非线性动力学系统在诸多工程领域中普遍存在，揭示其复杂的全局动力学行为对于系统实际工况的研究至关重要。然而，动力学系统的非线性特性使得建立精确数学模型并开展定量分析面临巨大挑战。传统的广义胞映射方法其一步转移概率矩阵的计算需要大量的样本，其计算是十分耗时的，甚至对于高维（三维以上）系统来说已经成为了研究的瓶颈。非线性系统的全局动力学结构包含吸引子、鞍等不变集及其不变流形，这些内在属性决定了响应的发生、发展和最终状态，因此有效开展全局分析是工程系统研究和设计的关键。与此同时，工程设计中往往需要了解系统关键参数区间内全局结构的改变，特别是分岔前后的全局动力学行为，这给传统广义胞映射带来了更加严峻的挑战。

本文将深度学习方法融入广义胞映射的映射提取，并采用数据驱动模型来有效预测短期响应，分析全局胞到胞映射的能力。广义胞映射方法的一个显著优势是能够在离散胞的尺度内弥补数据模型预测的误差。针对不同的动力学系统，可以通过所提的深度神经网络框架内灵活替换其循环结构单元以提高数据模型的拟合度。该方法利用有限的数据样本，直接从中学习系统初始状态到短时响应的动力学规律，避免了显式的数学建模和微分方程求解。通过数据模型预测系统的全局动力学响应，构建系统的一步转移概率矩阵，从而高效开展非线性系统的全局动力学分析。

在此基础上，引入数据增量算法，通过补充少量新数据对初级训练模型进行渐进式调优，实现对系统全局拓扑结构的分析。每当参数改变，对上一步数据模型逐步训练，进而获得新参数下的系统代理模型。从短时响应的角度揭示少量样本如何实现系统的拓扑分析，并研究样本量对初级训练模型和渐进训练模型泛化能力的影响。所提方法能在可控精度下，以极少样本预测系统拓扑结构的改变，包括倍周期分岔、鞍结分岔、边界危机等局部和全局分岔的发生。文中以三个典型动力学系统为研究对象，从定性和定量角度验证了方法的有效性和高效性。

为了应对参数变化导致的系统响应和分岔行为的复杂性，本文进一步提出了参数关联的深度学习胞映射方法。通过将系统参数也纳入神经网络的输入，提出全量数据和增量数据两种训练方式。该方法不仅能预测任意参数和初值下的动力学响应，还可预测系统的全局拓扑结构及其随参数变化的分岔过程，揭示分岔行为发生的内在机制。研究结果表明，所提算法在有限数据条件下能够较准确捕捉系统的吸引子、分岔等全局动力学特征，并揭示了样本量、参数采样分布等关键因素对算法性能的影响规律。

这些研究将有助于数据驱动的动力学系统研究，在避免了建模和大量实验数据需求的同时，能够从较少数据样本中提取系统的动力学本质规律，实现精准全局动力学行为分析。

Nonlinear dynamical systems are prevalent in various engineering fields, and revealing their complex global dynamical behaviors is crucial for understanding the practical operating conditions of systems. However, the nonlinear nature of dynamical systems poses significant challenges in establishing accurate mathematical models and conducting quantitative analyses. The traditional generalized cell mapping (GCM) method requires a vast amount of sample data to calculate the one-step transition probability matrix, making the calculation process highly time-consuming. For high-dimensional systems (above three dimensions), this has even become a bottleneck in research. The global dynamical structure of nonlinear systems encompasses invariant sets such as attractors, saddles, and their associated invariant manifolds. These intrinsic properties determine the initiation, evolution, and ultimate state of the response. Therefore, effectively conducting global analyses is crucial for engineering systems research and design. Simultaneously, engineering design often requires understanding changes in the global structure within critical parameter ranges of the system, especially the global dynamic behavior before and after bifurcations, posing even greater challenges for the conventional GCM.

A deep recurrent neural network is integrated into the derivation of mapping in GCM method, enabling efficient prediction of short-term responses and analysis of the global cell-to-cell mapping capability by data-driven models. One significant advantage of the GCM method is its ability to compensate for prediction errors within the scale of discrete cells. It is possible to improve the fitting of the data model by flexibly replacing its recurrent structural units within the proposed deep neural network framework according to the different dynamical systems. This method utilizes a limited number of data samples to directly learn the dynamical behavior from the initial state to its short-term response of system, thereby avoiding explicit mathematical modeling and differential equation computing. By utilizing the data model to predict the global dynamical response of the system and to construct the one-step transition probability matrix, efficient global dynamical analysis of nonlinear systems is conducted.

Building upon this foundation, the data incremental algorithm is introduced, which progressively finetunes the primary training model by supplementing a small amount of sample, enabling the analysis of the global topological structure. Whenever the parameters change, the previous data model is progressive trained to obtain a proxied model of the system under the new parameter. From the perspective of short-term responses, revealing how topological analysis of the system can be achieved with a small number of samples. Furthermore, the impact of sample size on the generalization capabilities of the primary training model and progressive training model is investigated. The proposed method can predict changes in the topological structure of system, as well as the occurrence of local and global bifurcations such as period-doubling bifurcations, saddle-node bifurcations, and boundary crises, with controllable accuracy using an extremely small number of samples. Three traditional dynamical systems are simulated as illustrations to qualitatively and quantitatively validate the effectiveness and efficiency of the proposed algorithm.

To address the complexity of system responses and bifurcation behaviors resulting from parameter variations, a parameter-associated deep learning cell mapping method is further proposed. By incorporating system parameters into the neural network inputs, two training modes are introduced: full data and incremental data. This method predicts not only the dynamical responses under arbitrary parameters and initial conditions but also the global topological structure and its bifurcation process with parameter variations, revealing the underlying mechanisms driving bifurcation behaviors. The research results demonstrate that the proposed algorithm can accurately capture global dynamical features, such as attractors and bifurcations with limited data conditions. It also shows the influence of critical factors on the performance of algorithm, such as sample size and parameter sampling distribution.

These investigations will contribute to data-driven investigations of dynamical systems. While avoiding the need for explicit modeling and extensive experimental data, they enable extracting the essential dynamical laws of systems from a relatively small number of data samples, thereby enabling precise analyses of global dynamical behaviors.