An Introduction to Dynamical Systems and Chaos

✍ Scribed by G. C. Layek

• Publisher: Springer Nature Singapore
• Year: 2024
• Tongue: English
• Leaves: 701
• Series: University Texts in the Mathematical Sciences
• Edition: 2
• Category: Library

✦ Synopsis

This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering.

The second edition of the book is thoroughly revised and includes several new topics: center manifold reduction, quasi-periodic oscillations, Bogdanov–Takens, period-bubbling and Neimark–Sacker bifurcations, and dynamics on circle. The organized structures in bi-parameter plane for transitional and chaotic regimes are new active research interest and explored thoroughly. The connections of complex chaotic attractors with fractals cascades are explored in many physical systems. Chaotic attractors may attain multiple scaling factors and show scale invariance property. Finally, the ideas of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors are discussed.

✦ Table of Contents

Preface to the Second Edition
Preface to the First Edition
Contents
1 Continuous Dynamical Systems
1.1 Dynamics: A Brief History
1.2 Dynamical Systems
1.3 Flows
1.4 Evolution
1.5 Fixed Points of a System
1.6 Linear Stability Analysis
1.7 Analysis of One-Dimensional Flows
1.8 Conservative and Dissipative Dynamical Systems
1.9 Some Definitions
1.10 Exercises
References
2 Linear Systems
2.1 Linear Systems
2.2 Eigenvalue–Eigenvector Method
2.3 Fundamental Matrix
2.3.1 General Solution of Linear Systems
2.3.2 Fundamental Matrix Method
2.3.3 Matrix Exponential Function
2.4 Solution Procedure of Linear Systems
2.5 Nonhomogeneous Linear Systems
2.6 Exercises
References
3 Phase Plane Analysis
3.1 Plane Autonomous Systems
3.2 Phase Plane Analysis
3.3 Local Stability of Two-Dimensional Linear Systems
3.4 Linearization and Its Limitations
3.5 Nonlinear Simple Pendulum
3.6 Linear Oscillators
3.7 Exercises
References
4 Stability Theory
4.1 Stability of Linear Systems
4.2 Methods for Stability Analysis
4.3 Stability of Linearized Systems
4.4 Topological Equivalence and Conjugacy
4.5 Linear Subspaces
4.6 Hyperbolicity and Its Persistence
4.6.1 Persistence of Hyperbolic Fixed Points
4.6.2 Center Manifolds
4.6.3 Normal Form Analysis with Center Manifold Reduction
4.7 Basin of Attraction and Basin Boundary
4.8 Exercises
References
5 Oscillations
5.1 Oscillatory Solutions
5.2 Theorems on Linear Oscillatory Systems
5.3 Nonlinear Oscillatory Systems
5.4 Periodic Solutions
5.4.1 Gradient Systems
5.4.2 Poincaré Theorem
5.4.3 Bendixson’s Negative Criterion
5.4.4 Dulac’s Criterion
5.5 Limit Cycles
5.5.1 Poincaré–Bendixson Theorem
5.5.2 Liénard System
5.5.3 van der Pol Oscillator
5.5.4 Quasiperiodic Oscillations
5.6 Applications
5.6.1 Glycolysis
5.6.2 Predator–Prey Models
5.7 Exercises
References
6 Theory of Bifurcations
6.1 Bifurcations
6.2 Bifurcations in One-Dimensional Systems
6.2.1 Saddle-Node Bifurcation
6.2.2 Pitchfork Bifurcation
6.2.3 Transcritical Bifurcation
6.3 Bifurcations in One-Dimensional Systems: A General Theory
6.3.1 Conditions for One-Dimensional Bifurcations
6.3.2 Normal Forms
6.4 Imperfect Bifurcation
6.5 Bifurcations in Two-Dimensional Systems
6.5.1 Saddle-Node Bifurcation
6.5.2 Transcritical Bifurcation
6.5.3 Pitchfork Bifurcation
6.5.4 Hopf Bifurcation
6.5.5 Projection Method for Calculating the First Lyapunov Coefficient
6.5.6 Homoclinic and Heteroclinic Bifurcations
6.5.7 Bogdanov–Takens Bifurcation
6.5.8 Layek and Pati Model
6.6 Lorenz System and Its Properties
6.7 Applications
6.7.1 Interacting Species Model
6.7.2 Convection of Couple-Stress Fluid Layer and Intermittent Chaos
6.8 Exercises
References
7 Hamiltonian Systems
7.1 Generalized Coordinates
7.2 Classification of Systems
7.3 Basic Problem with Constraints
7.3.1 Lagrange Equation of Motion of First Kind
7.3.2 Lagrange Equation of Motion of Second Kind
7.3.3 Cyclic Coordinates (Ignorable Coordinates)
7.3.4 Routh’s Process for Ignoration of Coordinates
7.4 Hamilton Principle
7.5 Noether’s Theorem
7.6 Legendre Dual Transformations
7.7 Hamilton Equations of Motion
7.8 Hamiltonian Flows
7.8.1 Critical Points of Hamiltonian Systems
7.8.2 Hamiltonian and Gradient Systems
7.9 Symplectic Transformations
7.9.1 Symplectic Forms
7.9.2 Symplectic Transformations
7.9.3 Derivation of Hamilton’s Equations from Symplectic Form
7.10 Poisson Brackets
7.11 Hamilton–Jacobi Equation
7.12 Exercises
References
8 Symmetry Analysis
8.1 Symmetry
8.2 Symmetry Analysis of Dynamical Systems
8.3 Group of Transformations
8.3.1 Symmetry Group of Transformations
8.3.2 Infinitesimal Transformations
8.3.3 Infinitesimal Generator
8.3.4 Extended Infinitesimal Operator
8.3.5 Invariance Principle
8.4 Canonical Parameters
8.5 Lie Group Theoretic Method for First-Order ODEs
8.6 Multiparameter Groups
8.6.1 Lie Algebra
8.6.2 Subalgebra and Ideal
8.6.3 Solvable Lie Algebra
8.7 Group Method for Second-Order ODEs
8.8 Method of Differential Invariant
8.9 Group Method for PDEs
8.10 Symmetry Analysis for Boundary Value Problems
8.11 Noether’s Theorems and Symmetry Groups
8.12 Symmetry Analysis of Korteweg-de Vries (KdV) Equation
8.13 Boundary Layer Flows of Power-Law Fluids: Lie Symmetry Approach
8.14 Logarithmic Velocity Near a Wall in Turbulence Regime
8.15 Exercises
References
9 Discrete Dynamical Systems
9.1 Maps and Flows
9.2 Composition of Maps
9.3 Orbits
9.4 Phase Portrait
9.5 Fixed Points
9.6 Existence and Uniqueness of Fixed Points
9.7 Stable and Unstable Fixed Points
9.8 Basin of Attraction and Basin Boundary
9.9 Linear Stability Analysis
9.10 Cobweb Diagram
9.11 Periodic Points
9.12 Periodic Cycles
9.13 Stability of Periodic Point and Periodic Cycle
9.14 Eventually Fixed Point, Periodic Point, Periodic Orbit
9.15 Superstable Fixed Point and Superstable Periodic Point
9.16 Hyperbolic Points
9.17 Nonhyperbolic Points
9.18 The Schwarzian Derivative
9.19 Exercises
References
10 Some Maps
10.1 Tent Map
10.2 Logistic Map
10.2.1 Some Properties of Logistic Map
10.2.2 Iterative Solutions of Logistic Equation
10.3 Dynamics of Quadratic and Cubic Maps
10.3.1 The Quadratic Map
10.3.2 The Cubic Map
10.4 Symbolic Maps
10.5 Shift Map
10.6 Euler Shift Map
10.7 Decimal Shift Map
10.8 Gaussian Map
10.9 Hénon Map
10.10 Skinny-Baker Map
10.11 Bifurcations in Discrete Systems
10.11.1 Saddle-Node (Fold) Bifurcation
10.11.2 Period-Doubling (Flip) and Period-Bubbling Bifurcations
10.11.3 Neimark-Sacker (Torus) Bifurcation
10.12 Exercises
References
11 Conjugacy of Maps
11.1 Conjugacy
11.1.1 Topological Semi-conjugacy
11.1.2 Homeomorphism
11.1.3 Topological Conjugacy
11.2 Properties of Conjugacy/Semi-conjugacy Relations
11.3 Conjugacy Between Tent and Logistic Maps
11.4 Dynamics on the Circle, Its Homeomorphisms, and Conjugacy
11.5 Irrational Rotation Number on Circle Maps
11.6 Denjoy’s Theorem: Diffeomorphism with Irrational Rotation Number and Conjugacy
11.7 Exercises
References
12 Chaos
12.1 Mathematical Theory of Chaos
12.2 Dynamics of Logistic Map
12.3 Symbolic Dynamics
12.4 Quantifying Chaos
12.4.1 Universal Sequence
12.4.2 Feigenbaum Number
12.4.3 Renormalization Group Theory and Superstable Cycle
12.4.4 Lyapunov Exponent
12.4.5 Invariant Measure
12.4.6 Sharkovskii Order and Coexistence of Periodic Cycles
12.4.7 Period-3 Implies Chaos
12.5 Chaotic Maps
12.5.1 Poincaré Map
12.5.2 Stroboscopic Map
12.5.3 Circle Map
12.5.4 Smale Horseshoe Map
12.6 Routes to Chaos
12.7 Organized Structures in Chaos
12.7.1 Bi-parameter Dynamics
12.7.2 Multiple Coexisting Attractors
12.8 Universality in Chaos
12.9 Hyperchaos
12.10 Bi-parameter Dynamics in Predator–Prey Model
12.11 Exercises
References
13 Fractals
13.1 Fractals
13.2 Self-similarity and Scaling
13.3 Self-similar Fractals
13.4 Constructions of Self-similar Fractals
13.5 Dimensions of Fractals
13.6 Iterative Function Systems
13.7 Strange Attractor
13.8 Strange Repeller
13.9 Multifractals and Global Spectrum
13.9.1 The f(α) Spectrum of Weighted Cantor Set
13.9.2 The f(α) Spectrum of General Cantor Set
13.9.3 The f(α) Spectrum of Period-Doubling Cascade
13.9.4 The f(α) Spectrum in Turbulent Energy-Eddies Cascade
13.10 Simple Model for Navier–Stokes’ Equations
13.11 Exercises
References
Index

✦ Subjects

Dynamical Systems, Stability Theory, Oscillations, Bifurcations, Hamiltonian Systems, Symmetry Analysis, Chaos, Fractals