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