《非線性科學概論（Introduction to Nonlinear Science）》構建了從基礎到前沿的完整知識體系，系統闡釋了非線性科學的理論脈絡與應用前景。開篇以線性代數、泛函分析與微分方程為數學基石，為深入理解非線性問題築牢分析基礎。隨後，本書深入剖析非線性動力系統的內在機制，闡釋混沌理論對初始條件的極端敏感性、分形結構在尺度變換下的自相似性，以及孤立子作為特殊波動解的穩定性原理，揭示非線性現象背後的普遍規律。最後，本書聚焦交叉學科前沿，探討非線性理論在神經網路動力學與機器學習優化演算法中的關鍵作用，展現其在複雜系統建模與智能計算中的強大潛力。全書貫通數學基礎、理論核心與跨學科應用，為讀者提供了探索非線性世界的方法地圖與思想工具。
作者介紹
編者:梅建琴|責編:王曉歷
目錄
1 Introduction 2.1 Linear Algebra 2.2 Functional Analysis 2.3 Ordinary Differential Equations 2.4 Approximate Analytical Methods 2 Mathematical Foundations 2.1 Linear Algebra 2.2 Functional Analysis 2.3 Ordinary Differential Equations 2.4 Approximate Analytical Methods 3 Nonlinear Dynamical Systems 3.1 Introduction to Dynamical Systems 3.2 Stability Theory 3.3 Limit Cycle 3.4 Bifurcation Theory 4 Chaos 4.1 Introduction to Chaos 4.2 Lyapunov Exponent 4.3 Lorenz Attractor 4.4 Chaos Control 4.5 Quantum Chaos 5 Fractals 5.1 Introduction to Fractals 5.2 Construction of Fractals 5.3 Fractal Dimension 6 Solitons 6.1 Introduction to Solitons 6.2 Travelling Waves Method 6.3 Inverse Scattering Transformation 6.4 Backlund Transformation 6.5 Bilinear Method 6.6 Symmetry Reduction Method 7 Neural Networks and Machine Learning 7.1 Introduction to Neural Networks 7.2 Convolutional Neural Networks 7.3 Recurrent Neural Networks 7.4 Introduction to Machine Learning 7.5 Algorithms in Machine Learning 7.6 Physics-Informed Neural Networks References
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