Fractal Patterns in Nonlinear Dynamics and Applications

Preface 
Scaling, Scale-invariance and Self-similarity 
Dimensions of physical quantity 
Buckingham Pi-theorem 
Examples to illustrate significance of P-theorem 
Similarity 
Self-similarity 
Dynamic scaling 
Scale-invariance: Homogeneous function 
Power-law distribution 
Fractals 
Introduction 
Euclidean geometry 
Fractals 
Space of Fractal 
Construction of deterministic fractals 
Stochastic Fractal 
Introduction 
A brief description of stochastic process 
Dyadic Cantor set (DCS): Random fractal 
Kinetic Dyadic Cantor set 
Stochastic dyadic Cantor set 
Numerical Simulation 
Stochastic fractal in aggregation with stochastic self-replication 
Discussion and summary 
Multifractality 
Introduction 
The Legendre transformation 
Theory of multifractality 
Multifractal formalism in fractal 
Cut and paste model on Sierpinski carpet 
Stochastic multifractal 
Weighted planar stochastic lattice model 
Algorithm of the weighted planar stochastic lattice (WPSL) 
Geometric properties of WPSL 
Geometric properties of WPSL 
Multifractal formalism in kinetic square lattice 
Fractal and Multifractal in Stochastic Time Series 
Introduction 
Concept of scaling law, monofractal and multifractal time series 
Stationary and Non-stationary time series 
Fluctuation analysis on monofractal stationary and non-stationary time series 
Fluctuation analysis on stationary and non-stationary multifractal time series 
Discussion 
Application in Image Processing 
Introduction 
Generalized fractal dimensions 
Image thresholding 
Performance analysis 
Medical image processing 
Mid-sagittal plane detection 
References