This textbook introduces the language and the techniques of the theory of dynamical systems of finite dimension for an audience of physicists, engineers, and mathematicians at the beginning of graduation. Author addresses geometric, measure, and computational aspects of the theory of dynamical systems. Some freedom is used in the more formal aspects, using only proofs when there is an algorithmic advantage or because a result is simple and powerful.
The first part is an introductory course on dynamical systems theory. It can be taught at the master's level during one semester, not requiring specialized mathematical training. In the second part, the author describes some applications of the theory of dynamical systems. Topics often appear in modern dynamical systems and complexity theories, such as singular perturbation theory, delayed equations, cellular automata, fractal sets, maps of the complex plane, and stochastic iterations of function systems are briefly explored for advanced students. The author also explores applications in mechanics, electromagnetism, celestial mechanics, nonlinear control theory, and macroeconomy. A set of problems consolidating the knowledge of the different subjects, including more elaborated exercises, are provided for all chapters.
Author(s): Rui Dilão
Series: UNITEXT for Physics
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Pages: 326
City: Cham
Tags: Dynamical Systems, Chaos, Hamiltonian Systems, Attractors, Invariant Manifolds, Bifurcations, Synchronisation, Celestial Mechanics, Nonlinear Control
Preface
Contents
Part I Introduction to Dynamical Systems
1 Differential and Difference Equations as Dynamical Systems
1.1 Differential Equations as Dynamical Systems
1.2 Stability of Fixed Points
1.3 Difference Equations as Dynamical Systems
1.4 Classification of Fixed Points
1.5 Poincaré Maps
1.6 Numerical Methods
2 Hamiltonian Systems
2.1 The Geometry of the Harmonic Oscillator
2.2 Denjoy Theory
2.3 The KAM Theorem
3 Strange Attractors, Interval Maps and Invariant Manifolds
3.1 The Lorenz Attractor
3.2 Interval Maps, Ergodicity, and Chaos
3.3 Some Exact Results on the Dynamics of Interval Maps
3.4 Strange Attractors
3.5 Stable, Unstable and Centre Manifolds
3.5.1 Stable and Unstable Manifolds Theorem
3.5.2 Dynamics in the Centre Manifold
4 Qualitative Theory of Dynamical Systems
4.1 Chaos
4.2 Lyapunov Exponents and Oseledets Theorem
4.3 Bifurcations of Differential Equations
4.4 Bifurcations of Difference Equations
5 Special Topics in Dynamical Systems
5.1 The Poincaré-Bendixon Theory
5.2 Complexity of Strange Attractors
5.3 Intermittency
5.4 Stochastic Iteration of Function Systems
5.5 Maps of the Complex Plane
5.6 Cellular Automata
5.7 Limit Sets
5.8 Linear Maps on the Torus and Symbolic Dynamics
Part II Applications
6 Examples of Nonlinear Systems
6.1 Parametric Resonance: The Swing
6.2 Singular Perturbations and Ducks
6.3 Strange Attractors in Delay Equations
6.4 Chaos in the Störmer Problem
6.4.1 Equations of Motion and Conservation Laws
6.4.2 Motion in the Equatorial Plane of the Dipole Field
6.4.3 Three-Dimensional Motion
7 Synchronisation of Clocks and Pendulums
7.1 Synchronisation of Clocks
7.1.1 A Synchronisation Model of Two Pendulum Clocks
7.1.2 A Simple Model for a Pendulum Clock
7.1.3 Synchronisation of Two Identical Clocks
7.1.4 Synchronisation of Two Clocks with Different Parameters: Robustness
7.2 Synchronisation of Pendulums
7.2.1 The Synchronisation Equations
7.2.2 Synchronisation of Two Pendulums
7.2.3 Libration Dynamics of Nge3 Pendulums
8 Introduction to Celestial Mechanics
8.1 The N-Body Problem
8.2 The Kepler Two–Body Problem
8.3 The Three–Body Problem
8.4 Three–Body Central Orbits
8.5 The Restricted Three–Body Problem
8.6 Sitnikov Problem and Chaotic Motions
8.7 Keplerian Dumbbell and the Spin-Orbit Interaction
9 Introduction to Nonlinear Control
9.1 The Pontriaguine Maximum Principle
9.2 The Acrobat
9.3 Controlling the Trajectory of a Satellite
9.4 Optimal Control of a Macroeconomic Model
Appendix Mathematical Appendix
A.1 Elementary Topology Concepts
A.1.1 Topology
A.1.2 Manifolds
A.1.3 Measure
A.2 Linear Differential Equations
A.2.1 Autonomous Equations
A.2.1.1 General Solution of the Linear Differential Equation (A.1)
A.2.1.2 General Solution of the Linear Differential Equation (A.2)
A.2.2 Non-autonomous Equations
A.3 Linear Difference Equations
A.4 Scale Transformations
A.5 Mathematica by Examples
A.6 Lyapunov Exponents with Mathematica
Appendix References
Index
