Chaos, Fractals and Complexity (Springer Proceedings in Complexity)

Organization
Preface
Contents
Contributors
Chaos
Diffusion Without Spreading of a Wave Packet in Nonlinear Random Models
	1 Introduction
		1.1 Difference Between Spreading and Diffusion
		1.2 Discussion of the Boltzmann Ergodic Hypothesis
	2 Dynamics of Hamiltonians with a Finite Number of Degrees of Freedom
		2.1 KAM Tori in Finite Nearly Integrable Hamiltonian Systems
		2.2 Arnold Diffusion Conjecture and Extension
		2.3 Anti Integrable Limit
	3 Dynamics of Hamiltonians on Infinite Lattices
		3.1 KAM Tori in Infinite Hamiltonian Systems on Lattices with Linear Discrete Spectrum
		3.2 Numerical Observation of KAM Tori in Large Non Integrable Hamiltonian Systems
	4 Long Time Behavior of Wave Packets
		4.1 A Criterion for the Long Time Behavior of a Wave Packet
		4.2 Discussion of the Long Time Behavior of Wave Packets
	5 Exact Results for Ding Dong Models
		5.1 Definition
		5.2 Regular Trajectories and a Lemma
		5.3 Some Exact Results
		5.4 Non Genericity of DD Models
	6 Concluding Remarks
	References
Nonlinear Phenomena Shaping the Structure of Spiral Galaxies
	1 Introduction
	2 Order and Chaos
		2.1 Two Dimensional (2D) Models
		2.2 Three Dimensional (3D) Models
	3 Discussion
	References
Phase Space Transport and Dynamical Matching in a Caldera-Type Hamiltonian System
	1 Introduction
	2 Hamiltonian Model
	3 Lagrangian Descriptors in Caldera-Type Hamiltonian Systems
	4 Numerical Results
	5 Conclusions
	References
The Building Blocks of Spiral Arms in Galaxies
	1 Introduction
	2 Spiral Arms in Grand Design Galaxies
		2.1 The Model
		2.2 Spiral Density Waves from Precessing Ellipses
	3 Spiral Arms in Barred Spiral Galaxies
		3.1 The Model
		3.2 Manifold Spirals
	4 Conclusion
	References
Ordered and Chaotic Bohmian Trajectories
	1 Introduction
	2 Nodal Point-X-Point Complex Mechanism
	3 Chaos and Entanglement in Bohmian Qubits
	4 Summary
	References
A Brief Introduction to Quantum Chaos of Generic Systems
	1 Introduction
	2 Quantum Mechanics of Classically Integrable Systems
	3 Quantum Chaos in Classically Ergodic Systems
	4 Quantum Chaos in Generic Systems
		4.1  The Wigner Functions
		4.2 The Principle of Uniform Semiclassical Condensation (PUSC) of Wigner Functions
		4.3 Spectral Statistics in the Mixed-type Systems
		4.4 Quantum Localization of the Chaotic Eigenstates
	5 The Poincaré-Husimi Functions in Billiards
	6 The Phase Space Localization Measures of Chaotic Eigenstates
	7 The Distribution of the Localization Measures
	8 Semiclassical Limiting Approach to the Regime of PUSC
	9 Discussion and Conclusions
	References
Dynamics and Statistics of Weak Chaos in a 4-D Symplectic Map
	1 Introduction
	2 Statistical Analysis of Weak Chaos
	3 Evidence of Weak Chaos in 4DMM Maps
		3.1 Weak Chaos in an Example of the 4DMM Map
		3.2 HE Cases of the 4DMM Map
		3.3 HH Cases of the 4DMM Map
		3.4 Close to the Instability Transition
	4 Conclusions
	References
Fractals
Multifractal Analysis of SEM Images of Multiphase Materials: The Case of OPC Clinker
	1 Introduction
	2 Multifractal Analysis
	3 Data
	4 Results
	5 Summary
	References
Fractal Dimensional Analysis for Retinal Vascularization Images in Retinitis Pigmentosa: A Pilot Study
	1 Introduction
	2 A Brief Introduction to Fractal Dimension
	3 Retinitis Pigmentosa
		3.1 First Methodology: Digital Fundus Photography
		3.2 Second Methodology: Optical Coherence Tomography Angiography
	4 Discussion
	References
Extending the Bayesian Framework from Information to Action
	1 Introduction
	2 Biological Versus Artificial Information Processing
	3 Extended Bayesian Inference: Two Modes in One Loop
	4 Outlook and Forthcoming Experiments
	References
Complexity
Fokas on Medical Imaging: Analytic Reconstructions for Emission Tomography
	1 Introduction
	2 Radon Transform and Its Attenuated Version in Two Dimensions
	3 Inversion of the Radon Transform in Two Dimensions
		3.1 Fourier-Based Inversion of the Radon Transform in Two Dimensions
		3.2 Complex Analysis Tools
		3.3 The Inversion of the Radon Transform in Two Dimensions Without the Fourier Transform: The Fokas Approach
	4 Inversion of the Attenuated Radon Transform in Two Dimensions
		4.1 The Inversion of the Attenuated Radon Transform in Two Dimensions Without the Fourier Transform: The Fokas Approach
		4.2 A Novel Method for the Inversion of the Attenuated Radon Transform in Two Dimensions
	References
Inverse EEG Problem, Minimization and Numerical Solutions
	1 Neurophysiology of EEG
	2 The Importance of EEG
	3 Head Model
	4 Basic Equations for the EEG Formulation
	5 Configuration of the Numerical Solution
	6 Future Work
	References
Traveling Waves in Flowing Sand: The Dynamical Systems Approach
	1 Introduction
	2 Steady Uniform Flow and the Monoclinal Flood Wave
	3 One Equation to Rule All Traveling Waveforms
		3.1 Focusing on Traveling Wave Solutions
		3.2 Non-dimensional Form
	4 Dynamical Systems Approach
	5 From Monoclinal Waves to Roll Waves
		5.1 Stage 1: The Monoclinal Wave
		5.2 Stage 2: The Undular Bore
		5.3 Stage 3: Roll Waves
	6 Conclusion
	References
Identifying Discrete Breathers Using Convolutional Neural Networks
	1 Introduction
	2 Creation of Breather and Phonon Samples
	3 Machine Learning Training Process and Results
	4 Conclusions
	References
Subthreshold Oscillations in Multiplex Leaky Integrate-and-Fire Networks with Nonlocal Interactions
	1 Introduction
	2 The Model
		2.1 The Coupled LIF Dynamics in the Multiplex
		2.2 Kuramoto Order Parameter
	3 Results
		3.1 Subthreshold Oscillations
	4 Conclusions and Open Problems
	References
Networks\' Modulation: How Different Structural Network Properties Affect the Global Synchronization of Coupled Kuramoto Oscillators
	1 Introduction
	2 Methods and Materials
		2.1 Connectivity Measurements
		2.2 Neural Networks as Graphs
		2.3 The Kuramoto Model
	3 Results
	4 Summary and Discussion
	References
Neural Correlates of Human-Machine Trust in Autonomous Vehicles Context
	1 Introduction
	2 Understanding Trust-in-Automation
		2.1 Cognitive Component
		2.2 Affective Component
		2.3 Behavioural Component
	3 Psychological and Cognitive Factors of Trust Between Drivers and AVs
	4 Current Approaches to Objective Trust Measures Using Brain Signals
		4.1 Approaches Based on EEG
		4.2 Approaches Based on fNIRS
	5 Conclusions and Future Directions
	References
Fokas and€Mathematics
High-Order Localized Wave Solutions of the New (3+1)-Dimensional Kadomtsev-Petviashvili Equation
	1 Introduction
	2 Soliton Solutions of the (3+1)-Dimensional KP Equation
	3 Rational Solutions of the (3+1)-Dimensional KP Equation
	4 Semi-rational Solutions of the (3+1)-Dimensional KP Equation
	5 Discussion
	References
Progress in Initial-Boundary Value Problems for Nonlinear Evolution Equations and the Fokas Method
	1 Introduction
	2 Initial-Boundary Value Problems via the Fokas Method
	3 A Higher Dispersion KdV on the Half-Line
	References
Instabilities of Linear Evolution PDEs via the Fokas Method
	1 Introduction
	2 The Equation tu+xxxu=0
	3 The Equation tu-xxxu=0
	4 Conclusion
	References
Fokas Diagonalization
	1 Introduction
		1.1 The Classical Spectral Transform Method
		1.2 Derivation of Transforms
		1.3 Example Problems
		1.4 A More General Spectral Transform Method
	2 Results
		2.1 Half Line Neumann Problem for the Stokes Equation
		2.2 Finite Interval Problem for the Heat Equation with a Nonlocal Condition
		2.3 Half Line Dirichlet Problem for the Linearized BBM Equation
	3 Conclusion
		3.1 On Pedagogy
	References
A Novel Difference-Integral Equation Satisfied Asymptotically by the Riemann Zeta Function
	1 Introduction
	2 Review of Some of the Results of ch22F
	3 Computation of 3 and 4
	4 Sketch of the Derivation of (1.1)
	5 Numerical Evidence
	References
The Role of Periodicity in the Solution of Third Order Boundary Value Problems
	1 Introduction
	2 A Motivating Example and the Unified Transform of Fokas
	3 Non-periodic Boundary Value Problems
		3.1 Uncoupled BC of Dirichlet Type
		3.2 Mixed BC of Dirichlet Type
		3.3 Coupled BC: Pseudo-periodic
		3.4 An Outlier: Quasi-periodic BC
	References
The Fokas Method for the Well-posedness of Nonlinear Dispersive Equations in Domains with a Boundary
	1 Introduction
	2 Inverse Scattering for Linear Equations: Rediscovering the Fourier Transform
	3 A Novel Approach for the Well-posedness of IBVPs
	References
Athanassios S. Fokas: A€Renaissance Scientist
Athanassios S. Fokas: A Renaissance Scientist
	References
Author Index