Banerjee S., Hassan M.K., Mukherjee S., Gowrisankar A. Fractal Patterns in Nonlinear Dynamics and Applications

Boca Raton: CRC Press, 2019. — 207 p.Most books on fractals focus on deterministic fractals as the impact of incorporating randomness and time is almost absent. Further, most review fractals without explaining what scaling and self-similarity means. This book introduces the idea of scaling, self-similarity, scale-invariance and their role in the dimensional analysis. For the first time, fractals emphasizing mostly on stochastic fractal, and multifractals which evolves with time instead of scale-free self-similarity, are discussed. Moreover, it looks at power laws and dynamic scaling laws in some detail and provides an overview of modern statistical tools for calculating fractal dimension and multifractal spectrum.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
FractalsEuclidean geometry
Fractals
Space of Fractal
Construction of deterministic fractals
Stochastic FractalA 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
MultifractalityThe 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 SeriesConcept 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 ProcessingGeneralized fractal dimensions
Image thresholding
Performance analysis
Medical image processing
Mid-sagittal plane detection