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1 Introduction 1

References 3

2 Fractional Calculus 7

2.1 Special Functions 7

2.2 Definitions of Fractional Derivatives and Integrals 9

2.3 Grünwald-Letnikov Fractional Integrals and Derivatives 9

2.4 Riemann-Liouville Fractional Integrals and Derivatives 11

2.5 Caputo Fractional Derivatives 12

2.6 Laplace Transform Method 12

2.6.1 Basic Facts about the Laplace Transform 12

2.6.2 Laplace Transform of Fractional Integrals 14

2.6.3 Laplace Transform of Fractional Derivatives 14

2.7 Fourier Transform Method 15

2.7.1 Basic Facts about the Fourier Transform 15

2.7.2 Fourier Transform of Fractional Integrals 16

2.7.3 Fourier Transform of Fractional Derivatives 17

2.8 Some Properties of Fractional Derivatives and Integrals 18

2.9 Numerical Methods for Calculation of Fractional Derivatives and Integrals 19

2.10 Fractional Calculus and Electricity 29

2.10.1 Analogue Fractional-Order Circuits 35

2.10.2 Experimental Measurement 36

2.10.3 Additional Remarks 37

References 38

3 Fractional-Order Systems 43

3.1 Fractional LTI Systems 43

3.2 Fractional Nonlinear Systems 47

3.3 Fractional-Order Controllers 47

3.3.1 Definition of Fractional-Order Controllers 47

3.3.2 Properties and Characteristics of Controller 49

3.3.3 Design of Controller Parameters and Implementation 51

References 52

4 Stability of Fractional-Order Systems 55

4.1 Preliminary Consideration 55

4.2 Stability of Fractional LTI Systems 63

4.3 Stability of Fractional Nonlinear Systems 78

4.4 Robust Stability of Fractional-Order LTI Systems 82

4.4.1 Stability Check When the Fractional Orders Are Crisp and Commensurate 82

4.4.2 Stability Check When the Fractional Orders Are Also Interval Real Numbers 87

4.5 Stability of Fractional-Order Nonlinear Uncertain Systems 97

References 98

5 Fractional-Order Chaotic Systems 103

5.1 Introduction to Chaotic Dynamics 103

5.2 Concept of Chua's Circuit 104

5.2.1 Classical Chua's Oscillator 104

5.2.2 Fractional-Order Chua's Oscillator 107

5.2.3 Fractional-Order Chua-Podlubny's Oscillator 113

5.2.4 Fractional-Order Chua-Hartley's Oscillator 114

5.2.5 Fractional-Order Memristor-Based Chua's Oscillator 114

5.3 Fractional-Order Van der Pol Oscillator 127

5.4 Fractional-Order Duffing's Oscillator 130

5.5 Fractional-Order Lorenz's System 134

5.6 Fractional-Order Chen's System 138

5.7 Fractional-Order Lü's System 140

5.8 Fractional-Order Liu's System 142

5.9 Fractional-Order Genesio-Tesi's System 145

5.10 Fractional-Order Arneodo's System 148

5.11 Fractional-Order R？ssler's System 151

5.12 Fractional-Order Newton-Leipnik's System 154

5.13 Fractional-Order Lotka-Volterra System 160

5.14 Fractional-Order Financial System 165

5.15 Fractional-Order CNN 168

5.16 Fractional-Order Volta's System 171

References 181

6 Control of Fractional-Order Chaotic Systems 185

6.1 Preliminary Considerations 185

6.2 A Survey of Control Strategies 187

6.3 Examples:Feed-Back Control of Chaotic Systems 188

6.3.1 Sampled-DataControl of Chua's Oscillator 188

6.3.2 Sliding Mode Control of the Economical System 192

References 197

7 Conclusion 201

References 203

Appendix A A List of Matlab Functions 207

Appendix B Laplace and Inverse Laplace Transforms 211

Glossary 215

Index 217