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E-raamat: Working with Dynamical Systems: A Toolbox for Scientists and Engineers

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Working with Dynamical Systems: A Toolbox for Scientists and EngineersSuurem pilt
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"This book provides working tools for the study and design of dynamical systems, emphasizing qualitative analysis over complex mathematics. The author includes executable Mathematica notebooks and extensive examples for discussion of mechanical models, models of chemical reactions, dynamics of patterns, and dynamical systems in space"--

This book provides working tools for the study and design of nonlinear dynamical systems applicable in physics and engineering. It offers a broad-based introduction to this challenging area of study, taking an applications-oriented approach that emphasizes qualitative analysis and approximations rather than formal mathematics or simulation. The author, an internationally recognized authority in the field, makes extensive use of examples and includes executable Mathematica notebooks that may be used to generate new examples as hands-on exercises. The coverage includes discussion of mechanical models, chemical and ecological interactions, nonlinear oscillations and chaos, forcing and synchronization, spatial patterns and waves.

Key Features:

·         Written for a broad audience, avoiding dependence on mathematical formulations in favor of qualitative, constructive treatment.

·         Extensive use of physical and engineering applications.

·         Incorporates Mathematica notebooks for simulations and hands-on self-study.

·         Provides a gentle but rigorous introduction to real-world nonlinear problems.

·         Features a final chapter dedicated to applications of dynamical systems to spatial patterns.

The book is aimed at student and researchers in applied mathematics and mathematical modelling of physical and engineering problems. It teaches to see common features in systems of different origins, and to apply common methods of study without losing sight of complications and uncertainties related to their physical origin.

Preface xi
1 Whence Dynamical Systems
1(32)
1.1 Classical Mechanics
2(9)
1.1.1 Conservative Equations of Motion
2(2)
1.1.2 Systems with One Degree of Freedom
4(2)
1.1.3 Symmetries and Conservation Laws
6(2)
1.1.4 Interacting Particles
8(1)
1.1.5 Dissipative Motion
9(2)
1.2 Chemical Kinetics
11(7)
1.2.1 Mass Action
11(2)
1.2.2 Adsorption and Catalysis
13(3)
1.2.3 Autocatalysis and Self-Inhibition
16(1)
1.2.4 Thermal Effects
16(2)
1.3 Biological Models
18(4)
1.3.1 Population Dynamics
18(2)
1.3.2 Epidemiological Models
20(1)
1.3.3 Neural and Genetic Networks
21(1)
1.4 Electric Currents
22(4)
1.4.1 Electric Circuits
22(2)
1.4.2 Electrochemical Reactions
24(1)
1.4.3 Membrane Transport
25(1)
1.5 Spatially Extended Systems
26(3)
1.5.1 From Time to Coordinate Dependence
26(1)
1.5.2 Fourier Decomposition
27(2)
1.6 Continuous vs. Discrete
29(4)
1.6.1 Iterative Systems
29(1)
1.6.2 From Continuous to Discrete
30(1)
1.6.3 Poincare Maps
31(2)
2 Local Bifurcations
33(36)
2.1 Bifurcation of Stationary States
34(7)
2.1.1 Branches of Stationary States
34(1)
2.1.2 Bifurcation Expansion
35(2)
2.1.3 Fold and Transcritical Bifurcations
37(1)
2.1.4 Cusp Singularity
38(2)
2.1.5 Higher Singularities
40(1)
2.2 Stability and Slow Dynamics
41(8)
2.2.1 Linear Stability Analysis
41(3)
2.2.2 Stable and Unstable Manifolds
44(1)
2.2.3 Exchange of Stability
45(2)
2.2.4 Amplitude Equations
47(2)
2.3 Bifurcations of Periodic Orbits
49(7)
2.3.1 Hopf Bifurcation
49(3)
2.3.2 Derivation of the Amplitude Equation
52(2)
2.3.3 Instabilities of Periodic Orbits
54(2)
2.4 Example: Exothermic Reaction
56(6)
2.4.1 Bifurcation of Stationary States
56(2)
2.4.2 Hopf Bifurcation
58(3)
2.4.3 Branches of Periodic Orbits
61(1)
2.5 Example: Population Dynamics
62(7)
2.5.1 Prey-Predator Models
62(2)
2.5.2 Stability and Bifurcations
64(1)
2.5.3 Periodic Orbits
65(4)
3 Global Bifurcations
69(34)
3.1 Topology of Bifurcations
70(4)
3.1.1 More Ways to Create and Break Periodic Orbits
70(2)
3.1.2 Bifurcations in a System with Three Stationary States
72(2)
3.2 Global Bifurcations in the Exothermic Reaction
74(6)
3.2.1 Basin Boundaries
74(1)
3.2.2 Saddle-Loop Bifurcations
75(3)
3.2.3 Sniper Bifurcation
78(2)
3.3 Bifurcation at Double-Zero Eigenvalue
80(5)
3.3.1 Locating a Double Zero
80(1)
3.3.2 Quadratic Normal Form
81(1)
3.3.3 Expansion in the Vicinity of Cusp Singularity
82(3)
3.4 Almost Hamiltonian Dynamics
85(7)
3.4.1 Weak Dissipation
85(2)
3.4.2 Hopf and Saddle-Loop Bifurcations
87(1)
3.4.3 Bifurcation Diagrams
88(3)
3.4.4 Basin Boundaries
91(1)
3.5 Systems with Separated Time Scales
92(5)
3.5.1 Fast and Slow Variables
92(1)
3.5.2 Van der Pol Oscillator
93(1)
3.5.3 FitzHugh--Nagumo Equation
94(2)
3.5.4 Canards
96(1)
3.6 Venturing to Higher Dimensions
97(6)
3.6.1 Dynamics Near Triple-Zero Eigenvalue
97(3)
3.6.2 Double Hopf Bifurcation
100(1)
3.6.3 Blue Sky Catastrophe
101(2)
4 Chaotic, Forced, and Coupled Oscillators
103(60)
4.1 Approaches to Hamiltonian Chaos
104(10)
4.1.1 Hiding in Plain Sight
104(2)
4.1.2 Resonances and Small Divisors
106(2)
4.1.3 Example: Henon-Heiles Model
108(3)
4.1.4 Quantitative Measures of Chaos
111(3)
4.2 Approaches to Dissipative Chaos
114(9)
4.2.1 Distilling Turbulence into Simple Models
114(2)
4.2.2 Chaotic Attractors
116(1)
4.2.3 Period-Doubling Cascade
117(3)
4.2.4 Strange, Chaotic, or Both?
120(3)
4.3 Chaos Near a Homoclinic
123(13)
4.3.1 Shilnikov's Snake
123(2)
4.3.2 Complexity in Chaotic Models
125(4)
4.3.3 Lorenz Model
129(7)
4.4 Weakly Forced Oscillators
136(6)
4.4.1 Phase Perturbations
136(2)
4.4.2 Forced Harmonic Oscillator
138(2)
4.4.3 Weakly Forced Hamiltonian System
140(2)
4.5 Effects of Strong Forcing
142(10)
4.5.1 Universal and Standard Mappings
142(3)
4.5.2 Forced Dissipative Oscillators
145(2)
4.5.3 Forced Relaxation Oscillator
147(5)
4.6 Coupled Oscillators
152(11)
4.6.1 Phase Dynamics
152(1)
4.6.2 Coupled Pendulums
153(3)
4.6.3 Coupled Relaxation Oscillators
156(2)
4.6.4 Synchronization in Large Ensembles
158(5)
5 Dynamical Systems in Space
163(64)
5.1 Space-Dependent Equilibria
163(7)
5.1.1 Basic Equations
163(2)
5.1.2 Stationary Solution in One Dimension
165(3)
5.1.3 Systems with Mass Conservation
168(2)
5.2 Propagating Fronts
170(10)
5.2.1 Advance into a Metastable State
170(5)
5.2.2 Propagation into an Unstable State
175(3)
5.2.3 Pushed Fronts
178(2)
5.3 Separated Time and Length Scales
180(18)
5.3.1 Two-Component Reaction-Diffusion System
180(3)
5.3.2 Stationary and Mobile Fronts
183(4)
5.3.3 Stationary and Mobile Bands
187(5)
5.3.4 Wave Trains
192(6)
5.4 Symmetry-Breaking Bifurcations
198(9)
5.4.1 Amplitude Equations
198(2)
5.4.2 Bifurcation Expansion
200(2)
5.4.3 Interacting Modes
202(2)
5.4.4 Plane Waves and their Stability
204(3)
5.5 Resonant Interactions
207(8)
5.5.1 Triplet Resonance
207(3)
5.5.2 Stripes--Hexagons Competition
210(2)
5.5.3 Standing Waves
212(3)
5.6 Nonuniform Patterns
215(12)
5.6.1 Propagation of a Stationary Pattern
215(3)
5.6.2 Self-Induced Pinning
218(3)
5.6.3 Propagating Wave Pattern
221(2)
5.6.4 Nonuniform Wave Patterns
223(4)
Bibliography 227(6)
Online Files 233(2)
Illustration Credits 235
Len Pismen is Emeritus Professor of Fluid Mechanics at the Technion Israel Institute of Technology. His other books include Vortices in Nonlinear Fields (1999), Patterns and Interfaces in Dissipative Dynamics (2006), and general audience books The Swings of Science (2018) and Morphogenesis Deconstructed (2020).
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