基于状态空间离散的非线性动力系统全局分析方法进展: 从模型驱动到数据驱动

李自刚; 洪灵; 江俊

doi:10.6052/1000-0992-25-002

基于状态空间离散的非线性动力系统全局分析方法进展: 从模型驱动到数据驱动

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

李自刚¹,
洪灵²,
江俊^2, ,

1.
西安科技大学理学院, 西安 710054
2.
西安交通大学复杂服役环境重大装备结构强度与寿命全国重点实验室, 西安 710049

基金项目: 国家自然科学基金项目 (12472031, 12172279, 12172267) 资助

详细信息

作者简介:
李自刚, 西安科技大学教授, 项目博导. 全国徐芝纶力学优秀教师. 现任理学院副院长, 动力学与控制学科学术带头人, 信息交叉力学创新团队负责人. 主要从事非线性系统全局振动控制与机器学习方法、转子动力学等研究. 在国内外学术刊物上发表论文40余篇. 主持国家自然科学基金项目3项. 获陕西省自然科学奖一等奖、三等奖, 陕西高等学校科学技术奖特等奖, 西安市科学技术二等奖等. 担任国际动力学与控制学术期刊“International Journal of Dynamics and Control”副主编

洪灵, 西安交通大学教授, 博士生导师. 主要从事非线性复杂系统全局动力学分析方法,模糊非线性系统动力学与最优控制以及机器学习方法等研究. 在国内外学术刊物上发表论文90余篇, 主持国家自然科学基金面上项目4项. 获陕西省自然科学一等奖和教育部自然科学二等奖, 陕西高等学校科学技术奖特等奖等, 2004年全国百篇优秀博士论文提名. 担任国际动力学与控制学术期刊“International Journal of Dynamics and Control”副主编

江俊, 西安交通大学教授, 博士生导师. 主要从事非线性动力学的全局分析方法、非线性模态及其降价模型、智能结构及振动控制; 多体与转子动力学、流固耦合振动等研究. 在国内外学术刊物上发表论文150余篇, 主持完成国家自然科学基金重点项目和面上项目6项. 获陕西省自然科学奖一等奖1项, 教育部科学技术奖二等奖2项, 陕西高等学校科学技术奖特等奖1项. 担任“International Journal of Dynamics and Control” “动力学与控制学报” “应用力学学报”和“西安交通大学学报” 等学术杂志编委

通讯作者:
jun.jiang@xjtu.edu.cn

中图分类号: O322
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出版历程
- 收稿日期: 2024-11-30
- 录用日期: 2024-11-30
- 网络出版日期: 2024-11-30

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

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

More Information

Corresponding author: jun.jiang@xjtu.edu.cn

摘要

摘要: 非线性动力系统的一切响应行为均受制于其内在的全局结构, 诸如多稳吸引子及其影响域的形貌和空间分布, 不稳定不变集和不变流形等. 因而, 在指定状态空间内开展全局分析, 不仅可以获得认识和预测系统响应的全部信息, 还能深刻揭示诱发系统复杂分岔、激变或边界蜕变等众多动力学现象的内在机制. 目前, 数值方法仍是非线性动力系统全局分析的最有效手段. 相较于点尺度的数值积分方法或点映射法, 基于状态空间离散思想的方法 (如: 胞映射方法等), 其采用子集覆盖来逼近系统的不变集, 一方面可以高效刻画系统的全局结构形貌, 另一方面可以实现对相邻轨道动态特征的集合表征. 胞映射方法经历40余年的发展, 其功能不断增强, 计算效率和精度已显著提升, 应用场景也逐渐拓宽. 本综述第二节将从当前的视角对状态空间离散方式进行简要归类, 以便于读者更好地了解在全局分析实施过程中该框架体系的本质及优势. 文章第三节将着重介绍近些年提出的一系列状态空间离散方法, 展示在非线性系统全局结构的高效刻画和内在特征的数据表征两方面已取得的最新进展, 突出全局分析从模型驱动向数据驱动的思维模式转变. 在文章第四节将总结本综述的意义和价值, 并就如何在状态空间离散框架下进一步泛化全局分析的概念, 以及应对未来发展和应用需求可能面临的问题和可以拓展的方向提出见解.
- 状态空间离散 /
- 全局分析 /
- 胞映射方法 /
- 模型驱动 /
- 数据驱动
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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图 1 非线性动力系统全局结构的基本框架示意图, 如鞍性不变集 (红色“△”)、稳定不变流形 (蓝色线)、不稳定不变流形 (黄色线)、吸引子 (两侧低地势区)、吸引域 (两侧盆地区域), 虚有向线表示吸引域内部遵循于全局不变结构的响应轨迹

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图 2 面向集合方法的胞尺寸精细化过程

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图 3 两尺度胞参考点映射方法的原理示意图. 其中$x_m^{(i)}$表示第i条轨迹的第m个映射状态点, 网格和数字代表胞的区间和编码, 浅黄和浅蓝区域分别对应周期1 (“○”) 和周期3 (“*”) 两个吸引子的全局吸引域, 浅灰色结构为周期3吸引域内部的鞍性混沌, 边界上“△”为不稳定鞍点. 当第j条轨迹进入第i条轨迹的“已访问”胞且触发$ {d^{(i,j)}} \lt {d_{{\text{threshold}}}} $, 则后续轨迹无需计算 (彩色虚线) 并更新相应的$ R(Z) $; 当第i条轨迹的第n个映射状态点再次进入该条轨迹的“访问中”胞且与第m个映射状态点触发$ d_{m,n}^{(i)} \lt \varepsilon $时, 则标记该轨迹的周期性并更新相应的$ R(Z) $

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图 4 转子/定子碰摩系统的全局结构和动力学分岔(Fan et al. 2022). 其中左图为诱发两种间歇性准周期碰摩行为发生的混沌激变, 左图为导致两种间歇性准周期碰摩行为消亡的突变分岔, 中图为上述现象发生、发展和消亡的动力学分岔图.

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图 5 PSSCM方法的子域划分方式和虚拟不变集(Li et al. 2022). 其中红色胞{8}代表入胞, 蓝色胞{2}代表出胞, 黄色胞{5}代表相交胞, 深灰色胞集合{1,9}是待处理子域1内的瞬态胞, 浅灰色胞集合{6,7}是待处理子域外的潜在 (未知) 转移胞. 集合{5,8,9}即为子域1内的虚拟不变集, 它是跨子域不变集{3,4,5}在子域1中的覆盖集

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图 6 Mathieu–Duffing系统状态空间子域分割与合成 (空间切分位置为X_d = 0.18)(Li, Jiang, Li, et al. 2019): (a) 子域划分形成的标记胞; (b) 子域综合后恢复的全局不变集. 其中红色胞为入胞, 蓝色胞为出胞, 黄色胞为相交胞, 深灰色胞为子域内的瞬态胞, 紫色为恢复后的全局不变集结构

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图 7 PSSCM方法采用的GPU集群计算架构(Li et al. 2022). 其中调度主机负责子域分割、合成以及计算资源调配. 每个计算节点由1个CPU线程和1个GPU设备组成, 独立节点中GPU负责处理子域中数据映射并传给对应的CPU执行子域内全局分析, 实现GPUs设备层级的子域并行化. GPU中的每个线程执行一次胞−胞映射, 实现GPU线程层级的胞并行化

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图 8 转子/定子碰摩诱发干摩擦反向涡动的全局响应和12维极限环分岔 (状态空间3维投影展示)(Li et al. 2022): (a)-(c) 多稳态全局响应相图; (d) 无碰摩周期响应Hopf分叉诱发间歇性准周期碰摩响应 (${\varOmega } \approx 0.65$), 间歇性准周期碰摩 (极限环) 由突变分岔而失稳(${\varOmega } \approx 0.74$), 响应最终进入干摩擦反向涡动大幅振荡 (${\varOmega } = 0.76 \to 0.80$)

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图 9 GCMSAI方法中插值参考点的配置: (a)邻接胞参考点(Liu et al. 2018); (b)插值胞内参考点(Wang et al. 2020)

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图 10 阻尼摆系统六个共存吸引域间的Wada边界(Liu et al. 2018). 其中黑色“▲”代表共存的多个吸引子, 彩色点为吸引域的影响区域, 不同吸引域间的边界表现出典型分形特征

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图 11 2014年3月8日—14日印度洋海域洋流短期运动特征及对周围浮漂的吸引性(李自刚等 2021): (a) 印度洋海域布置的浮漂数据及运动轨迹; (b) 短期涡旋中心(黑色点)和涡旋域 (彩色区域); (c)-(f) 不同涡旋中心对周围浮漂运动轨迹 (真实数据) 的吸引性

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图 12 非线性系统渐进式学习及预测能力的进化过程(Li, Ma, et al. 2024): (a)-(d) 样本提取区域的精细化 (胞分辨率分别为2 × 2, 4 × 4, 8 × 8, 16 × 16); (e)-(h) 数据模型对全局吸引域学习和预测能力的进化过程

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图 13 少数据量下追踪非线性全局拓扑结构的渐进式学习方法(Li, Jiang, et al. 2024)

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图 14 离散子空间扩展方法示意图

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表 1 全局动力学结构 (不变特性) 对动力学行为的影响及部分典型应用

全局动力学结构		典型应用	应用场景
吸引子		共存动态特性	机电系统(Kim et al. 2014, Wu et al. 2018, Zhu et al. 2022), 生态共存(Beisner et al. 2003, Schmitt et al. 2019, 杨永均等 2019), 非线性模型(Li, Jiang, et al. 2020, Mitra et al. 2017, Niroomand et al. 2022)
		多目标优化	优化设计(Sun et al. 2015, Sun, Zheng, et al. 2017, Wang et al. 1995)
		复杂行为认知	稀有事件(Hu et al. 2021, Klokov et al. 2011), 混沌特性(Dudkowski et al. 2018, Y. Li et al. 2024, Sahoo et al. 2022), 复杂系统(Heino et al. 2022, Kahana et al. 2023, Rodrigues 2017)
吸引域		稳定性和鲁棒性测度	振动评价(Deng et al. 2019, Yang et al. 2017, Yue et al. 2024), 仿生结构(Andonovski et al. 2022, 柳宁等 2008), 生物种群(Dubey et al. 2019, Pattanayak et al. 2021, Pattanayak et al. 2021), 工程安全域(Lenci et al. 2019, Yan et al. 2021)
		能量势阱刻画	俘能结构(Sun et al. 2024, F. Yang et al. 2023, Zhang et al. 2023), 抑振分析(Mamaghani et al. 2016, Q. Yang et al. 2023)
		几何特征	内在结构特征表征(Brzeski et al. 2018, Danca et al. 2019, Wang et al. 2021, Zhang et al. 2021)
		响应分类与控制	分类指标(Daza et al. 2022, Rega et al. 2005), 疾病诊断(Conforte et al. 2020), 行为预测(Molčan et al. 2023, Orlando et al. 2018), 控制策略(De Rengervé et al. 2015, Qin et al. 2020, Şimşek et al. 2020, Z. Sun et al. 2019)
鞍型不变集		危险分岔	边界分岔(Liu, Jiang, Hong & Tang 2019, Ponce et al. 2018, Szemplinska-Stupnicka et al. 1997)
		早期预警	临界点(Bauch et al. 2016, Grziwotz et al. 2023, Lohmann et al. 2024, Scheffer et al. 2009, Xu et al. 2023)
		特定属性	瞬态混沌(Baroni et al. 2023, Gawroński et al. 2024, Lai et al. 2011), 混沌巡游(郭旭等 2009)
不变流形	不稳定	响应跃迁	三体问题(An et al. 2024, Gómez et al. 2001, Tamakoshi et al. 2019), 路径控制(Yu et al. 2001)
		流形导向	Lagrangian 相干结构(Allshouse et al. 2015, Haller 2015), 特征识别(Bollt 2005, Huang et al. 2015, Nolan et al. 2020)
		特征降维	谱子流形(Alora et al. 2023, Cenedese et al. 2022, Jain et al. 2022)
		瞬态动力学	转迁轨道(Koch et al. 2024, Revuelta et al. 2018, Duguet et al. 2008), 短期演化(Mujica et al. 2018, Nave et al. 2019, Serra et al. 2020)
	稳定	逃逸壁垒	逃逸路径设计(Cilenti et al. 2022, Henkelman et al. 2002, Jin et al. 2017), 噪声驱动响应(Capala et al. 2020), 结构失效(Naik et al. 2017, Zhong et al. 2021)
		形貌刻画	边界属性(Guckenheimer et al. 2015, Huang et al. 2022, Osinga 2018)
		参数优化	运动优化(Horibe et al. 2018, 武朋玮等 2017)

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表 2 数据模型对全局吸引域各区域的预测精度

典型区域	胞分辨率
典型区域	2 × 2	4 × 4	8 × 8	16 × 16
总体	75.14%	85.96%	93.54%	98.87%
吸引域内部	79.21%	90.62%	95.77%	99.06%
吸引域边界	21.47%	30.72%	62.33%	96.40%
状态空间边缘	62.39%	55.67%	91.21%	98.13%

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