State space discretization based methods for global analysis of nonlinear dynamic systems from model-driven to data-driven: A review

LI Zigang; HONG Ling; JIANG Jun

doi:10.6052/1000-0992-25-002

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Li Z G, Hong L, Jiang J. State space discretization based methods for global analysis of nonlinear dynamic systems from model-driven to data-driven: A review. Advances in Mechanics, in press doi: 10.6052/1000-0992-25-002

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Li Z G, Hong L, Jiang J. State space discretization based methods for global analysis of nonlinear dynamic systems from model-driven to data-driven: A review. Advances in Mechanics, in press doi: 10.6052/1000-0992-25-002

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State space discretization based methods for global analysis of nonlinear dynamic systems from model-driven to data-driven: A review

doi: 10.6052/1000-0992-25-002 cstr: 32046.14.1000-0992-25-002

LI Zigang¹,
HONG Ling²,
JIANG Jun^{2
,
,}

1.
College of Sciences, 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

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Corresponding author: jun.jiang@xjtu.edu.cn
Received Date: 2024-11-30
Accepted Date: 2024-11-30

Available Online: 2024-11-30

Abstract

Abstract

Response behaviors of nonlinear dynamic systems are subject to their inherent global structures, such as the morphology and distribution of multi-stable attractors and basins of attraction in state space, unstable invariant sets as well as invariant manifolds. Therefore, conducting a global analysis within the specified state space can not only obtain all the information for understanding and predicting the responses, but also profoundly reveal the internal mechanisms that induce numerous dynamic phenomena, like complex bifurcations, catastrophes, or boundary transitions in the system. Currently, numerical methods remain the most effective means for the global analysis of nonlinear dynamic systems. Compared with the pointwise numerical integration or point mapping approaches, the methods based on the state space discretization, such as the cell mapping method, approximate the invariant sets by covering subsets (cells). This pattern, on the one hand, can efficiently depict the morphology of underlying global structure, and on the other hand, it characterizes the dynamics of adjacent orbits. After 40 years of development, the function of cell mapping method is continuously enhanced, the computational efficiency and accuracy are significantly improved, and its scenarios of application are also being broadened. In the second section of this review, the manners of state space discretization will be briefly classified from the perspective of current research, so that readers can understand the essence and advantages of this framework in global analysis clearly. Focusing on a series of state space discretization methods proposed in recent years, in the third section, we follow the shift in the idea of global analysis from model-driven to data-driven, introducing the latest progress achieved in two aspects: The efficient characterization of global structure and the data mining of inherent features. In the fourth section, the significance of this review is summarized, and insights will be put forward on how to further generalize the concept of global analysis within the state space discretization framework, as well as on the possible problems and expandable directions that may be faced in response to future development and application requirements.
- state space discretization,
- global analysis,
- cell mapping,
- model-driven method,
- data-driven method

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