<p><P>This book gives a general, basic understanding of the mathematical structure "nonlinearity" that lies in the depths of complex systems. Analyzing the heterogeneity that the prefix "non" represents with respect to notions such as the linear space, integrability and scale hierarchy, "nonlinear s
Nonlinear Science: The Challenge of Complex Systems (Springer Complexity)
β Scribed by Zensho Yoshida
- Publisher
- Springer
- Year
- 2010
- Tongue
- English
- Leaves
- 218
- Category
- Library
No coin nor oath required. For personal study only.
β¦ Synopsis
Modern science has abstracted, as compensation for establishing rigorousness, the complexity of the real world, and has inclined toward oversimpli?ed ?ctitious n- ratives; as a result, a disjunction has emerged between the wisdom of science and reality. Re?ecting on this, we see the need for science to recover reality; can it reveal new avenues for thought and investigation of the complexity? The study of science is the pursuit of clarity and distinctness. Physics,after Galilei placed it in the realm of mathematics, has been trying to establish clearness by mathematical logic. While physics and mathematics, respectively, have different intellectual incentives, they have intersected in history on countless occasions and have woven a ?awless system of wisdom. The core of rigorous science is always made of mathematical logic; the laws of science cannot be represented without the language of mathematics. Conversely, it is undoubtedly dif?cult to stimulate ma- ematical intellect without a reference to the interests of science that are directed to the real world. However, various criticisms have been raised against the discourses of sciences that explain the events of the real world as if they are βgovernedβ by mathematical laws. Sciences, being combined with technologies, have permeated, in the form of technical rationalism, the domain of life, politics, and even the psychological world. The criticisms accuse seemingly logical scienti?c narratives of being responsible for widespread destruction and emergence of crises, unprecedented suffering of hum- ity.
β¦ Table of Contents
Preface
Contents
1 What Is NONLINEAR?
Nature and Science
Natura Vexata
Syndrome
DΓ©construction of Linear Theory
The Scale of Phenomenon / Theory with Scale
The Role of Scale in Scientific Revolutions
The Mathematical Recognition of Scale
The Territory of Linear Theory
Linear Space ---The Horizon of Mathematical Science
The Mathematical Definition of Vectors
Graphs---Geometric Representation of Laws
Exponential Law
Nonlinearity---Phenomenology and Structures
Nonlinear Phenomena
The Typology of Distortion
Nonlinearity Emerging in Small Scale---Singularity
Nonlinearity Escaping from Linearity---Criticality
Bifurcation (Polyvalency) and Discontinuity
Notes
Problems
Solutions
References
2 From Cosmos to Chaos
The Order of Nature---A Geometric View
Galileo's Natural Philosophy
Geometric Description of Events
Universality Discovered by Newton
Function---The Mathematical Representation of Order
Motion and Function
Nonlinear Regime
Beyond the Functional Representation of Motion
Decomposition---Elucidation of Order
The Mathematical Representation of Causality
Exponential Law---A Basic Form of Group
Resonance---Undecomposable Motion
Nonlinear Dynamics---An Infinite Chain of Interacting Modes
Chaos---Motion in the Infinite Period
Separability/Inseparability
Invariance in Dynamics
Constants of Motion
Chaos---True Evolution
Collective Order
Complete Solution---The Frame of SpaceEmbodying Order
The Difficulty of Infinity
Symmetry and Conservation Law
Symmetry in Dynamical System
The Deep Structure of Dynamical System
The Translation of Motion and Non-motion
Chaos---The Impossibility of Decomposition
Notes
Problems
Solutions
References
3 The Challenge of Macro-Systems
The Difficulty of Prediction
Chaos in Phenomenological Recognition
Stability
Attractors
Stability and Integrability
Randomness as Hypothetical Simplicity
Stochastic Process
Representation of Motion by Transition Probability
H-Theorem
Statistical Equilibrium
Statistically Plausible Particular Solutions
Collective Phenomena
Nonequilibrium and Macroscopic Dynamics
A Model of Collective Motion
A Statistical Model of Collisions
Notes
Problems
Solutions
References
4 Interactions of Micro and Macro Hierarchies
Structure and Scale Hierarchy
Crossing-Over Hierarchies
Connection of Scale Hierarchies---Structure
Topology---A System of Differences
The Topology of Geometry
Scale Hierarchy and Topology
Fractals---Aggregates of Scales
The Scale of Event / The Scale of Law
Scaling and Representation
Scale Separation
Spontaneous Selection of Scale by Nonlinearity
Singularity---Ideal Limit of Scale-Invariant Structure
Connections of Scale Hierarchies
Complexity---Structures with Multiple Aspects
Singular Perturbation
Collaborations of Nonlinearity and Singular Perturbation
Localized Structures in Space--Time
Irreducible Couplings of Multi-Scales
Notes
Problems
Solutions
References
Index
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