Fractal patterns in nonlinear dynamics and applications Cover Title Page Copyright Page Dedication Preface Table of Contents Preface Symbols 1: Scaling Scale invariance and Self similarity 1.1 Dimensions of physical quantity 1.2 Buckingham Pi theorem 1.3 Examples to illustrate the significance of Π theorem 1.4 Similarity 1.5 Self similarity 1.6 Dynamic scaling 1.7 Scale invariance: Homogeneous function 1.7.1 Scale invariance: Generalized homogeneous functions 1.7.2 Dimension functions are scale invariant 1.8 Power law distribution 1.8.1 Examples of power law distributions 1.8.1.1 Euclidean geometry 1.8.1.2 First return probability 1.8.2 Extensive numerical simulation to verify powerlaw first return probability 2: Fractals 2.1 Introduction 2.2 Euclidean geometry 2.3 Fractals 2.3.1 Recursive Cantor set 2.3.2 von Koch curve 2.3.3 Sierpinski gasket 2.4 Space of fractal 2.4.1 Complete metric space 2.4.2 Banach contraction mapping 2.4.3 Completeness of the fractal space 2.5 Construction of deterministic fractals 2.5.1 Iterated function system 3: Stochastic Fractal 3.1 Introduction 3.2 A brief description of stochastic process 3.3 Dyadic Cantor Set (DCS): Random fractal 3.4 Kinetic dyadic Cantor set 3.5 Stochastic dyadic Cantor set 3.6 Numerical simulation 3.7 Stochastic fractal in aggregation with stochastic self replication 3.8 Discussion and summary 4: Multifractality 4.1 Introduction 4.2 The Legendre transformation 4.3 Theory of multifractality 4.3.0.1 Properties of the mass exponent τ(q) 4.3.1 Legendre transformation of τs(q):