Nonlinear Science (eBook)

Professor Yoshida's contributions range from the mathematical physics to leading an experimental team of plasma physics. He has studied the self-organization of structures in plasmas using various theoretical methods such as variational principles, singular perturbation theory, operator theory, functional analysis, topological methods, etc., and, in particular, is well known for important contributions to the mathematical theory of the curl operator, which he has developed to extend and deepen the understanding of nonlinear structures in general vortex dynamics systems.

Preface 5
Contents 8
1 What Is NONLINEAR? 11
Nature and Science 11
Natura Vexata 11
Syndrome 14
Déconstruction of Linear Theory 15
The Scale of Phenomenon / Theory with Scale 16
The Role of Scale in Scientific Revolutions 16
The Mathematical Recognition of Scale 18
The Territory of Linear Theory 20
Linear Space ---The Horizon of Mathematical Science 20
The Mathematical Definition of Vectors 22
Graphs---Geometric Representation of Laws 25
Exponential Law 30
Nonlinearity---Phenomenology and Structures 34
Nonlinear Phenomena 34
The Typology of Distortion 35
Nonlinearity Emerging in Small Scale---Singularity 37
Nonlinearity Escaping from Linearity---Criticality 39
Bifurcation (Polyvalency) and Discontinuity 41
Notes 43
Problems 51
Solutions 51
References 53
2 From Cosmos to Chaos 54
The Order of Nature---A Geometric View 54
Galileo's Natural Philosophy 54
Geometric Description of Events 55
Universality Discovered by Newton 57
Function---The Mathematical Representation of Order 61
Motion and Function 61
Nonlinear Regime 63
Beyond the Functional Representation of Motion 65
Decomposition---Elucidation of Order 67
The Mathematical Representation of Causality 67
Exponential Law---A Basic Form of Group 69
Resonance---Undecomposable Motion 71
Nonlinear Dynamics---An Infinite Chain of Interacting Modes 74
Chaos---Motion in the Infinite Period 76
Separability/Inseparability 78
Invariance in Dynamics 83
Constants of Motion 83
Chaos---True Evolution 89
Collective Order 90
Complete Solution---The Frame of SpaceEmbodying Order 92
The Difficulty of Infinity 94
Symmetry and Conservation Law 95
Symmetry in Dynamical System 95
The Deep Structure of Dynamical System 96
The Translation of Motion and Non-motion 100
Chaos---The Impossibility of Decomposition 103
Notes 106
Problems 114
Solutions 115
References 118
3 The Challenge of Macro-Systems 120
The Difficulty of Prediction 120
Chaos in Phenomenological Recognition 120
Stability 121
Attractors 125
Stability and Integrability 128
Randomness as Hypothetical Simplicity 130
Stochastic Process 130
Representation of Motion by Transition Probability 132
H-Theorem 134
Statistical Equilibrium 136
Statistically Plausible Particular Solutions 140
Collective Phenomena 141
Nonequilibrium and Macroscopic Dynamics 141
A Model of Collective Motion 142
A Statistical Model of Collisions 146
Notes 149
Problems 156
Solutions 157
References 159
4 Interactions of Micro and Macro Hierarchies 161
Structure and Scale Hierarchy 161
Crossing-Over Hierarchies 161
Connection of Scale Hierarchies---Structure 162
Topology---A System of Differences 165
The Topology of Geometry 165
Scale Hierarchy and Topology 166
Fractals---Aggregates of Scales 167
The Scale of Event / The Scale of Law 169
Scaling and Representation 169
Scale Separation 172
Spontaneous Selection of Scale by Nonlinearity 175
Singularity---Ideal Limit of Scale-Invariant Structure 176
Connections of Scale Hierarchies 178
Complexity---Structures with Multiple Aspects 178
Singular Perturbation 179
Collaborations of Nonlinearity and Singular Perturbation 182
Localized Structures in Space--Time 186
Irreducible Couplings of Multi-Scales 191
Notes 195
Problems 209
Solutions 210
References 214
Index 216