivision, global zero finding Chinese Library Classification O324 2010 Mathematics Subject Classification 74H15, 74H50, 74S30 1 Introduction The engineering systems operated in complex environments are often nonlinear, exhibiting chaotic motions, bifurcation phenomena, instability, etc., which usually have a large number of * Citation: LI, Z. G., JIANG, J., HONG, L., and SUN, J. Q. A subspace expanding technique for global zero finding of multi-degree-of-freedom nonlinear systems. Applied Mathematics and Mechan- ics (English Edition ) (2020) https://doi.org/10.1007/s10483-020-2604-6 † Corresponding author, E-mail: jqsun@ucmerced.edu Project supported by the National Natural Science Foundation of China (Nos. 11702213, 11772243, 11572215, and 11332008) and the Natural Science Foundation of Shaanxi Province of China (No. 2018JQ1061) ©Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2020