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Table of contents
Preface ...................................................................................................................................... 9
Introduction ............................................................................................................................ 12
PART I. MODELS AND FORECAST
Chapter 1. The concept of model. What is remarkable in mathematical models ........... 15
1.1. What is called “model” and “modeling” ...................................................................... 15
1.2. Science, scientific knowledge, systematization of scientific models ........................... 21
1.3. Delusion and intuition. Rescue with mathematics ....................................................... 25
1.4. How many models for a single object can exist ........................................................... 29
1.5. Ways the models are born ............................................................................................ 31
1.6. Structural scheme for the process of mathematical modeling...................................... 33
1.7. Conclusions from historical practice of modeling. Indicative destiny of models
in mechanics.................................................................................................................. 35
Chapter 2. Two approaches to modeling and forecast ....................................................... 40
2.1. Basic concepts and peculiarities of dynamical modeling............................................. 41
2.1.1. Definition of dynamical system ........................................................................................ 41
2.1.2. Nonrigorous example. Variables and parameters ............................................................. 44
2.1.3. State space. Conservative and dissipative systems. Attractors, multistability, basins of
attraction ........................................................................................................................... 46
2.1.4. Characteristics of attractors............................................................................................... 50
2.1.5. Parameter space. Bifurcations. Combined spaces, bifurcation diagrams.......................... 57
2.2. Foundations to claim a process “random”.................................................................... 58
2.2.1. Set-theoretic approach....................................................................................................... 59
2.2.2. Signs of randomness traditional for physicists.................................................................. 70
2.2.3. Algorithmic approach ....................................................................................................... 71
2.2.4. Randomness as unpredictability........................................................................................ 71
2.3. Concept of partial determinancy .................................................................................. 72
2.4. Lyapunov exponents and limits of predictability ......................................................... 74
2.4.1. Practical estimate of prediction time................................................................................. 74
2.4.2. Prdictability and the Lyapunov exponent: The case of infinitesimal perturbations.......... 75
2.5. Scales of consideration: How they determine appraisal about process properties
(complex deterministic dynamics or randomness) ....................................................... 80
2.6. Example with coin........................................................................................................ 82
Chapter 3. Dynamical (deterministic) models of evolution................................................ 87
3.1. Terminology ................................................................................................................. 87
3.1.1. Operator, map, equation, evolution operator..................................................................... 87
3.1.2. Functions, continuous and discrete time ........................................................................... 88
3.1.3. Return map, iteration......................................................................................................... 89
3.1.4. Flows and cascades, Poincare section and Poincare map ................................................. 89
3.1.5. Illustrative example........................................................................................................... 90
3.2. Systematization of some kinds of model equations ..................................................... 91
3.3. Explicit functional dependencies.................................................................................. 96
3.4. Linearity and nonlinearity ............................................................................................ 99
3.4.1. Linearity and nonlinearity of functions and equations...................................................... 99
3.4.2. The nature of nonlinearity...............................................................................................100
3.4.3. Illustration with pendulums............................................................................................. 101
3.5. Models – ordinary differential equations ................................................................... 103
3.5.1. Kinds of solutions ........................................................................................................... 103
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