Nonlinear Dynamics - A Concise Introduction Interlaced with Code

Preface
In a Nutshell
Structure
Usage in a Lecture
Acknowledgements
Contents
1 Dynamical Systems
1.1 What Is a Dynamical System?
1.1.1 Some Example Dynamical Systems
1.1.2 Trajectories, Flows, Uniqueness and Invariance
1.1.3 Notation
1.1.4 Nonlinearity
1.2 Poor Man's Definition of Deterministic Chaos
1.3 Computer Code for Nonlinear Dynamics
1.3.1 Associated Repository: Tutorials, Exercise Data, Apps
1.3.2 A Notorious Trap
1.4 The Jacobian, Linearized Dynamics and Stability
1.5 Dissipation, Attractors, and Conservative Systems
1.5.1 Conserved Quantities
1.6 Poincaré Surface of Section and Poincaré Map
2 Non-chaotic Continuous Dynamics
2.1 Continuous Dynamics in 1D
2.1.1 A Simple Model for Earth's Energy Balance
2.1.2 Preparing the Equations
2.1.3 Graphical Inspection of 1D Systems
2.2 Continuous Dynamics in 2D
2.2.1 Fixed Points in 2D
2.2.2 Self-sustained Oscillations, Limit Cycles and Phases
2.2.3 Finding a Stable Limit Cycle
2.2.4 Nullclines and Excitable Systems
2.3 Poincaré-Bendixon Theorem
2.4 Quasiperiodic Motion
3 Defining and Measuring Chaos
3.1 Sensitive Dependence on Initial Conditions
3.1.1 Largest Lyapunov Exponent
3.1.2 Predictability Horizon
3.2 Fate of State Space Volumes
3.2.1 Evolution of an Infinitesimal Uncertainty Volume
3.2.2 Lyapunov Spectrum
3.2.3 Properties of the Lyapunov Exponents
3.2.4 Essence of Chaos: Stretching and Folding
3.2.5 Distinguishing Chaotic and Regular Evolution
3.3 Localizing Initial Conditions Using Chaos
4 Bifurcations and Routes to Chaos
4.1 Bifurcations
4.1.1 Hysteresis
4.1.2 Local Bifurcations in Continuous Dynamics
4.1.3 Local Bifurcations in Discrete Dynamics
4.1.4 Global Bifurcations
4.2 Numerically Identifying Bifurcations
4.2.1 Orbit Diagrams
4.2.2 Bifurcation Diagrams
4.2.3 Continuation of Bifurcation Curves
4.3 Some Universal Routes to Chaos
4.3.1 Period Doubling
4.3.2 Intermittency
5 Entropy and Fractal Dimension
5.1 Information and Entropy
5.1.1 Information Is Amount of Surprise
5.1.2 Formal Definition of Information and Entropy
5.1.3 Generalized Entropy
5.2 Entropy in the Context of Dynamical Systems
5.2.1 Amplitude Binning (Histogram)
5.2.2 Nearest Neighbor Kernel Estimation
5.3 Fractal Sets in the State Space
5.3.1 Fractals and Fractal Dimension
5.3.2 Chaotic Attractors and Self-similarity
5.3.3 Fractal Basin Boundaries
5.4 Estimating the Fractal Dimension
5.4.1 Why Care About the Fractal Dimension?
5.4.2 Practical Remarks on Estimating the Dimension
5.4.3 Impact of Noise
5.4.4 Lyapunov (Kaplan–Yorke) Dimension
6 Delay Coordinates
6.1 Getting More Out of a Timeseries
6.1.1 Delay Coordinates Embedding
6.1.2 Theory of State Space Reconstruction
6.2 Finding Optimal Delay Reconstruction Parameters
6.2.1 Choosing the Delay Time
6.2.2 Choosing the Embedding Dimension
6.3 Advanced Delay Embedding Techniques
6.3.1 Spike Trains and Other Event-Like Timeseries
6.3.2 Generalized Delay Embedding
6.3.3 Unified Optimal Embedding
6.4 Some Nonlinear Timeseries Analysis Methods
6.4.1 Nearest Neighbor Predictions (Forecasting)
6.4.2 Largest Lyapunov Exponent from a Sampled Trajectory
6.4.3 Permutation Entropy
7 Information Across Timeseries
7.1 Mutual Information
7.2 Transfer Entropy
7.2.1 Practically Computing the Transfer Entropy
7.2.2 Excluding Common Driver
7.3 Dynamic Influence and Causality
7.3.1 Convergent Cross Mapping
7.4 Surrogate Timeseries
7.4.1 A Surrogate Example
8 Billiards, Conservative Systems and Ergodicity
8.1 Dynamical Billiards
8.1.1 Boundary Map
8.1.2 Mean Collision Time
8.1.3 The Circle Billiard (Circle Map)
8.2 Chaotic Conservative Systems
8.2.1 Chaotic Billiards
8.2.2 Mixed State Space
8.2.3 Conservative Route to Chaos: The Condensed Version
8.2.4 Chaotic Scattering
8.3 Ergodicity and Invariant Density
8.3.1 Some Practical Comments on Ergodicity
8.4 Recurrences
8.4.1 Poincaré Recurrence Theorem
8.4.2 Kac's Lemma
8.4.3 Recurrence Quantification Analysis
9 Periodically Forced Oscillators and Synchronization
9.1 Periodically Driven Passive Oscillators
9.1.1 Stroboscopic Maps and Orbit Diagrams
9.2 Synchronization of Periodic Oscillations
9.2.1 Periodically Driven Self-Sustained Oscillators
9.2.2 The Adler Equation for Phase Differences
9.2.3 Coupled Phase Oscillators
9.3 Synchronization of Chaotic Systems
9.3.1 Chaotic Phase Synchronization
9.3.2 Generalized Synchronization of Uni-Directionally Coupled Systems
10 Dynamics on Networks, Power Grids, and Epidemics
10.1 Networks
10.1.1 Basics of Networks and Graph Theory
10.1.2 Typical Network Architectures
10.1.3 Robustness of Networks
10.2 Synchronization in Networks of Oscillators
10.2.1 Networks of Identical Oscillators
10.2.2 Chimera States
10.2.3 Power Grids
10.3 Epidemics on Networks
10.3.1 Compartmental Models for Well-Mixed Populations
10.3.2 Agent Based Modelling of an Epidemic on a Network
11 Pattern Formation and Spatiotemporal Chaos
11.1 Spatiotemporal Systems and Pattern Formation
11.1.1 Reaction Diffusion Systems
11.1.2 Linear Stability Analysis of Spatiotemporal Systems
11.1.3 Pattern Formation in the Brusselator
11.1.4 Numerical Solution of PDEs Using Finite Differences
11.2 Excitable Media and Spiral Waves
11.2.1 The Spatiotemporal Fitzhugh-Nagumo Model
11.2.2 Phase Singularities and Contour Lines
11.3 Spatiotemporal Chaos
11.3.1 Extensive Chaos and Fractal Dimension
11.3.2 Numerical Solution of PDEs in Spectral Space
11.3.3 Chaotic Spiral Waves and Cardiac Arrhythmias
12 Nonlinear Dynamics in Weather and Climate
12.1 Complex Systems, Chaos and Prediction
12.2 Tipping Points in Dynamical Systems
12.2.1 Tipping Mechanisms
12.2.2 Basin Stability and Resilience
12.2.3 Tipping Probabilities
12.3 Nonlinear Dynamics Applications in Climate
12.3.1 Excitable Carbon Cycle and Extinction Events
12.3.2 Climate Attractors
12.3.3 Glaciation Cycles as a Driven Oscillator Problem
Appendix A Computing Lyapunov Exponents
Appendix B Deriving the Master Stability Function
Appendix References
Index