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Intro -- Preface -- Contents -- Abbreviations -- Symbols -- 1 Introduction -- 2 Preliminaries -- 2.1 Limit Cycle and Semi-stable Limit Cycle -- 2.2 Criterion of Semi-stable Limit Cycle -- 2.2.1 Limit Cycles of Oscillatory Approach and Monotone Approach -- 2.2.2 Criterions -- 2.3 Feature Scale, Scale-Free Domain, Fractal, Random Fractal, Dimension -- 2.4 Iterative Function System and Fractal -- 2.5 Dissipative System -- 2.6 Attractor, Attracting Set, Basin of Attraction, Strange Attractor, and Semi-strange Attractor -- 2.7 Relationship between Semi-stable Limit Cycles and Semi-strange Attractors -- 2.8 Elementary Reaction and Reaction Rate -- 2.9 Lagrangian Particle Dynamic System -- 3 Universal Mathematical Model of Mesoscale Eddy -- 3.1 Mesoscale Eddy -- 3.2 Mathematical Model of Mesoscale Eddy -- 3.2.1 Bounded Motion -- 3.2.2 Movement Asymptotic Unity and Uniform Tendency -- 3.3 Universal Mathematical Model of Mesoscale Eddy -- 3.3.1 Momentum of a Stochastic Ellipse -- 3.3.2 Elementary Reaction Rate -- 3.3.3 Basic Mathematical Model of Mesoscale Eddy -- 3.3.4 Universal Mathematical Model of Mesoscale Eddy -- 4 Semi-stable Limit Cycle in Mathematical Model of Mesoscale Eddy -- 4.1 Analysis of Parameter Distribution of Stable and Unstable Limit Cycles -- 4.1.1 α and β are Positive and m is Odd -- 4.1.2 α and β are Negative and m is Odd -- 4.1.3 α is Positive, β is Negative and m is Odd -- 4.1.4 α is Negative, β is Positive and m is Odd -- 4.1.5 m is a Decimal -- 4.2 Stable Limit Cycle -- 4.3 Unstable Limit Cycle -- 4.4 Parameter Distribution Analysis of Semi-stable Limit Cycle -- 4.4.1 Special System -- 4.4.2 General System -- 4.4.3 Different Internal and External Stability -- 4.5 Externally Unstable and Internally Stable Semi-stable Limit Cycle -- 4.6 Externally Stable and Internally Unstable Semi-stable Limit Cycle.
5 Semi-stable Limit Cycles and Mesoscale Eddies -- 5.1 Semi-stable Limit Cycles and Mesoscale Cold Eddies -- 5.2 Semi-stable Limit Cycles and Mesoscale Warm Eddies -- 6 Example Verification -- 6.1 Basic Method -- 6.2 Numerical Experiment -- 6.2.1 Value in Special Circumstances -- 6.2.2 Full Parameter Case -- 6.2.3 Clockwise Model -- 6.2.4 Anti-clockwise Model -- 6.2.5 Algorithm Parallelization and Model Checking in Global Oceans -- 7 Spatiotemporal Structure of Mesoscale Eddies: Self-similar Fractal Behavior -- 7.1 Spatiotemporal Fractal Structure of Mesoscale Warm Eddy -- 7.2 Spatiotemporal Fractal Structure of Mesoscale Cold Eddy -- 7.3 Self-similar Fractal Structure under Affine Transformation -- 7.3.1 Transformation Relations of Spatial Coordinates -- 7.3.2 Spatial Structure -- 8 Mesoscale Eddies: Disk and Columnar Shapes -- 8.1 The Specific Implementation Process of Water Particle Motion ... -- 8.1.1 Specific Transformation -- 8.1.2 Disk-Shaped Mesoscale Cold Eddy -- 8.2 Specific Implementation Process of Water Particle Motion Transformation ... -- 8.2.1 Specific Transformation -- 8.2.2 Disk-Shaped Mesoscale Warm Eddy -- 8.3 Approximate Approximation of Mesoscale Disk-Shaped Mesoscale Eddy -- 9 Fractal Analysis and Prediction for Spatiotemporal Complexity of Mesoscale Eddy -- 9.1 Spatiotemporal Structure of Mesoscale Eddies Based on Universal Model -- 9.1.1 Mesoscale Cold Eddy -- 9.1.2 Mesoscale Warm Eddy -- 9.2 Mathematical Model and Complexity Analysis of Spatiotemporal Fractal Structure of Mesoscale Eddies -- 9.2.1 Fractal Model of Snowflake -- 9.2.2 Fractal Model of Random Snowflake -- 9.2.3 Mesoscale Eddies and Spatiotemporal Fractal Structures of Cantor Self-Similar Fractal Sets -- 9.3 Spatiotemporal Fractal Analysis and Prediction of the Complexity of Mesoscale Eddies -- 9.3.1 Data -- 9.3.2 Fractal Dimension of Mesoscale Eddy.
9.3.3 Fractal Processing of Mesoscale Eddies Profile of the Ocean -- 9.3.4 Three-Dimensional Fractal Structure of Abnormal Salinity -- 9.3.5 Comprehensive Analysis -- 10 Nonlinear Characteristics of Universal Mathematical Model of Mesoscale Eddy -- 10.1 Dissipation of Nonlinear Systems -- 10.2 Chaotic Behavior of Universal Nonlinear System of Mesoscale Eddy -- 10.3 Singularity of Mesoscale Eddy and its Physical Meaning -- 11 Same Solution Between Momentum Balance Equations and Mesoscale Eddies -- 11.1 Navier-Stokes Equation -- 11.2 Same Solution Between the Mathematical Model of Mesoscale Eddy ... -- 11.3 Necessary Conditions for Existence of Mesoscale Eddies in Special Model -- 11.4 Sufficient and Necessary Conditions for the Existence of Mesoscale Eddies in the General Model -- 11.4.1 No Stickiness -- 11.4.2 Stickiness -- 11.4.3 Perturbation Terms of Parameters with Pressure Change Rate -- 11.4.4 Necessary Conditions -- 12 Momentum Balance Equation Based on Truncation Function and Mathematical Model of Mesoscale Eddies -- 12.1 Sufficient Conditions of Mesoscale Eddies for the Two-Dimensional ... -- 12.2 Existence of Mesoscale Eddies in Two-Dimensional Momentum Balance ... -- 12.2.1 β-Plane Approximation and Viscosity -- 12.2.2 β-Plane Approximation and Nonviscosity -- 12.3 Mesoscale Cold and Warm Eddies Produced by Truncation Function and Circulation Factor -- 13 Interpolation Prediction of Mesoscale Eddies -- 14 Random Elliptic Curve and Brownian Motion Trajectory of Mesoscale Eddy -- 14.1 Trajectory of Elliptical Arc -- 14.1.1 Mesoscale Cold Eddy -- 14.1.2 Mesoscale Warm Eddy -- 14.2 Trajectory of Brownian Curve -- 15 Mathematical Model for Edge Waves of Mesoscale Eddies and Its Spatio-Temporal Fractal Structures -- 15.1 Mathematical Model of Edge Waves -- 15.1.1 Poincaré Cross-Section -- 15.1.2 Edge Wave Motion and Its Duffing Dynamical System.
15.2 Mathematical Model of Edge Waves Based on Poincaré ... -- 15.2.1 Generators of Edge Waves -- 15.2.2 A Mathematical Model for Random Fractal of Edge Waves -- 15.3 Fractal Analysis of Internal Structure Complexity of Edge Waves -- 15.4 New Problems Arising from Random Fractal Models of Edge Waves -- Appendix References.
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Fractal Control Theory / / by Shu-Tang Liu, Pei Wang
